Analysis of Limit Order Markets by Joshua Slive B.A. (Honours), Queen's University, Kingston, 1995 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF Doctor of Philosophy in T H E FACULTY OF GRADUATE STUDIES (Faculty of Commerce and Business Administration, Finance Division) We accept this thesis as conforming to the required standard The University of British Columbia October 2002 © Joshua Slive, 2002 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department of I ^O<-v^u-v-\ (? /~C j 6 r^. The U n i v e r s i t y o f B r i t i s h Columbia Vancouver, Canada Date 0 ' I / •^ i vcf ^ 0^2- Abstract This dissertation consists of three essays analyzing the strategies of traders who participate in limit order markets from both theoretical and empirical perspectives. In the first essay, I examine the implications of asymmetric information on the strategies of traders facing a multi-period limit order market. In the second essay, we examine liquidity provision on a limit order market using a simple order-choice model where traders with extreme liquidity needs place market orders and traders with less extreme liquidity needs place limit orders or stay out of the market. In the third essay, I solve the dynamic problem of an investor in a limit order market who manages his order over the trading day. ii Contents Abstract ii Contents iii List of Tables vi List of Figures viii Acknowledgements 1 2 ix Introduction 1 1.1 Overview 1 1.2 Limit Order Markets 2 1.3 The Vancouver Stock Exchange 5 Asymmetric Information in Limit Order Markets 7 2.1 Introduction 7 2.2 The Model 9 2.3 The Optimization Problem 12 2.4 The Solution 14 2.5 An Example 17 2.5.1 Finite Horizon 18 2.5.2 Infinite Horizon 20 2.6 Possible Extensions 22 iii 3 Liquidity Supply and Demand in Limit Order Markets 3.1 Introduction 23 3.2 Description of the Sample 27 3.3 Model 30 3.4 Empirical Results 37 3.4.1 The Price of Immediacy 37 3.4.2 The Arrival Rate of the Traders and the Private Value Distribution 3.4.3 Interpreting the Findings 3.5 4 23 . 42 44 Conclusions 48 Dynamic Strategies in a Limit Order Market 50 4.1 Introduction 50 4.2 Model 53 4.3 4.2.1 A Market With Exogenous Order Flow 53 4.2.2 The Optimizing Trader 56 Optimal Strategy 60 4.3.1 One-Period Strategy 60 4.3.2 Multi-period Solution 63 4.4 Empirical Analysis 69 4.5 Conclusion 72 References 73 Appendices 75 Appendix A Market order frequency 76 Appendix B Proofs for Liquidity Supply and Demand in Limit Order M a r kets 78 B.l The conditional log-likelihood function 79 B.2 Execution probabilities in the Weibull competing risks model 79 B.3 The picking off risk 80 iv Appendix C Tables and Figures for Asymmetric Information in Limit Order Markets 81 Appendix D Tables and Figures for Liquidity Supply and Demand in Limit Order Markets Appendix E 85 Tables and Figures for Dynamic Strategies in Limit Order M a r - kets 107 v List of Tables Cl Optimal bid for the uninformed trader 82 C. 2 Cross-Type Limit Fill Probabilities 83 D. l Summary Statistics 85 D.2 Order size 86 D.3 Conditioning Variables 87 D.4 Weibull Hazard Rate Model for Order Submissions 88 D.5 Ordered Probit Models for Order Submissions 90 D.6 Competing Risks Models for BHO 91 D.7 Competing Risks Models for ERR 92 D.8 Competing Risks Models for W E M 93 D.9 Picking Off Risk Regressions 94 D.10 Estimation Results for the Discrete Choice Model 96 D . l l Expected Utility by Trader Valuation 97 D. 12 Order Submission Probabilities by Trader Valuation 98 E. l Base case parameter values 108 E.2 Base case solution 109 E.3 No management solution 110 E.4 Optimal limit order across spreads 110 E.5 Fixed limit order across spreads Ill E.6 Short horizon solution . Ill E.7 High volatility solution 112 E.8 Basic Summary Statistics 112 vi E.9 Submissions Conditional on Spread 113 E.10 Submissions Conditional on Midquote Volatility 114 E . l l Submissions Conditional on Recent Volume 115 E.12 Outcome Conditional on Distance from Execution at Submission 116 E.13 Outcome Conditional on Spread at Submission 117 E.14 Outcome Conditional on Recent Volume at Submission 118 vii List of Figures 2.1 Timeline for the Limit Order Game 12 2.2 Trader Types 18 4.1 Time-line for the dynamic limit order market problem 59 C.l Bidding Strategies for High Liquidity Shock Types 82 C.2 Limit Fill and Market Order Probabilities for High Liquidity Shock Types . . 84 C. 3 Limit Fill and Market Order Probabilities for the Market as a Whole 84 D. l Optimal order submission strategy 100 D.2 Comparative statics for BHO 101 D.3 Comparative statics for ERR 102 D.4 Comparative statics for W E M 103 D.5 Comparative statics for BHO: lagged return 104 D.6 105 Comparative statics for ERR: lagged return D. 7 Comparative statics for W E M : lagged return 106 E. l 108 Histogram of buy order prices for CLV viii Acknowledgements This work would not have been possible without the help and encouragement of many people. In particular, the training that I received from the faculty of the Finance Division at the University of British Columbia's Faculty of Commerce and Business Administration was invaluable. Of particular note was the assistance of members of my committee: Murray Frank, Gerald Garvey, Ron Giammarino, and Burton Hollifield. Others who have provided extremely valuable input include Alan Kraus, Raman Uppal, Vasant Naik, and especially my colleague Lorenzo Garlappi. The third chapter of this dissertation is part of a research project involving Burton Hollifield, Robert A. Miller, and Patrik Sandas. My part of this work includes important contributions to the empirical analysis and writing of the paper. Finally, I would like to thank my wife Genevieve Bassellier, who not only made it possible for me to complete this work through her patience and advice, but above all makes it all worthwhile. JOSHUA The University of British Columbia October 2002 ix SLIVE Chapter 1 Introduction 1.1 Overview The presence of an efficient and liquid market for the exchange of securities is an assumption that underlies much of our financial theory. In order to finance new investments, to allow individuals to optimally manage their income and consumption, and to allow resources in general to be allocated as efficiently as possible, we must have well-functioning financial markets. The study of the microstructure of financial markets is thus important as a foundation for other areas of finance. Further, the microstructure of financial markets is an important field in its own right for the guidance it can give to market planners and participants in their work. This dissertation consists of three essays analyzing the strategies of traders who participate in a particular type of financial market: the limit order market. In the first essay, I examine the implications of asymmetric information on the strategies of traders facing a multi-period limit order market. I present some ideas for analyzing the decision problem of a trader who participates in a market where other traders may have more information about the true value of the security. Such information-based trading can have two important effects. First, when traders with information participate in the market, they make the market price more informative by revealing part of their information through the act of trading. Second, informed traders can drive-out liquidity providers by picking-off uninformed orders, thereby reducing the liquidity of the market and making it a less attractive place for uninformed traders. 1 In the second essay, we model a trader's decision to supply liquidity by submitting limit orders or demand liquidity by submitting market orders in a limit order market. The best quotes and the execution probabilities and picking off risks of limit orders determine the price of immediacy. The price of immediacy and the trader's willingness to pay for immediacy determine the trader's optimal order submission, with the trader's willingness to pay for immediacy depending on the trader's valuation for the stock. We estimate the execution probabilities and the picking off risks using a sample from the Vancouver Stock Exchange to compute the price of immediacy. The price of immediacy changes with market conditions — a trader's optimal order submission changes with market conditions. We combine the price of immediacy with the actual order submissions to estimate the unobserved arrival rates of traders and the distribution of the traders' valuations. High realized stock volatility increases the arrival rate of traders and increases the number of value traders arriving — liquidity supply is more competitive after periods of high volatility. An increase in the spread decreases the arrival rate of traders and decreases the number of value traders arriving — liquidity supply is less competitive when the spread widens. In the third essay, I solve the dynamic problem of an investor in a limit order market who manages his order over the trading day. A trader on a limit order market has the ability to actively monitor his order and use cancellations and order changes to mitigate the adverse selection and execution risks inherent in limit orders. I find that the ability to implement a dynamic strategy has a large impact on the payoffs to submitting limit orders and on the limit order submission strategy of a trader. After calibrating the parameters to a stock on the Vancouver Stock Exchange, profits from limit order submission are 48% higher when implementing a dynamic strategy compared to a one-shot strategy. Cancellations and order changes are used to avoid adverse selection by moving orders when the underlying value changes. Order changes are used to mitigate execution risk by converting to a market order when the probability of execution declines. 1.2 Limit Order Markets A limit order market is a continuous double auction where buyers and sellers meet to exchange financial securities. It differs from traditional securities markets in that there are 2 no dealers, specialists, or formal market-makers with responsibilities to manage liquidity. Instead, liquidity is provided exclusively by the submission of limit orders by traders. Limit orders constitute offers to trade a given quantity of shares at a given price, while the limit order book is a queue of outstanding offers. The ordering of the queue, referred to as the priority, plays a crucial role in determining how the market operates, since orders at the head of the queue are more likely to trade than orders in the back of the queue. Many different types of priority are used on exchanges. A common set of priorities are Price Priority: The buy order with the highest price or the sell order with the lowest price is given preference. Time Priority: The first order submitted is given preference. Disclosure Priority: Orders that disclose their volume receive preference over hidden orders. In general, these priorities operate in the order given here, so that an order that improves price will gain priority over all orders at worse prices, regardless of the time of submission. If traders were allowed to place prices along a continuous pricing grid, then time priority would obviously lose significance, since price priority could be obtained by minuscule price improvements. Therefore, limit order markets usually specify a minimum price variation, or tick, that defines a discretely spaced grid of available prices. The matching of buy and sell orders is usually performed by a computer algorithm that examines each incoming order to see if it fulfills the conditions set by some order on the book. If the new order fails to match with any existing order, it becomes part of the limit order book for future traders. If the incoming order could trade with more than one order on the book, then the priority rules are used to determine which order will be filled. Traders may change or cancel orders on the book at any time during the trading day. Under these circumstances, the traditional dichotomy between market and limit orders becomes much less clear. Since there is no specialist willing to quote prices for immediate trades, the price available for a market order depends on the contents of the limit order book. An order receives immediate execution if and only if it meets the requirements to trade against an order already on the book. Hence there is no significant distinction between a limit order 3 priced to receive immediate execution (which is sometimes referred to as a marketable limit order) and a true market order. These two types of orders are observationally equivalent in 1 the data and will be treated as identical in the theoretical models. The decision problem faced by a trader arriving at a limit order market is often examined in the context of the trade-off between limit order and market order submission. A trader choosing to submit a limit order trades at a more favorable price than a trader who submits a market order. But this price improvement is offset by the execution risk and adverse selection cost inherent in limit orders. Execution risk captures the probability that a limit order will fail to execute if the market price moves away from it. Adverse selection cost—often called picking-off risk or winner's curse—captures the fact that a limit order is more likely to execute in the future when the underlying value of the asset has moved against the trader. The use of a limit order book in the trading of securities has become an important feature of many of the world's financial markets. A large number of markets, including the New York Stock Exchange and NASDAQ, employ hybrid systems where both a limit order book and a specialist play a role in the exchange. In addition, pure limit order markets such as the Paris Bourse, the Toronto Stock Exchange and the Vancouver Stock Exchange rely on a computer-driven limit order book as the sole means of exchanging securities. 2 Along with the increasing prevalence of limit order markets comes an increasing interest in the economic issues surrounding these systems. Beyond their importance as a new way of exchanging securities, these markets provide a particularly interesting resource to researchers who wish to bring theoretical models to empirical data. Because of the structure of the trading system, a complete and accurate picture of the trading process is collected as a by-product of the operation of the exchange. There is no specialist with proprietary knowledge of the book, nor is there a need to manually collect data. However, both the theoretical and empirical analysis of limit order markets are still young fields in the finance literature. The results that I obtain through my study of limit order markets shed light not only on the In fact, under the rules of the Vancouver Stock Exchange, it is possible to submit a limit order that has a higher probability of immediate and complete execution than a market order. This is because market orders are restricted from walking up the book by a one tick rule, whereas limit orders face no such restriction. Domowitz (1993) details over fifty exchanges which employ electronic limit order books of one form or another. Most of these automated trade execution systems have come on-line since 1988. 1 2 4 operation of this particular type of market, but also on the larger issues offinancialmarkets in general. 1.3 The Vancouver Stock Exchange In my empirical work in this thesis, I focus on a particular stock market: the Vancouver Stock Exchange. This is a junior equity market that lists mostly smaller firms and has a great many firms from the resource industry. It was one of the first exchanges to convert to a pure limit order system when it adopted the Vancouver Computerized Trading system in 1989. My sample was obtained from the audit tapes of the Vancouver Computerized Trading system over a period from 1989 through 1993. Since the end of this sample period, the Vancouver Stock Exchange has ceased to exist under that name. It was first involved in an amalgamation of Canadian exchanges to become a part of the Canadian Venture Exchange, which was then, after a second transaction, renamed the TSX Venture Exchange. The trading system of the TSX Venture Exchange is essentially the same as the Vancouver Computerized Trading system. At the time of the sample, the Vancouver Stock Exchange consisted of forty-five member firms. Each of these firms is directly connected to the exchange by computer-datalink. Only the member firms can submit orders directly to the system. These firms may act as brokers submitting orders on behalf of their customers and as dealers submitting orders on their own behalf. When orders are received at the exchange, they are matched based on price and then time priority. Orders must be multiples of a particular ticks size: one cent for orders below $3, five cents for orders between $3 and $4.99 and twelve and a half cents for orders at $ 5 and above. In addition, orders to the main board lot book must be multiples of a fixed size ranging from 1000 shares for lower-priced orders to 100 shares for higher-priced orders. If a trader wishes to submit a large order without revealing all the quantity, he may submit a hidden order. For these types of orders, a minimum of 1000 shares or 50% of the total order must be visible. The remaining quantity remains hidden on the book. Hidden quantities lose their time priority, so all orders at the same price will be executed before any hidden quantities. But once the disclosed portion of an order is executed, an additional 5 quantity becomes revealed equal to the amount of the original disclosure. Although hidden orders can play a large role on some limit order markets, I find that hidden orders are rarely used on the Vancouver Stock Exchange. Limit order markets are generally very transparent trading systems, and the Vancouver Stock Exchange is no exception. Member firms can view the entire limit order book at any time, while other parties may purchase data-feeds from the exchange that list the top quotes on each side of the book with the corresponding depth as well as the top ten best individual orders on each side of the book. While memberfirmscan also view trader identification codes for orders, this information is not available in my data set. To make use of this data, it is necessary to reconstruct the individual order histories and the time-series of the limit order books. Although no record is included in the data for orders added to the book, it is possible to extract the limit orders by examining the trade, cancellation, and change records. These records include initial submission information that allow the identification of the exact time that each order was first received by the exchange. Along with the book from the opening of the trading day, this allows me to follow each individual order from submission through to trade, cancellation, or expiration. As I do this, I apply the trading rules of the exchange. In less than one percent of cases, I find events in the data that I cannot reconcile with the rules of the exchange. I drop these orders from my analysis. 6 Chapter 2 Asymmetric Information in Limit Order Markets 2.1 Introduction This paper attempts to further our knowledge in the area of limit order markets by studying how asymmetric information will affect the exchange of securities on limit order markets. A simple model of a fixed-order-size limit order market is discussed and numerically solved. The solution is then used to derive empirical implications which can be tested using available tick-by-tick microstructure data. A large quantity of empirical literature has shown us that insider trading and other forms of asymmetric information play a large role in the trading of securities. Yet there remains a great deal of controversy on both the normative and positive issues involved. Informed traders earn profits at the expense of uninformed traders by exploiting their information in the timing of trades. This may serve to reduce market liquidity by driving uninformed traders out of the market. However, informed traders also play an important role in assuring that securities prices accurately reflect the underlying value of the security. The method by which informed traders exploit their information, and the process by which this information is incorporated into prices, are important areas of research. While the overall quantity of microstructure literature has seen a huge increase over the last decade, very little of the analysis has focused on empirically testable aspects of limit order order markets. This analytical gap has begun narrowing in the last several years, starting 7 with the studies of the institutional characteristics of limit order exchanges in Domowitz (1990, 1992, 1993) and the authoritative descriptive study of limit order book characteristics contained in Biais, Hillion, and Spatt (1995). The theoretical analysis of limit orders has included the studies of the competition between dealers and limit order books in Glosten (1994), Seppi (1997), Parlour (2002), and Viswanathan and Wang (1998). Kumar and Seppi (1993) and Charkravarty and Holden (1995) have examined the choice between market and limit orders in an equilibrium context. On the empirical side, Harris and Hasbrouk (1996), Handa and Schwartz (1996), Sandas (2001), and Hollifield et al. (2002) analyze the limit order submission strategy. Some of these issues have also been addressed from an experimental perspective in Schnitzlein (1996). Most of the theoretical work cited thus far treats the limit order market from a largely static context. The work in this paper follows most closely on recent extensions of this literature that focus on the dynamic analysis of limit order markets. This dynamic analysis allows a more complete description of how the order flow and the limit order book interact in the trading process. Parlour (1998) develops a dynamic equilibrium model of a limit order market with afixedhorizon and a one tick pricing grid. She is able to analyze how the state of the limit order book influences order choice from period to period. Foucault (1999) solves a model where traders may place orders of any price, but are restricted to afixedorder size as in Glosten arid Milgrom (1985). By solving his model in an infinite horizon context, Foucault obtains time-invariant results which can be used to make statements about overall market characteristics. This paper contributes to the literature by analyzing the dynamic submission problem under asymmetric information with a finite minimum price variation (tick size). I follow closely the model in Foucault (1999) and Parlour (1998). While Parlour solves the limit order submission problem with an arbitrary order size and afixedprice, Foucault solves the problem with a fixed order size but an arbitrary price. I follow most closely the model of Foucault (1999) where orders are offixedsize, but the trader may choose an optimal price. I extend the model by constraining the traders to a discrete pricing grid, and adding the possibility of asymmetrically informed traders. By doing so, I am able to analyze important problems that result from the dispersion of information among traders. I can compare the trading strategies and market outcomes under conditions where information is possessed by 8 only a very few traders to conditions where many traders are informed. Developing a dynamic model of a limit order market under asymmetric information is a step towards the following research goals: • Develop and test a theory of how asymmetric information affects the trading process in limit order markets and in hybrid markets which include limit order books. • Create methods to test for the presence of asymmetric information in a market by examining the characteristics of the limit order book and the order flow. • Compare the ability to incorporate information into prices of limit order and traditional dealer markets. 2.2 The Model The game is played by risk-neutral traders who are indexed by the single period when they arrive at the market, t. The trading day lasts for T periods where T is random. The probability of the day ending between any two periods is p. Treating the length of the day as random seems somewhat unsatisfactory since, in reality, this quantity is fixed and easily observable. This formulation, however, allows the solution to be characterized in a time-independent manner, which greatly eases the interpretation of results. In addition, by varying p, the model is able to address the issues of price dynamics as the end of the day approaches that have been raised in Parlour (1998). Furthermore, the trading day could be reinterpreted as the life of the security. Then T is the date of asset liquidation, which is a random value in reality. Or alternatively, the end of the trading day can be interpreted 1 as the point where uncertainty about the value of the security is resolved — for example, the time of the release of a news announcement that was anticipated, but whos timing was unknown. Traders participate in the market for an asset whos underlying value follows a random walk: v = v -i+e t t t (2.1) This interpretation is much more satisfactory on exchanges where the limit order books carries over from one day to the next, and less satisfactory when all outstanding orders are canceled at the end of the day. The Vancouver Stock Exchange is an example of the former type. 1 9 where it is an independently and identically distributed random variable that represents innovations in the asset value. When the market closes, traders realize a payoff of VT dollars 2 for each unit of the asset they hold. Traders arriving at the market at time t < T value the asset according to their expectation of the future payoff of the asset, plus an individual liquidity shock component that is independently and identically distributed with realization yt- The period T utility for the trader with liquidity shock y is 3 t U (y ) = (VT + yt- PB)IB + (Ps -ytt v )Is T (2.2) where PB and Ps are buy and sell execution prices, and Is and 1$ are indicator variables for the execution of the buy and sell orders. 4 In addition to the liquidity shock y , traders differ based on their knowledge of the t evolution of the asset value innovations. All traders arriving at the market at time t know the current innovation e . Some traders, who will be referred to as informed, also know next t period's asset value innovation et+i- The amount of information possessed by the trader entering the market at time t will be represented by 6 , which will equal I for an informed t trader, and U for an uninformed trader. The probability of an informed trader arriving at the market during period t is Pr(0 = I) = <f>. t In this framework, it is possible to place dominant strategy restrictions on the price at which a trader would be willing to buy or sell the asset. The reservation price for a trader at time t can be written Rt = vt + yt + E(et+i\9 ). (2.3) t For uninformed traders, the last term will be equal to the unconditional expectation, while for informed traders it will equal the realized value of et+\. No period t trader will ever be willing to buy at a price greater than R or sell at a price less than R , since this would result t t in a lower final utility than no action at all. The possibility of negative values is ignored here. It could be easily dealt with at the expense of some additional structure on the model. Liquidity shocks of some sort are necessary in all models of this type to avoid degenerate no-trade results. In a more complete model, the liquidity component could be derived from differences in the discount factor for different individuals. This utility function abstracts from many complexities such as trader endowments and time preference. However, this simple partial equilibrium framework allows the model to focus on the key issues in a limit order framework. 2 3 4 10 Traders arrive at the market one at a time and submit orders to buy and sell one unit of the asset at prices bt and at respectively. These orders indicate the traders willingness to buy at any price less than or equal to bt and sell at any price greater than or equal to a . The t prices submitted are restricted to fall on a pre-defined pricing grid that begins at zero and specifies available prices separated by a fixed tick size of Q; ie. the bid and ask prices must each be a member of the set {0, Q, 2Q, 3Q,...}. It is easy to show that b < Rt < a . The t t assumption that agents trade on both sides of the book is made for convenience only and does not constrain the choice set of the agents. A trader can choose not to trade on one or both sides of the book by placing limit orders sufficiently out of the money. If an order submitted by a trader fails to execute immediately, it will be referred to as a limit order, while all other orders will be referred to as market orders. Contrary to reality, all orders in this market last for only a single period and may not be changed or canceled after submission. The state of the market in each period can be represented using the state variable St = (st,Vt,yt,E[et+i\9t\). The only part of the state variable not yet defined is the book, St = (A ,Bt). t The values A and Bt are, respectively, the ask price and bid price available t on the book for the trader who arrives at time t. Because of the restriction to one unit, one period orders, there can never be more than one buy order and one sell order on the book at any time. To be specific, the ask price will be equal to the price of the limit buy order, at-i, if that order does not execute at time t — 1, and will be equal to oo otherwise. Formally, At = <! a -i if a -i > oo otherwise, h-i if h-i < t B -i t t A-i (2.4) , n . (2.5) 0 otherwise. With the book defined, it is possible to rephrase the feasible actions available to the agent who arrives at period t. By choosing a bid or ask price, the trader is implicitly making two inter-related decisions. First, she chooses whether to take up an order on the book by submitting a market order for immediate execution, or alternatively, post an new order on the book by submitting a limit order. Second, in the case where she chooses to post a limit order, she must choose the limit order price. Both these decisions can be summarized together in the chosen bid and ask prices, since a bid price greater than or equal to the ask price on the book (bt > At) defines a market buy order, and analogously at < Bt defines a market sell 11 • The time t — 1 trader submits at-\ and bt~\. • If they did not execute, then the orders on the book At-i and Bt-\ are canceled. This makes the new book At and Bt determined by equations (2.4) and (2.5). • The asset value innovation et is revealed to all traders, giving the public knowledge asset value vt• With probability p the game ends and the asset is liquidated at price VT = v . t • The time t trader enters the market. • The information type of the time t trader 9t is drawn. • The asset value innovation tt+i is drawn. • The time t liquidity shock y is drawn. t • If 6 = I, then et+\ is revealed to the trader. t • The trader picks bid and ask prices bt and at• If they did not execute, the orders At and Bt are cancelled. • ... Figure 2.1: Timeline for the Limit Order Game order. By conventional limit order exchange rules, the actual transaction price is determined by the order on the book, so the price is A for any b > At and similarly for the other side t t of the book. A time line for the game is presented in figure 2.1. 2.3 The Optimization Problem Consider a trader who arrives in the market at time t when the state is St. This trader will submit a buy order and a sell order. The expected utility this trader receives from submitting the buy order bt can be written J (S , b ) B t t = I(b < A )pPx{a i t t t+ < b \S ) (y + E [vr|S a t t t t> +I{bt > A ) (y + E [v \S ] - A ) t t T 12 t t t + 1 < b] - b) t t (2.6) where I(bt > A ) is an indicator variable for market order execution and I(bt < At) indit cates limit order states. While the sell order submitted at time t is not considered here, an analogous equation can be easily defined using the symmetry of the problem. This formulation of expected utility highlights most of the important trade-offs agents face in choosing between an order with immediate execution (a market order) and one which will go on the book (a limit order). These trade-offs can be classified in three general categories: Execution Risk: Traders who choose market orders can guarantee immediate execution, while traders who choose limit orders face some probability that their orders will not execute. The execution risk is made up of two components. With exogenous probability 1 — p, the market closes before next period and the limit order has no opportunity to trade. In addition, the endogenous probability Pr(at+i < bt\St) determines whether the trader next period will find it optimal to submit a sell order that will execute against the buy order with price btWinner's Curse: As the public value of the asset evolves between period t and period t + 1, it is easy to show that limit buy orders on the book will execute more often in states where the public value has declined than in states where it has increased. This implies E V \St,a i<b T t+ t <E V \S T t , (2.7) or, in other words, execution of a limit order is bad news about the evolution of the asset value. Comparing equations (2.7) and (2.6) shows that limit order traders face an additional cost of being picked-off after adverse information revelations. Price Improvement: The disadvantages of limit order submission are weighed against the possibility of obtaining a more advantageous price. The price paid in the market order case is equal to At, while the price in the limit order case is bt which is constrained to be less than At. In a simplified model, it is possible to think of limit bid orders getting to trade at the bid price while market bid orders must trade at the ask price. Price improvement can be traded off against execution probability, since a higher bid price will attract more trades in the following period. 13 For a trader arriving at time t, the optimization problem can be written max{J (S ,a ) A t + J (S ,bt)} t B (2.8) t at,bt where JA(-) is defined analogously to JB(')- Under the special conditions of this model, the problem is actually separable, so the trader can optimize each side of the book independently. 2.4 The Solution A standard Nash Equilibrium concept is used to identify solutions of the limit order market order submission game. Definition 1 A sub-game perfect equilibrium of the limit order market consists of order placement strategies a*(St) and b*(St) that solve (2.8) for each state of the world St given that all other players employ the same order placement strategies. The additional complexity of single-period asymmetric information along with a discrete pricing grid makes an analytical solution to this problem intractable. Instead, a bruteforce numerical solution is obtained. In order to make a brute-force approach feasible, it is necessary to restrict the size of the state space in the problem. Two different facts are useful here. First, the problem can be normalized such that the actual level of the common asset valuation v plays no role in agents choices. Instead, they worry only about the evolution of t the asset value from period to period. Second, the extent of the pricing grid is restricted by invoking the reservation values of the traders of the extreme types to bound the possible bid and ask prices. Using these facts, the solution algorithm works as follows. 1. For each possible St+i in the next period, arbitrarily pick a bid and ask price chosen by the traders. 2. Define a transition probability matrix that gives the probability of moving from each period t state to each period t +1 state. This matrix depends on the current state only through the informed traders knowledge of the evolution of the common asset value. 3. Using the transition probability matrix, calculate the probability of execution for each possible bid and ask price bt and a . For the bid side, this probability is defined as the t sum of the probabilities of reaching each t + 1 state where at+i < b . t 14 4. Calculate the change in asset valuation conditional on execution by finding the asset value change for all the execution states. 5. Apply equation (2.6) and the analogous equation on the ask side for each possible b t and at to obtain the expected utility. 6. Find the optimal bid and ask by applying equation (2.8). 7. Transform the optimal bid and ask for each current period state into corresponding values for next period states by adjusting for changes in common asset value. 8. Repeat from step two until step 6 yields the same results for two consecutive iterations. The solution to the model consists of, for each type of trader, a bid price for each possible ask price on the book, and an ask price for each possible bid price on the book. Contrary to the model in Parlour (1998), the one period nature of the orders implies traders do not care about the buy side of the book when submitting buy orders, or the sell side of the book when submitting sell orders. These orders will disappear before next period, and therefore do not provide competition against the orders submitted this period. The presentation of the solution is much simplified by the fact that the entire bidding strategy on each side of the book can be completely characterized with two numbers: the worst order on the book which will induce a market order this period, and the limit order submitted when the market order cutoff price is not met. The following proposition formalizes that notion. Proposition 1 The choice of an optimal bid (ask) price by a trader depends on the state of the book only through a comparison between the ask (bid) price on the book and a cutoff price which determines whether a market order or a limit order is submitted. Furthermore, conditional on a limit order being submitted, the price of the limit order does not depend on the book. Proof of Proposition 1: Define the value C (6t, vt, yt) as the solution for A in the equation b pPr(a Utility of Market buy m t < b \S ) (y + E [v \S , a i < b ] -) • b t t t T •v Utility of Limit buy 15 t t+ t t (2-9) Then, for any value of At > C (0t,vt,yt), the utility from submitting a market order will b be less than the limit order utility. This must be the case since the limit order utility does not depend on the existing book, and the market order utility depends on the book only through A . In particular, E [vV|5 ,a +i < &t] = E Vr\9t,vt,yt,at i t t t + < b , Pr(a i < t i+ b \S ) = P r ( a i < b \9t,v ,yt), and E ^Vr\S = E [Vr|0 ,ut , none of which depend on A t t or B . t t+ t t t Similarly, for A < C (6,vt,yt) t b t t the market order utility will exceed the limit order utility. A similar proof can be applied for sell orders, and the remainder of the proposition is straightforward. • Using this proposition allows a better understanding of how individual traders solve the optimization problem. For a trader arriving in period t, the decision to submit a limit order depends on the expected earnings of that limit order in period t+1, which in turn depend on the submission strategy of the trader arriving next period. The entire bidding strategy, however, is not relevant. The period t trader cares only about the cutoff prices chosen by period t + 1 traders, since these are the prices which effect the execution probability of (b ,a ) t t and therefore the expected profit in period t. In equilibrium, the period t trader knows the cutoff prices chosen by each period t + 1 type. Therefore, the optimization process in period t consists of choosing which t + 1 traders that she wishes to execute against, and setting the bid (ask) price equal to the highest (lowest) cutoff value in the group. This choice is non-trivial since trading against some traders implies executing after a positive value revelation, while trading against others implies trading after a negative value revelation. A trader submitting a bid may prefer to trade only after positive value revelations. However, since she must trade with any trader whose cutoff is less than or equal to her bid, she will be forced to accept some trades which imply negative value revelations in order to get other trades. In fact, since the sell cutoff values will be lower after a value decrease, bidders will always execute after value decreases with a (weakly) greater probability than after value increases. The incentive to trade provided by liquidity shocks and possible inside information must therefore be balanced against this adverse selection cost in choosing which types to trade against. The problem is further complicated by the fact that there is no clear ordering of cutoff values. Depending on the parameter values, an uninformed trader with a high liquidity shock after a value decrease may have a higher or lower cutoff than an uninformed trader with a 16 low liquidity shock after a value decrease, and so on. Therefore, the optimization problem for each set of parameter values involves first ordering the types according to cutoff values, and then identifying how high (or low) to allow the limit order to execute. This decision uniquely determines the limit order price. 2.5 A n Example In order to actually solve the model, it is necessary to place considerably more structure on the important random variables that determine traders' valuations of the asset: y and ett The structure builds on Foucault (1999). The evolution of the common value of the asset follows a simple, binomial process where et — +cr with probability 1/2 and et = —a with probability 1/2. Similarly, the liquidity shocks are defined such that yt = +L or — L with equal probabilities. Nothing inherent in the solution technique employed requires such a simple structure. Although a discretization will be necessary, neither the restriction to two possible values, nor the symmetry around zero are required for a solution. Departing from earlier work, the model also allows the consideration of a discrete tick size, which is fixed at Q. In the world described, there are four possible traders who could arrive in each period. Each trader is either informed or uninformed, and each has either a high or a low liquidity shock. The informed traders can be further subdivided into traders who know next period's information revelation will be positive, and traders who know next period's information revelation will be negative. Trader types are summarized in figure 2.2. This figure makes it clear that the only difference between an uninformed trader and an informed trader is that the informed trader knows the value of et+i prior to choosing the bid and ask price, while the uninformed player onlyfindsout this value after committing to a strategy. For illustrative purposes, consider the following set of parameter values: a = .1, L = .2, p = .99, v = 10, Q = .01. In words, this characterizes a market where, period by t period, liquidity shocks are twice as large as common asset value shocks, the probability of the market ending is small, and the tick size is very small relative to the shocks. 17 v t Figure 2.2: Trader Types 2.5.1 Finite Horizon In order to gain additional insight into the structure of the model and the solution technique, it is possible to solve a simpler finite horizon version. For this sub-section only, assume that the liquidation date T is known with certainty, so the probability of the game ending in any t <T is 0 (p = 1). Then it is possible to solve the model with standard backward induction techniques as described below. Since the game ends in period T, the period T — 1 trader knows with certainty that any limit order submitted in period T — 1 will expire unexecuted. Therefore, the only necessary decision in this period is whether to take-up an order on the book by submitting a market order, or to refrain from trading by submitting a limit order. By definition, the reservation value of the trader given in equation (2.3) gives the value to the trader of no transaction. The trader wishes to execute, therefore, if she can obtain any value greater than her reservation value. In this case, the trader will take up any bid price on the book which is greater than her reservation or any ask price less than her reservation. If neither of these conditions are met, she will not trade. Then the strategy of the T — 1 trader is Set both the bid and ask cutoff price equal to Rt- Conditional on not submitting market orders, the bid price is 0 and the ask price is oo. 18 The period T—2 trader does not have such a simple order submission strategy. Instead, he must consider the profitability of both limit order submission and market order submission. To solve the problem, the trader must first consider what the optimal limit orders will be, conditional on not executing a market order in period T — 2. To do this, he looks ahead to the cutoff strategy of the period T — 1 trader. In what follows, the bidding strategy of the uninformed, high liquidity shock type in period T — 2 is examined. Table C . l gives the cutoff prices for each possible trader who could come to market in period T — 1 calculated using the strategy given above (Cutoff = i?r-i)- Notice that, from a period T — 2 perspective, the T — 1 traders are identified not only by their type, but also by ex-i, the evolution of the asset value between periods T — 2 and T — 1. The numerical calculations use the assumptions that c/> = .5 and VT-2 = 10. It is easily shown that an optimal bidding strategy will involve placing a bid equal to one of the cutoff values. 5 Any bid between two cutoff prices in the table could be reduced without changing execution probabilities. Therefore, the bidding problem of the T — 2 trader is reduced to picking at which of the cutoff prices to set his bid. Increasing the bid results in higher execution probability, as shown by the values in CumProb. But the problem is not quite as simple as balancing execution probability and price improvement. The column AE(VT) gives the change in the expected terminal asset payoff for the T —2 trader conditional on executing a bid at a given cutoff price. This column summarizes the adverse selection cost of limit orders. The last column gives the profit from submitting a limit order at each cutoff price, taking into account the execution probability, execution price, and adverse selection obtained from trading with all types with cutoffs at that price and below. The optimal limit price is 9.9 or 10 and the optimal profit (above the reservation price) is . l . The trader is 6 shading his bid below his reservation price of VT-2 + L = 10.2 both to exploit his market power, and to compensate for the adverse selection. Knowing the profit from the optimal bid, it is possible to find the T — 2 bidding cutoff price for this trader type. This trader will wish to submit a market order if the profit obtained from it is greater than the .1 expected profit obtained by submitting a limit order. The complexities of discrete pricing are ignored here. For this set of parameters, a value of Q = .1 or Q = .01 would both lead to the results outlined below. The solution algorithm used in the numerical solutions picks the minimum bid price from any set of bid prices which yield the same payoff. 5 6 19 Solving for A in equation (2.9) yields a cutoff value of 10.1, which is below his reservation t price. This differs from the T — 1 strategy of setting the cutoff equal to the reservation price because of the possibility of submitting profitable limit orders in period T — 2. An analogous technique could be used for finding the optimal strategy on the ask side of the book for this trader, and for finding the bid and ask strategies for each trader type in period T — 2. Once the T — 2 cutoff prices are known, then the T — 3 problem can be solved, and the solution could be continued backward indefinitely. Unfortunately, the solution obtained is not easily generalizable. In particular, an analytic solution cannot be easily obtained, because the nature of the solution depends crucially on the ordering of the cutoff values, and this ordering depends in turn on the magnitude of the parameter values. By dividing the problem into a sufficient number of parameter regions, an analytic solution in the finite horizon should be possible, although the complexity of the algebra would likely become quickly unmanageable. 2.5.2 Infinite Horizon The infinite horizon solution technique is similar to the one used in the finite horizon. However, instead of starting with a known set of cutoff values in T — 1, the goal is to solve for a fixed point in the cutoff values and limit order prices. In addition to the complexity of cutoff ordering faced in the finite horizon game, any attempt to analytically solve the infinite horizon game is broken by the fact that cutoff values and optimal bid prices must be identified simultaneously. Unfortunately, there are too many degrees of freedom to solve the resulting system of equations. Instead, the numerical solution technique discussed earlier is employed. Figure C . l shows the equilibrium bidding strategy for traders with a high liquidity shock over a range of possible probabilities of facing informed traders (4>). From the results in Proposition 1, we can characterize the entire trading strategy with two numbers. The Market Cutoff is the price such that, if the ask order on the book is less than or equal to this price, the trader will submit a market order. The Limit Bid is the bid price chosen if the ask order on the book exceeds the market cutoff. This figure describes an informed trader who will hide among the uninformed by submitting an identical limit order, even though she has a reservation value greater than the uninformed trader. As the market becomes more heavily populated with informed traders, the bidding strategies diverge. Even though these 20 two traders have reservation values that differ by only a = .1, they submit bids that differ by .2. Table C.2 shows, for each type of trader, the probability of executing against each type of trader in the next period, conditional on submitting a limit order this period. Columns represent trader types in the current period and rows represent trader types next period. For example, when the probability of an informed trader (0) is .65, then an uninformed trader (Information=0) with a high liquidity shock who submits a limit order this period will execute next period against an uninformed low liquidity shock seller after an increase in the common asset value with probability .09. Summing down the column reveals that this type will execute with probability .42 conditional on submitting a limit order. By examining section a and b of table C.2, it is easy to explain the jump observed in the limit order price. As the probability of facing an informed trader increases, the increased adverse selection causes traders to shade their bids further from the reservation values. Eventually, the uninformed trader optimally chooses to forgo trading with one trader type, and therefore lowers the bid price until it matches the cutoff of the next highest seller. At this point, it is no longer optimal for the informed trader to hide among the uninformed, so she adjusts her bid upwards, to better reflect her reservation value. Other interesting features of the model are revealed through the examination of the limit order fill rates and market order probabilities for each type of trader and for the market as a whole. Figure C.2 shows these values for the traders with the high liquidity shock. The market order probability is the probability of seeing a market order this period, conditional on knowing the type this period. The limit order fill rate is the probability of seeing a market order next period, conditional on knowing the type this period, and knowing a limit order will be submitted this period. These numbers do not emerge easily from the optimization process. The technique for calculating the market order probability is discussed in appendix A, and the limit fill rate emerges from this in a straightforward way. Thisfiguregives further insight into the jump occurring at p = .65 in the limit order bids. We can see at this point, that the uninformed traders step out of the market and the informed traders take over. Since it is rare to be able to observe the type of trader on a trade-by-trade basis in empirical data, it will be very useful to see how the market looks on an aggregate basis. Figure C.3 shows the market-wide limit order fill and market order probabilities. In other words, 21 this figure shows the unconditional probability of seeing a market order at any particular point in the order flow, as well as the unconditional probability that any particular limit order will execute. 2.6 Possible Extensions Extending the model to a world with a variety of different kinds of liquidity shocks and a more complicated public value process would be easy within the present technical specification. However, this type of change would likely yield few interesting results, other than smoothing the choices observed over a variety of parameter values. The basic insights obtained in the binomial world most likely extend quite easily into a world with more complicated distributions. A much more rewarding, though surely much more difficult extension would involve allowing limit order to remain on the book for more than one period. This extension drastically changes the way that traders view the book in several different ways. Most importantly, a trader submitting a buy order can no longer ignore the current orders on the buy side of the book, since these orders provide potential competition to a limit order submitted this period. This parallels the reasoning in the one-tick world of Parlour (1998). For the researcher, this extension would open up a variety of additional issues that play important parts in limit order markets. For example, even a simple model with two-period orders could allow a basic analysis of how the depth of the book affects trader decisions. The analysis of time priority and tick size also plays a much greater role in such a model. In the long term, a model with long-lived orders could allow an examination of the role of order cancellations. This is a relatively unexplored, and potentially empirically important area. 22 Chapter 3 Liquidity Supply and Demand in Limit Order Markets 3.1 Introduction Market liquidity is used by exchanges, regulators, and investors to evaluate trading systems. In a limit order market, all traders with access to the trading system can supply liquidity by submitting limit orders or demand liquidity by submitting market orders. Market liquidity is determined by the traders' order submission strategies. Understanding the determinants of liquidity in a limit order market therefore requires understanding the determinants of the traders' order submission strategies. A market order transacts immediately at a price determined by the best quotes in the limit order book: a market order offers immediacy. A limit order offers price improvement relative to a market order, but there are costs to submitting a limit order rather than a market order. The limit order may take time to execute and may not completely execute before it expires; we call the probability that the order executes the execution probability. Since the limit order may not execute immediately, there is chance that the underlying value of the stock changes before the limit order executes; we call the resulting risk the picking off risk. The best quotes and the price improvements, execution probabilities and picking off risks of limit orders determine the price of immediacy. A trader's optimal order submission depends on the price of immediacy, and the trader's willingness to pay for immediacy. Why do traders' optimal order submissions vary? For example, the bottom panel 23 of Table 3 in Harris and Hasbrouck (1996) reports that on the NYSE, 42% of the order submissions are market orders when the spread is $1/8 and 30% of the orders submissions are market orders when the spread is $1/4. The change in the order submission frequency depends on the change in the price of immediacy and the distribution of the traders' willingness to pay for immediacy. But we do not directly observe the price of immediacy, nor the traders' willingness to pay for immediacy. Instead, we only observe the traders' order submissions. We model a trader's decision to supply liquidity by submitting limit orders or demand liquidity by submitting market orders. In our model, a trader's willingness to pay for immediacy depends on his valuation for the stock. Traders with extreme valuations for the stock lose more from failing to execute than traders with moderate valuations for the stock. Traders with extreme valuations therefore have a higher willingness to pay for immediacy than traders with moderate valuations. We interpret traders with extreme valuations as liquidity traders and traders with moderate valuations as value traders. A trader's valuation along with the price of immediacy determines whether the trader submits a market order, a limit order, or no order. We use a sample from the Vancouver Stock Exchange to estimate the price of immediacy and we estimate the unobserved distribution of traders' valuations and the unobserved arrival rates of traders. We estimate the price of immediacy by estimating the execution probabilities and picking off risks for alternative order submissions under the identifying assumption that traders have rational expectations. We estimate the distribution of the traders' valuations and the arrival rates of the traders by combining the estimated price of immediacy with the traders' actual order submissions under the identifying assumption that traders make their order submissions to maximize their expected utility. In our sample, when the proportional spread is 2.5%, approximately 37% of the orders submissions are market orders and when the proportional spread is 3.5%, approximately 30% of the order submissions are market orders. We use our estimates to compute the valuations for the traders who submit market orders in both cases. When the proportional spread is 2.5%, traders with valuations at least 4.9% away from the average valuation submit market orders, and when the proportional spread is 3.5%, traders with valuations at least 7.1% away from the average valuation submit market orders. The change in the spread changes the price of immediacy by changing the best quotes, and the execution probabilities and picking 24 off risks for limit orders. The magnitude of the change in the price of immediacy exceeds the change in the spread because a limit order offers relatively more immediacy for the same price improvement when the spread is wider. We also use our estimates of the price of immediacy to compute the expected utilities for liquidity and value traders in different market conditions. Traders can increase their expected utility by submitting different orders in different market conditions. Liquidity traders can increase their expected utility by up to 40% by submitting a limit order rather than a market order when the spread is wide and depth is low. Value traders can increase their expected utility by up to 10% by submitting a limit order rather than submitting no order when the spread is wide and the depth is low. The idea that the price of immediacy and the traders willingness to pay for immediacy determine trading activity goes back to Demsetz (1968). In Glosten (1994), Seppi (1997) and Parlour and Seppi (2001), liquidity is provided by a large number of risk neutral value traders who are restricted to submit limit orders. The equilibrium price of immediacy is determined by a zero-expected profit condition for the value traders. Sandas (2001) empirically tests and rejects the zero-expected profit conditions using a sample from the Stockholm Stock Exchange. Biais, Bisiere and Spatt (2001) estimate a model of imperfect competition based on Biais, Martimort and Rochet (2000), finding evidence of positive expected profits before decimalization and zero afterward using a sample from the Island ECN. Both studies use models where multiple limit orders are first submitted, followed by a single market order submission. We focus instead on how the order book evolves in real time from order submission to order submission. In our sample, value traders with a valuation within 2.5% of the average value of the stock account for between 32% and 52% of all traders. The value traders typically submit limit orders or no orders at all. The average expected time until the arrival of a value trader is approximately 23 minutes. The average time between orders submissions is 6 minutes. Profit opportunities for value traders are competed away slowly relative to the frequency of order submissions. We allow for the possibility that any trader can submit a limit order in our model; liquidity traders may compete with the value traders in supplying liquidity. In this respect, our model is similar to the models in Cohen, Maier, Schwartz and Whitcomb (1981), Foucault 25 (1999), Foucault, Kadan, and Kandel (2001), Handa and Schwartz (1996), Handa, Schwartz, and and Tiwari (2002), Harris (1998), Hollifield, Miller and Sandas (2002), and Parlour (1998) . We extend Hollifield, Miller and Sandas (2002) to allow for a stochastic arrival process for traders and a non-zero payoff to the traders at order cancellation. Several empirical studies document that traders' order submissions respond to market conditions. Biais, Hillion, and Spatt (1995) find that traders on the Paris Bourse react to a large spread or a small depth by submitting limit orders. Similar results hold in other markets. For example, see Ahn, Bae and Chan (2001) for the Stock Exchange of Hong Kong; Al-Suhaibani and Kryzanowksi (2001) for the Saudi Stock Market; Coppejans, Domowitz and Madhavan (2002) for the Swedish OMX futures market; and Chung, Van Ness and Van Ness (1999) and Bae, Jang and Park (2002) for the NYSE. Harris and Hasbrouck (1996) measure the payoffs from different order submissions on the NYSE for a trader who must trade and for a trader who is indifferent to trading. For a trader who must trade, submitting limit orders at or inside the best quotes is optimal, while for a trader indifferent to trading, submitting no order is optimal. Griffiths, Smith, Turnbull and White (2000) measure the payoffs from different order submissions on the Toronto Stock Exchange, finding that limit orders submitted at the quotes are optimal submissions for a trader who must trade. Al-Suhaibani and Kryzanowski (2001) find similar results for the Saudi Stock Market. A number of empirical studies examine the timing of orders. Biais, Hillion and Spatt (1995) document that traders submit limit orders in rapid succession when the spread widens on the Paris Bourse. Russell (1999) estimates multivariate autoregressive conditional duration models for the arrival of market and limit orders using a sample from the NYSE. Hasbrouck (1999) finds that the arrival rate of market and limit orders is negatively correlated over short horizons using a sample from the NYSE. Easley, Kiefer and O'Hara (1997) and Easley, Engle, O'Hara and Wu (2002) develop and estimate structural models relating the time between trades and the bid-ask spread to the arrival rates of informed and uniformed traders on the NYSE. Our work contributes to the literature by developing a comprehensive model of liquidity supply and demand that incorporates the profitability of various order submission strategies, the preferences of traders who arrive at the market, and the timing of the arrival 26 of traders. Using our structural model, we are able to simultaneously analyze these three factors, and extract them independently from limit order data. Our analysis is new in technique; no previous paper has been able to simultaneously identify all these factors. Our work is also original for the richness of the data set that it analyses; we are able to fully track the complete limit order book in three stocks. Our contribution is a new techinque to measure and assess liquidity in a market, as well as empirical evidence on how liquidity evolves in a particular market. 3.2 Description of the Sample Our sample was obtained from the audit tapes of the Vancouver Computerized Trading system. The sample contains order and transaction records from May 1990 to November 1993 for three stocks in the mining industry. Table D . l reports the stock ticker symbols, stock names, the total number of order submissions, and the percentage of buy and sell market and limit orders submitted in our sample. The bottom panel of the table reports the mean and standard deviation of the percentage bid-ask spread, and the mean and standard deviation of the depth in the limit order book at or close to the best bid and ask quotes, measured in units of thousands of shares. The depth measure is calculated as the average of the number of shares offered on the buy and the sell side of the order book within 2.5% of the mid-quote. We have detailed information, but there are limitations. First, we cannot separate the trades that a member firm makes on its own behalf from those it makes on behalf of its customers. Second, we cannot link different orders submitted by the same customer or member firm at different times. Third, we do not observe the identification codes the member firms observe. The first limitation causes us to focus on how a representative trader makes order submission decisions. Table D.2 reports the mean order size for buy and sell limit orders and market orders. The mean depth reported in Table D . l corresponds to a little more than three times the mean order size for all three stocks. The second row in each panel of Table D.2 reports t-tests of the null hypothesis of equal mean order sizes for market and limit orders, with pvalues in parentheses. The test rejects the null hypothesis for six out of nine pairs of means. 27 Despite evidence of statistically significant differences between market and limit order sizes, the economic significance of the differences is small. The relative difference between the mean order size for market and limit orders reported in the last column of the table is between one-half and four percent. To determine if traders' order submission decisions change in systematic ways as conditions change, we estimate models to predict the timing and type of order submissions, using conditioning variables reported in Table D.3. We divide the conditioning variables into five groups: book, activity, market-wide, value proxies, and time dummies. The book variables measure the current state of the limit order book, and include the bid-ask spread, and measures of depth close to the quotes and away from the quotes. Biais, Hillion, and Spatt (1995) and Engle and Russell (1998) document that in the Paris Bourse and the New York Stock Exchange, periods of high order submission activity are likely to be followed by periods of high order submission activity, and similarly for periods of low order submission activity. We include the number of recent trades, the sum of the time elapsed between the last ten order book changes, and the volatility of the mid-quote over the previous 10 minutes to capture such effects. We include market-wide conditioning variables to capture any market-wide effects on order submissions. We use the absolute values of the changes in the market-wide variables to proxy for their volatility. Because of data availability, all of our market-wide conditioning variables are computed at a daily frequency. Changes in the Toronto Stock Exchange (TSE) market index measure the overall information flow into the market. We use the T S E mining index to capture any industry effects. The change in the Canadian overnight interest rate is included because frictions such as margin requirements depend on the overnight interest rate. The change in the Canadian/US dollar exchange rate is a proxy for news about the Canadian economy. We include the absolute value of the lagged open to open mid-quote return of each stock to measure realized stock volatility. We compute a centered moving average of the midquotes over a twenty minute window as a proxy for the underlying value of the stock. We use a moving average to reduce any mechanical price effects arising from market orders using up all liquidity at the best quotes and changing the mid-quotes. We include the distance between the current mid-quote and the centered moving average as a measure of temporary 28 order imbalances in the order book. We also include six hourly dummy variables to capture any deterministic time effects. Table D.4 reports the results from estimating a Weibull model for the hazard rate of order submissions: Pr (Order submission in [t, t + dt)) — exp (~l'z ) a(t — U) t a ti dt, (3.1) 1 where the subscript t denotes conditioning on information available at t, ti is the time of the previous order submission, and Zt is a vector of conditioning variables. The point estimates 1 t of a are all less than one; the conditional probability of an order submission is decreasing in the length of time since the previous order submission. The parameters on the spread are negative — a wider spread predicts a longer time to the next order submission. The depth variables have mixed effects on the predicted time to the next order submission. The parameters are positive for recent trades and negative for duration: short time between order submissions predicts short time between order submissions in the future. The signs of the parameters on market-wide variables vary from stock to stock and many are not statistically different from zero. The parameters on lagged return are all positive; periods of high stock volatility predict shorter time between order submissions in the future. The parameters on the hourly dummies indicate that in general, the time between order submissions is longer in the first three hours of the day than during the last few hours of the day. To determine whether the conditioning variables predict the time between order submissions, we report chi-squared tests of the null hypothesis that all parameters are jointly equal to zero. The test statistic is reported below each group of conditioning variables with the corresponding p-value in parenthesis. Except for the market-wide variables for BHO, we reject the null in all cases. Table D.5 reports the estimation results for six ordered probit models of buy and sell order submissions. We condition on the variables in Table D.3 but use only close depth on Equation (3.1) is interpreted as 1 lim AtiO Pr (Order submission in [t, t + Ai)| previous order submission at U) ^ , ^ + \a-i = exp (7'z ) a(t - U) At t f u u 29 the opposite side of the order, and include the log of order size. We model the traders' choice between three types of orders: a market order, a limit order at one tick from the best quote, and a limit order at two or more ticks from the best quote. The dependent variable is zero for a market order, one for a limit order at one tick from the best quotes, and two for all other limit orders. The parameters on the spread are positive: traders are more likely to submit limit orders when the spread is large. The parameters for the close ask depth for sell orders and close bid depth for buy order are both negative; traders are less likely to submit limit orders when the depth on the same side as the order is high. The parameters on order size indicate that traders submitting larger orders are more likely to submit limit orders than traders submitting smaller orders. The last row of each panel reports chi-squared test statistics for a test of the null hypothesis that the estimated parameters on the conditioning variables are jointly equal to zero. The null hypothesis is rejected for all groups of conditioning variables but the market-wide variables. For the market-wide variables we reject the null for sell orders for BHO and for buy and sell orders for ERR. Overall, the conditioning variables predict the traders' decisions to supply liquidity by submitting limit orders or to demand liquidity by submitting market orders, as well as the timing of the order submissions. 3.3 Model We model the traders' order submission strategies. Traders arrive sequentially and differ in their valuations for the stock. The probability that a trader arrives is Pr (Trader arrives in [t, t + dt)) = \ dt. t t (3.2) The subscript t denotes conditioning on information available at time t. Information available at time t includes the time since the last order submission, the history of order submissions, general market conditions, and the current limit order book. Once a trader arrives, he can submit a market order for q shares, a limit order for q shares, or no order. Although we assume a fixed order size, we condition on the observed order size in our empirical work to allow for the possibility that the optimal order submission depends on q. The decision indicator variables dyf for s = 0,1,..., S; 30 for b = 0,1,..., B; and denote the trader's decision at t. If the trader submits a sell market order, dg /' — 1; e if the trader submits a sell limit order at the price s ticks above the bid quote, d l = 1; if a e l s the trader submits a buy market order, <i^ = 1; if the trader submits a buy limit order b y ticks below the ask quote, = 1; and if the trader does not submit any order, df° = 1. The trader is risk neutral and has a valuation per share for the stock of v , equal to t the sum of a common value and a private value: v = y + u. t t (3.3) t The common value, yt, is the trader's time t expectation of the liquidation value of the stock. The common value changes as the traders learn new information. Traders who arrive at t' > t therefore have more information about the common value than a trader who arrives at t. The private value, ut, is drawn i.i.d. across traders from the continuous distribution Pr (u <u) = G (u), t t t (3.4) with continuous density g . The distribution is conditional on information available at t, with t a mean of zero. Once the trader arrives, his private value is fixed until an exogenous random resubmission time t + Tresyfrmit > t. At t + r^y^it, the trader cancels any unexecuted limit orders and receives a fixed utility of V per share for any unexecuted shares, where V is the expected utility of a new order submission at t + T bmitresu The trader does not know the realization of the resubmission time when he arrives at the market. The resubmission time is bounded b y t + T where the constant T satisfies T < oo. Suppose that a trader with valuation v = y + ut submits a buy order b ticks below t the ask quote at price p^: d!^ = 1. Define 0 < t Qt+r < 1 as the cumulative fraction of the order executed by time t + T, and dQt+r = Qt,t+r — Qt+r- (3.5) as the fraction of the order that executes at time t + r . If the order is canceled at time t " i " Tresubmit^ dQt+r - 0, for r > T t. resubmi 31 (3.6) Ignoring the cost of submitting the order and the utility of any resubmission, the utility that the trader receives from executing dQt,t+r shares at t + T at price p t is 0t {yt+r + u - Pb,t) dQt+r = (,v - p ) dQt+r + (yt+T ~ yt) dQt+Tt t (3-7) bit Here, yt+ is the common value at t + T; (vt — Pb,t) dQt+r is the utility from executing dQt+ T T with the common value unchanged; and (yt+ ~ yt) dQt+ T is the utility from any common T value changes between t and t + T. Integrating over the possible execution times for the order, including the resubmission utility and the cost of the submission, the realized utility from submitting the order is U ,t+T= [ t {v -Pb,t)dQt+r + ( t 7r=0 (y +T-yt)dQt+T+ V{l-Qt+ )-c. T T (3.8) JT=0 Define (3.9) Qt+T d j y = i as the execution probability for the order. For a market order, the execution probability is one. Further, define = Et\f T (yt+T - = ll yt) dQt+T (3.10) as the picking off risk for the order. The picking off risk is the covariance of changes in the common value and the fraction of the order that executes. For a market order, the picking off risk is zero. The trader's expected utility from submitting a buy order at price p b is the expected tj value of equation (3.8), conditional on the trader's information, which using the definitions of the execution probability and picking off risk is equal to Uit,t+T < ? = 1, t * ] = ( t * - ,t) + V ( l - < f ) - c. + Pb (3.11) Similarly, the expected utility of submitting a sell order at p t is St t,t+T df = s s l,v t (p ,t - v ) Vsf ~ Slf + V ( l - r f) s t a - c (3.12) The trader's order submission strategy maximizes his expected utility, max W}.{dM }.dT° W s Y. st A > dS E s=0 Ut t+T + dfV, b=0 32 (3.13) subject to: d f € {0,1}, s s a= 0 , S , <$> e {0,1}, b = 0 , B , E<t" + s=0 d?° € {0,1}, (3.14) E< + ^° = 13/ t (3-15) 6=0 Equation (3.15) is the constraint that at most one submission is made at t. Let d *l *(v), d^*(v), s s d^°*(v) l be the optimal strategy, describing the trader's optimal order submission as a function of his information and valuation. Lemma 1 shows that the optimal order submission strategy is monotone in the trader's valuation. Lemma 1 Suppose that a buyer with valuation v optimally submits a buy order at price b > 0 ticks below the ask quote, so that d*^ (v) — 1. y If the execution probabilities are strictly decreasing in the distance between the limit order price and the best ask quote, b<b + l implies that ipffi > ip^ , for b = 0,..., B - 1, lt (3.16) then a trader with valuation v' > v submits a buy order at a price py weakly closer to the ask quote: > <t ™d b' < b. (3.17) Similar results hold on the sell side. Lemma 1 implies that all traders whose valuations are in the same interval submit the same order. We assume that sell market orders, sell limit orders between 1 and St ticks above the bid quote, buy market orders, and buy limit orders between 1 and B ticks below t the ask quote are all optimal submissions for the trader depending on his valuation. The assumption holds if the thresholds defined below form a monotone sequence. Define the threshold valuation 6^ (b, b') as the valuation of a trader who is indifferent v between submitting a buy order at price ^ - ^ + and a buy order at price py it V > - f f ^ - » ) . «3,S, The threshold valuation for a buy order at price pb t and not submitting an order is t ^(b,m)=p^ +V - ^ 33 s r . (3.19) The threshold valuation for a sell order at price p t and a sell order at price p ' t is St 6f l (s, s>) = s t -V Patt ' }, (3.20) e The threshold valuation for a sell limit order at price p t and not submitting any order is S: Of {s, NO) = p 1 -V- Sit ^r . 1 (3.21) The threshold valuation for a sell order at price p t and a buy order at price p t is St 0i Traders with high private values submit buy orders with high execution probabilities and prices. Traders with low private values submit sell orders with high execution probabilities and low prices. Traders with intermediate private values either submit no order or submit limit orders if the execution probabilities are high enough and the picking off risks are low enough. Define the marginal thresholds for sellers and buyers as (Marginal) = Of (Marginal) = 1 max (o (S , B ), 0^ iy (B , NO)) , min (o (S B ) ,6 s M t (St, NO)) . t t u t t t t (3.23) If the marginal threshold for the buyers is equal to the marginal threshold for the sellers, all traders find it optimal to submit an order. Otherwise, there are traders who find it optimal not to submit any order. Proposition 2 The optimal order submission strategy is s = 0, and -oo<y t + u < 6>f"(0,1), t or d f*{yt + u ) = 1, if < s s t 8 = 1 , S - 1 and 0f"(s - 1, s) < y + u < 6 t t t (s, s + 1) s ell t or ^ s = S , and e t (S - 1, S ) <yt + u < 0f"{Marginal), t = s ell 0, otherwise. t t t (3-24) 34 [ b = 0 and 6^ (0, l)<y y + u < oo, t t or b= 1 , B 4y*(y + u ) = 1, if < t t - 1 and 6^ (b -l,b)<y y + u < 9^ (b, b + 1), t y t t or b = B and 9^ (Marginal) <y + u < 6^ (B y t = t y t t - 1, B ), t 0, otherwise. °*(j/t + u ) t = = 1 (3.25) if e (Marginal) s eU t <y + u < Of t t 1 (Marginal), 0, otherwise. (3.26) Figure D . l provides a graphical representation of the trader's order submission problem. Here, buy market, one tick and two tick buy limit orders, sell market orders, and one tick and two tick sell limit orders are optimal for a trader with some valuation. The continuation value is equal to zero. The expected utility as a function of the trader's valuation from submitting different sell orders are plotted with dashed lines and the expected utility from submitting different buy orders are plotted with dashed-dotted lines. From equations (3.11) and (3.12), the trader's expected utility from submitting any particular order is a linear function of his valuation, with slope equal to the execution probability for that order. The dark solid line is the maximized utility function. Geometrically, the thresholds are the valuations where the expected utilities intersect. For example, the threshold for a sell market order and a one tick sell limit order is 9f (0,1); M a trader with a valuation less than 0$ (0,1) submits a sell market order. The thresholds ell associated with submitting any particular order and submitting no order are the valuations where the expected utilities cross the horizontal axis. Here, Of (2, NO) < 6 (2,2), 1 and 0 ^ ( 2 ^ 0 ) > 9 (2,2), t gbuy t so that if the trader's valuation is between 0f"(2,NO) and ^ NO), the trader does not submit any order. The threshold valuations measure the price of immediacy. Consider the threshold val- uation for two buy orders given in equation (3.18). A lower execution probability for a buy order at pyj implies a decrease in the threshold valuation. For the same price improvement and picking off risk, the higher priced buy order, now offers relatively more immediacy than the lower price buy order; the relative price of immediacy is lower. The traders willingness to pay for immediacy is determined by their valuations. Traders with valuation equal 35 to or above the threshold valuation in equation (3.18) submit buy orders at the price pb t or t higher. A lower price of immediacy as a result of a lower execution probability at p \t implies 0 that a larger fraction of the traders will submit buy orders at pi, or higher. tt Using the optimal order strategy given in Proposition 2, the distribution of traders' valuations and the arrival rates of traders, we compute the conditional probability of different order submissions. The conditional probability of a buy market order between t and t + dt is the probability that a trader who arrives finds it optimal to submit a buy market order times the probability that a trader arrives: Pr (Buy market order in [t, t + dt)) - Pr (y + u > 0^(0,1)) \ dt t t = [l-G t t t t (0^(0,1) - y )] Xtdt. (3.27) t Similarly, Pr (Buy limit order in [t,t + dt)) = [G* (^"(0,1) - y ) - G (^(Marginal) - y )] \ dt, t t t t t (3.28) Pr (Sell market order in [t, t + dt)) = [G* (^""(0,1) - yt)] Xtdt, (3.29) t Pr (Sell limit order in [t, t + dt)) = | G (of*"(Marginal) - y ) - G (e t ( ( b t {0,1) - y ) X dt. s ell t f t (3.30) No orders may be submitted between t and t + dt for two reasons. Either a trader does not arrive or a trader arrives and does not submit any order, Pr (No order submission in [t, t + dt)) f = 1 - X dt + [Gt (^(Marginal) - y ) - G (0f"(Marginal) - y )] X dt. (3.31) t t t t t Equations (3.27) through (3.31) show how the thresholds, the private values distribution and the arrival rate of the traders jointly determine the timing of order submissions. The arrival rate of traders and the distribution of traders' valuations determine the relative competitiveness of liquidity supply. Consider a stock with a high arrival rate of traders and many traders with private values close to zero. From equations (3.28) and (3.30), the model predicts a large probability of limit order submissions when the price of immediacy increases so that value traders find it optimal to submit limit orders. In this case, we expect the order 36 book to offer profit opportunities for value traders only for short periods of time. Liquidity supply is therefore relatively competitive. 3.4 Empirical Results The model described in the previous section provides a framework for our econometric approach. The model characterizes a trader's choice between submitting an order or not, and if he submits an order, what kind of order to submit. In our empirical work, we model the rate at which orders are executed and canceled parametrically. Our approach allows us to identify the unobserved arrival rate of traders and their distribution of valuations from the timing of market and limit order submissions in the sample. We use the conditional probabilities of observing different order submissions in equations (3.27) through (3.31) to compute the conditional log-likelihood function for buy and sell market and limit order submission times. The log-likelihood function is conditional on the variables reported in Table D.3 and the log of order size. The conditional log-likelihood function is reported in Appendix B . l . The conditional log-likelihood function depends on the common value at the time of order submission, the thresholds, the arrival rate of traders, and the private value distribution. The thresholds depend on the market and limit order prices; execution probabilities and picking off risks of market and limit order submissions; the expected utility of a resubmission; and the costs of submitting an order. 3.4.1 The Price of Immediacy We assume that the traders have rational expectations about the execution probabilities and picking off risks. We therefore use all order submissions and the realized execution histories in our sample to form estimates of the execution probabilities and picking off risks for orders that the traders could have submitted. In forming estimates of the picking off risks, we use a twenty minute centered average mid-quote to proxy for the common value. An order leaves the book when it is executed or when it is canceled. We use a Weibull independent competing risks model to estimate how long an order remains in the limit order book. Lancaster (1990) provides an introduction to the competing risks model. Suppose that a limit order is submitted at t\. The order leaves the book at the minimum of the hypothetical 37 cancellation time for the order and the hypothetical execution time for the order. The hazard rate for the hypothetical cancellation time is Prt (Order submitted at t\ cancels in [t, t + dt)) = ex.p(z' . y )a (t — ti) ~ dt, t , c ac c 1 and the hazard rate for the hypothetical execution time is Pr (Order submitted at t{ executes in [t,t + dt)) = exp(^.7 )a (t — £j) t e e ae_1 cft, (3.33) where z^ is a vector of conditioning information known when the order is submitted at t{. Lo, MacKinlay and Zhang (2001) estimate a Gamma model for the time to first execution and time to completion for limit orders, treating canceled orders as censored observations. Al-Suhaibani and Kryzanowski (2000) and Cho and Nelling (2000) estimate Weibull models for the time to execution of limit orders, also treating canceled limit orders as censored observations. Cho and Nelling (2000) compute the execution probabilities for limit orders based on the parameter estimates for the time to execution. We compute execution probabilities using the competing risks model, where we explicitly estimate the hazard rate of cancellations. Details of the computations of the execution probabilities are reported in Appendix B.2. We condition on the variables in Table D.3, including the log of order size. We track the orders for two-days, treating orders outstanding two days after submission as censored observations. We handle partial executions by assuming an order was an execution if at least 50% of the order size is executed, otherwise we treat the order as a cancellation. We estimate hazard.rates for execution and cancellation for buy and sell limit orders submitted one tick from the best quotes and for marginal limit orders. We chose the marginal limit order so that approximately 95% of the limit order submissions are closer to the quotes than the marginal order at any price level. Tables D.6 through D.8 report the estimation results for the Weibull competing risks models for the execution and cancellation hazard rates for buy and sell limit orders at one tick from the best quotes and for marginal limit orders. The models are estimated by maximum likelihood. The parameter estimates for the Weibull a parameters are between 0.544 and 0.817 for the execution hazard rates and between 0.468 and 0.652 for the cancellation hazard rates. The longer the order has been outstanding, the lower is the probability that the order either 38 (3.32) executes or cancels. The Weibull a parameter is lower for the cancellation hazard rate than for the execution hazard rate; the probability that a limit order leaves the book by cancellation rather than execution increases with the time the order spends in the book. Increasing the spread increases the hazard for execution for all one tick orders and for most marginal orders. Depth on the same side decreases the hazard for execution for all orders. Depth on the opposite side increases the hazard for execution for all but two order types: the exceptions are the marginal orders for BHO. The marginal impact of increasing the depth on the same side is greater for the marginal orders than for the one tick orders. Larger order size decreases the hazard for execution for all orders and for all stocks. Orders are executed and canceled more quickly following periods of frequent order submissions. For one tick orders, higher mid-quote volatility increases the hazard for execution and decreases the hazard for cancellation for all sell orders. The effect is reversed for buy orders. More recent trades increases the hazard for executions and cancellations and an increase in the durations for the last ten orders decreases the hazard for executions and cancellations. The market wide variables have small, but statistically significant, parameters on the hazards for execution and cancellation. Lagged return increases the hazard for execution for all but one order type, although many of the parameters are not significantly different from zero. When distance to mid-quote is positive so that the common value is above the mid-quote, sell orders have higher hazards for executions and cancellations. When distance to mid-quote is negative so that the common value is below the mid-quote buy orders have higher hazards for executions and cancellations. Most of the parameters on the hourly dummies are negative and statistically significant for the hazard for cancellation; orders submitted earlier during the day are less likely to be canceled. For execution times there is an offsetting effect for one tick away orders; orders submitted earlier in the day have a lower hazard for execution. We forecast execution probabilities for one tick and marginal limit orders at every order submission in our sample using the parameter estimates from the competing risks model. We compute the probability that the order executes within two days. We use a two day cutoff because the majority of executions occur within two days. For BHO, the average execution probability for marginal sell limit orders is approximately 16%, for one tick sell limit orders 61%, for marginal buy limit orders 13% and for one tick buy limit orders 63%. 39 The estimates for the other stocks are similar. We also compute the elasticities of the execution probabilities with respect to the conditioning variables, evaluated at the means of the conditioning variables. For all stocks, an increase in spread leads to the largest estimated increase in the execution probabilities among all conditioning variables. When the spread is more than one tick a limit order submitted at one tick away from the quotes undercuts the prevailing quotes and moves to the front of the order queue. The wider the spread, the more the one tick order undercuts the quotes and lowers the price of immediacy for traders on the opposite side. As a consequence, one tick away limit orders have execution probabilities that are increasing in the spread. We find smaller effects on marginal orders from an increase in the spread. Larger order size decreases the execution probabilities for all limit orders. Greater depth on the buy side of the book increases the execution probabilities for sell limit orders, and depth on the sell side of the book increases the execution probabilities for buy limit orders. Depth on the buy side of the book decreases the execution probabilities for buy limit orders, with a similar effect on the sell side. Proposition 1 in Parlour (1998) predicts that in equilibrium the execution probabilities for buy limit orders are decreasing in the depth on the own side and increasing in the depth on the other side, and the sensitivity to own side depth is greater than the sensitivity to other side depth. The estimated execution probabilities for one tick buy and sell limit orders are consistent with her prediction. For all stocks, we find that the execution probability is decreasing in own side depth and increasing in the other side depth, but the absolute magnitudes of the sensitivities are only consistent with Proposition 1 in Parlour (1998) for ERR and for buy orders for W E M . Increasing recent order submission activity as measured by increasing the number of recent traders, increasing mid-quote volatility and reducing lagged duration increases the execution probabilities for one tick and marginal sell limit orders, and decreases the execution probabilities for one tick and marginal buy limit orders. The market-wide variables have small effects on the execution probabilities. Lagged returns also have small effects on the execution probabilities. Execution probabilities are in general lower for the last hour of trading and they tend to decrease over the trading day although the trend is not monotone for all stocks. Using the competing risks model to compute execution probabilities, we assume that 40 either the order fully executes or fully cancels. We show in Appendix B.3 that the assumption implies that the picking off risk is buy _ c j-, rU+T j {yu+T - ytJdQti+r b,U = 1,Q U+T = 1 fltS. (3.34) We assume that the expected change in the common value conditional on an execution is a linear function: rU+T rU+T j (yu+T -yti)dQ +T u d . = l,Qti+r = i y bt = z' (3, TI (3.35) where zt is information known at the time of the order submission. t We report the results of estimating equation (3.35) with ordinary least squares in Table D.9. We condition on the variables in Table D.3, also including the log of order size. Using the parameter estimates, we compute the expected change in the common value, conditional on the limit order executing. At the mean values of the conditioning variables, the expected change is approximately zero for one tick limit orders, minus four cents for marginal buy limit orders and four cents for marginal sell limit orders. The expected change in the common value conditional on execution is decreasing in the spread for sell orders except for marginal sell orders for BHO and increasing in the spread for all buy limit orders. The only other conditioning variable with much impact on the expected change in the common value is the distance between the mid-quote and the common value. When the common value relative to the mid-quote increases, the expected change in the common value decreases for sell and buy limit orders, consistent with the distance capturing temporary imbalances in the order book. We form estimates of the picking off risk by substituting our estimates of the expected change in the common value conditional on execution and the execution probabilities in equation (3.34). At the mean values the picking off risk is close to zero for one tick limit orders, and of the order of one cent for marginal limit orders. A change in the distance to the mid-quote has the largest effect on the picking off risk. When the common value is one standard deviation above its mean value, the picking off risk for a marginal sell limit orders drops roughly by one-half. Similar results hold on the buy side. The impact of the spread is mixed. A one standard deviation increase in the spread leaves the average picking off risk for a marginal sell limit order unchanged and approximately doubles the picking off risk for 41 a marginal buy limit order. The other conditioning variables have smaller impact on the estimated picking off risks. 3.4.2 The Arrival Rate of the Traders and the Private Value Distribution The arrival rate of the traders is parameterized as a Weibull distribution, Xtdt = exp(r/ x )c_(t-t ) - dt, a ti i aa (3.36) 1 with xt denoting the market-wide variables and absolute value of the lagged return at U. t The private value distribution is parameterized as a mixture of two normal distributions with standard deviations depending on the common value and market-wide variables, ^ T - T ) + (1 - P ) $ ( Gt(u) = p$ ( \y aiexp(rxt)J "h^) » (3-37) \y a exp CT'x ) J t t 2 t where 3? denotes the standard normal cumulative distribution function; a\ ^ a_, 0 < p < 1, and Xt denotes the market-wide variables and absolute value of the lagged return at t. The mixture of normals allows both for more mass in the middle of the distribution and fatter tails than the normal distribution. We normalize valuations as percentages of the common value by parameterizing the standard deviation as proportional to the common value. Table D.10 reports the estimates of the discrete choice model, with associated standard errors reported in parentheses. The model is estimated by maximizing the conditional likelihood function. The likelihood function is relatively flat with respect to the order submission cost, for positive order entry costs. We therefore did not estimate the order entry cost but set it equal to 0.1 cents. The first row of the top panel reports estimates of the Weibull parameter, a . The a parameter estimate is less than one for all stocks: the longer the time since the last order submission, the lower the conditional probability that a new trader will arrive. The second through seventh rows of the top panel report estimates of the arrival rate parameters, j . a The conditioning variables are standardized by dividing by their sample standard deviations. Evaluated at the mean values of the conditioning variables, the expected time between traders arrivals is 148 seconds for BHO, 131 seconds for ERR and 396 for W E M . The mean times between order submissions in our sample are 367 seconds for BHO, 347 seconds for ERR 42 and and 425 seconds for W E M . This difference between the unconditional arrival rate of orders and the conditional arrival rate of traders shows that some traders who arrive at the market do not find it optimal to submit an order. This happens when a value trader (a trader with a private value close to the mean) arrives, but after analyzing the book and the market conditions, finds that supplying liquidity by submitting a limit order will not be profitable. Since this trader has no particular need to trade, he simply leaves the market. The fitted trader arrival rate does depend, to some extent, on the chosen parameterization of the private value distributions. For example, if the true private-value distribution is bimodal, our imposition of a unimodal distribution would have the effect of forcing weight towards the mean, thereby increasing the apparent percentage of traders who submit no orders,In most financial markets, however, it is evident that there are a large percentage of traders who participate in the market simply to earn trading profits. The existence of this mass of traders, with valuations near the mean, leads us to conclude that a uni-modal distribution is reasonable. By using a mixture of normals, we assure enough flexibility in the distribution to allow for thick tails of liquidity traders. The estimated parameters on many of the market-wide variables are positive; increases in the market-wide variables tend to increase the trader arrival rate. For example, the arrival rate of traders increases following periods of higher market-wide volatility, as measured by the TSE market index. The parameters on the absolute value of the lagged return are positive and larger in magnitude than the other arrival rate parameters. Higher realized stock volatility predicts an increase in the arrival rates. The null hypothesis of constant arrival rates is rejected for all three stocks. The second panel reports estimates of the valuation distribution parameters. For all stocks, the private value distribution is a mixture of two normal distributions, with approximately 85% weight on a distribution with a small standard deviation and 15% on a distribution with a large standard deviation. The estimates imply the standard deviation of private value distribution is 21% for BHO, 24% for ERR, and 21% for W E M . The estimated distributions differ from a normal. The kurtosis is 17 for BHO, 24 for ERR, and 18 for WEM; the extreme tails of the private values distribution are fatter than a normal distribution. The inter-quartile range is 9% for BHO, 6% for ERR, and 6% for W E M . The inter-quartile range is between one fifth to one third of what it would be for a normal distribution with the same 43 standard deviation; there is more mass in the middle of the distributions than for a normal distribution with the same standard deviation. The estimated parameters on many of the market-wide variables are statistically significant, although small. The parameters on the lagged return on all three stocks are negative — when realized stock return volatility is high, there tend to be more value traders. The hypothesis of constant variance for the valuation distribution is rejected for all three stocks. For all three stocks, the continuation value is approximately 2% of the common value. The continuation value is approximately the same as one-half the mean spread. 3.4.3 Interpreting the Findings Figures D.2 through D.7 provide plots of the effects of changing the conditioning information on the fitted probability of different order submissions conditional on an order being submitted and the expected time to an order submission. The probability of observing a sell market order is computed by substituting our estimates of the execution probabilities, the picking off risks, the order submission costs, the continuation values, the common value and the coefficients of the valuation distribution into equation (3.38) below: Pr (Sell market order in [t, t + dt) | Order submission in [t, t + dt)) t - , <M*f"(0,!)-»,) , (3.38) 1 - G (^"(Marginal) - y,J + G , (Of "(Marginal) - y ) t t with the other probabilities computed similarly. Thefittedexpected time to an order submission is computed using the implied hazard rate for order submissions from the discrete choice model, evaluated at the parameter estimates. The top left graphs in Figures D.2 through D.4 plot the probability of a sell market order submission and a sell limit order submission conditional on an order submission, as a function of the spread, keeping all other variables at their sample means. The top middle graphs in Figures D.2 through D.4 plot the conditional probabilities of buy market and buy limit order submissions conditional on an order submission, as a function of the spread, holding all other variables at their sample means. For all three stocks, a larger spread increases the probability that limit orders are submitted. The top right graphs of Figures D.2 through D.4 plot the fitted expected time to the 44 next order submission as a function of the spread, holding all other variables at their sample means. The expected time to the next order decreases for all stocks. According to our estimates, a wider spread implies a higher price of immediacy. A higher price of immediacy encourages value traders to supply liquidity rather than stay out of the market; the fitted expected time to the next order submission therefore decreases. But the estimates of the Weibull model for order submissions in Table D.4 show that a wider spread predicts a longer expected time to the next order submission. Since Figures D.2 through D.4 are comparative statics — holding all other variables constant as the spread changes, for example — the difference between their results and the results from the Weibull model point to other variables changing simultaneously with the spread and impacting the time between order submissions. In particular, for our model, the arrival rate of traders and the relative proportion of value traders to liquidity traders must decrease when the spread widens. There is less competition in supplying liquidity in wider spread markets, and this drives down the arrival rate of orders, independent of the fact that the wider spread itself actually encourages a higher arrival rate. The second row of Figures D.2 through D.4 plot the fitted probabilities of limit and market order submissions and the fitted expected time to the next order submission as a function of the bid depth, holding all other variables at their sample means. Higher bid depth leads to a higher execution probability for sell limit orders and a lower execution probability for buy limit orders; the price of immediacy is lower for sellers and higher for buyers. Sellers therefore are more likely to submit market orders and buyers are more likely to submit limit orders although the change is small compared with the change for the spread. The third row of Figures D.2 through D.4 plot the fitted probabilities of limit and market order submissions and the fitted expected time to the next order submission as a function of the distance between the common value and the mid-quote, holding all other variables at their sample means. The distance between the common value and the mid-quote reflects temporary imbalances in the order book. When the common value is higher than the mid-quote the price of immediacy is high for sellers and low for buyers, leading sellers to submit limit orders and buyers to submit market orders. Similar effects hold on the buy side. For both the bid depth and the distance to the mid-quote, the effects on the expected time to the next order submission are mixed. Figures D.5 through D.7 plot the order submission probabilities and the time to the 45 next order changing the absolute value of the lagged stock return. The top left and middle plots of the figures show that as the absolute value of the lagged return increases, more traders are likely to submit limit orders, conditional on an order submission. The sole exception is the buy order for BHO. In the second row, we plot the fitted probabilities and times holding the arrival rate and the valuation distribution at their mean values, while varying the estimated price of immediacy with the absolute value of the lagged return. In the third row, we plot the fitted probabilities and expected times holding the price of immediacy at its sample mean, while varying the valuation distributions and arrival rates to change with the absolute value of the lagged return. Comparing the top, middle and bottom plots for the conditional probability, both changes in the price of immediacy and changes in the valuation distribution causes the order submission probabilities to change. Changes in the expected time to the next order are mainly caused by changes in the arrival rate of traders. We performed similar computations for the effects of changing the market-wide variables. Generally, we found small changes in the fitted probabilities and expected times from changing the market-wide variables despite the statistically significant impact of market-wide variables on the arrival rates and the valuation distributions reported in Table D.10. Table D . U reports the expected utilities for traders with six different private values across three different market conditions: a low liquidity state with a wide spread and low depth, a high liquidity state with a narrow spread and high depth, and an order imbalance state where the common value is above the mid-quote. For each private value, we report the expected utility from submitting a market order, one tick limit order, marginal limit order, or no order. The private values are 1.25%, 2.5%, or 5% higher or lower than the common value; the corresponding private values in cents are reported in the second row. The reported expected utility is a lower bound on the trader's expected utility since we do not compute the expected utility for limit orders between one tick from the quotes and marginal limit orders. The maximum utility is indicated for each private value and market condition with a box. In the low liquidity state, the price of immediacy is high. A trader with a private value 2.5% from the common value optimally submits a marginal limit order and a trader with private value 5% from the common value optimally submits a one tick limit order. In the high liquidity state, the price of immediacy is lower than in the low liquidity state. A 46 trader with a private value 1.25% from the common value optimally submits no order in BHO and ERR and optimally submits a limit order in W E M . A trader with a private value 2.5% from the common value optimally submits limit orders for BHO and ERR and submits market orders for W E M . A trader with a private value 5% from the common value optimally demands liquidity by submitting a market order for all stocks. In the order imbalance state, the optimal order strategies are asymmetric. A high common value relative to the mid-quote implies that the price for immediacy is low for buyers and high for sellers. For traders with positive private values, the optimal strategy is therefore to submit market orders in ERR and W E M for all three valuations above the common value, picking off some sell limit orders. Traders who submit buy orders earn a higher surplus than they did in the other market conditions. For BHO, traders with private values 1.25% and 2.5% above the common value submit one tick limit orders rather than market orders. Overall, the expected utility calculations reported in Table D . l l show that traders can earn a higher expected utility by following the optimal order submission strategies rather than a naive strategy of always making the same order submission. Table D.12 reports summary statistics for the estimated optimal order submission strategies for five intervals for the private value. The first two rows in each panel report the mean and the standard deviation of the proportion of traders in each private value interval. The next five rows report the mean order submission probabilities for a sell market, a sell limit, a buy limit, a buy market, and no order. The bottom five rows report the standard deviations of the order submission probabilities. We interpret traders with valuations within 2.5% of the common value as value traders. Such traders represent 32% of the traders in BHO, 48% of the traders in ERR and 52% of the traders in W E M . The probability of the value traders submitting no order is 36% for BHO, 61% for ERR, and 12% for W E M . When value traders submit an order, it is typically a limit order. The probability of a market order submission for the value traders is between 0.7% and 3.6%. The value trader supply liquidity when the price of immediacy is high and do not submit orders when the price for immediacy is low. Traders with valuations between 2.5% and 5% from the common value compete with the value traders in supplying liquidity. Such traders represent 26% of the traders in BHO, 47 28% of the traders in ERR and 26% of the traders in in W E M . They submit either market or limit orders most of the time; the probability that they submit no order is between 0.4% and 8.9%. Their order submissions are fairly evenly split between market and limit orders. Liquidity traders with valuations more than 5% from the common value have the highest willingness to pay for immediacy. Such traders represent 42% of the traders in BHO, 23% of the traders in ERR, and 22% of the traders in W E M . Submitting no order is almost never an optimal strategy for liquidity traders: the probability of submitting no order is between 0.1% and 1.9%. The probability of submitting a market order is roughly 80% for BHO, 81% for ERR, and 88% for W E M . The standard deviations of the optimal order submission probabilities for all types of traders show that their optimal order submissions vary with market conditions. Using the estimated Weibull parameters from Table D.10 and the proportion of the traders that are value traders, we compute the expected time between the arrivals of value traders for each stock. The average time across all three stocks is approximately 23 minutes while the average time between order submissions is approximately 6 minutes. Value traders compete in supplying liquidity relatively slowly. 3.5 Conclusions We model a trader's decision to supply liquidity by submitting limit orders or demand liquidity by submitting market orders in a limit order market. The best quotes, and the execution probabilities and picking off risks of limit orders determine the price of immediacy. The price of immediacy and the traders valuation for the stock determine the optimal order submission. We estimate the execution probabilities and the picking off risks for a sample from the Vancouver Stock Exchange. We use a competing risks model to estimate the hazard rates for execution and cancellation. Both executions and cancellations are predictable. The predictable cancellations times suggest that one natural extension of our model would be to model the trader's decisions to cancel in more detail. We use the estimates to compute the price of immediacy. The price of immediacy changes with market conditions; a value trader and a liquidity trader can increase their expected utility by changing their order submissions with market conditions. Increasing the 48 spread increases the price of immediacy causing liquidity traders to switch from submitting market orders to submitting limit orders, and value traders to switch from submitting no orders to submitting limit orders. We combine our estimates of the price of immediacy with the actual order submissions to estimate the unobserved arrival rate of the traders and the distribution of the traders' valuations. The arrival rate of traders and the relative proportion of value traders to liquidity traders increase following periods of high stock volatility. The arrival rate of traders and the relative proportion of value traders to liquidity traders decrease when the spread is wide. We leave the question of how value traders' decide to allocate resources to supplying liquidity in different stocks for future work. Euronext, the recent amalgamation of the exchanges in Amsterdam, Brussels and Paris is a limit order market where liquidity providers supplement the liquidity in the limit order books for the less actively traded stocks. Liquidity providers are obligated to provide a specified level of liquidity in a stock, and in exchange their transaction fees are waived. Periodically Euronext evaluates whether or not the liquidity providers add enough liquidity to justify the loss in revenue from the waived transaction fees. Estimates of the arrival rates of the traders and the distribution of the traders' valuations provide useful information for such evaluations. We leave such computations for future work. 49 Chapter 4 Dynamic Strategies in a Limit Order Market 4.1 Introduction In many modern exchanges, liquidity in the market for securities is provided in whole or in part by limit orders placed on a limit order book. Liquidity suppliers submit limit orders that are added to the book, while liquidity demanders submit market orders that execute against orders on the book. But these are not one-shot decisions. Limit order traders have the opportunity to return to the market—to cancel an order or change their position in the book—at any time before their orders expire or are executed. Traders do not appear at the market with the opportunity to trade, make their order selection, and then passively realize the outcome of the trading process. The trading rules on limit order markets allow traders to continuously monitor the book and make changes to orders whenever necessary. A modern investor, whether professional trader or client, can have access to live data feeds from the exchange and the ability to alter orders almost instantly. This ability to implement dynamic order management strategies has an impact on how we think of the traditional trade-offs. Execution risk can be mitigated by converting from a limit order to a market order when the demand for immediacy rises. Adverse selection costs can be reduced by repricing or canceling orders when the underlying value of the asset changes. Continuously and actively monitoring the market in order to implement a dynamic order management strategy is difficult for many traders because of the costs involved. While 50 data feeds in themselves are relatively inexpensive, the direct costs and opportunity costs of spending time watching the trading screen may be substantial. In addition, time priority rules state that when two orders in the book are at the same price, the order that was submitted first takes the lead in the queue. This means that traders who change orders in the book lose position in the queue of limit orders. In this paper I characterize the dynamic order management strategies of limit order traders who have the ability to change or cancel orders, but who face costs from doing so. As in Glosten (1994), traders are not endowed with an exogenous motive to trade, but rather trade only when they find it profitable. As such, they fill the role of liquidity-supplying market makers on an exchange that does not have a formal market maker. The model that I develop allows a more thorough examination of the liquidity provision process on exchanges. In addition, I use a large and detailed data set from the Vancouver Stock Exchange to compare the predictions of the model to observed trader behavior. I develop a model that assess how traders use dynamic order management strategies to minimize the costs of limit order submissions. I model a finite-horizon, fixed order size market where traders are subject to execution risk, adverse selection costs, monitoring costs, and time priority. I find that the ability to implement a dynamic order management strategy has a large impact both on the initial submission decision of a trader and on the value to submitting limit orders. I calibrate the parameters of the model to match a particular stock on the Vancouver Stock Exchange. Then I find that allowing a trader to cancel or move his order increases the initial limit order value by 48 percent. Further, traders optimally manages their orders to reduce adverse selection costs and the risk of non-execution. The traders reduce risk of non-execution by converting limit orders to market orders when the common value moves away from the limit order price. They reduce adverse selection costs by moving or canceling limit orders if the common value moves toward the limit order price. A large literature has recently developed analyzing the choice problem for limit order market traders. Glosten (1994), Kumar and Seppi (1993), Chakarvarty and Holden (1995), Rock (1996), Seppi (1997), and Parlour and Seppi (2002) all develop static models of equilibrium in the limit order submission decision. Parlour (1998), Foucault (1999), and Foucault, Kadan, and Kandel (2001) look at the same problem in a dynamic framework, yet they view limit order submission as a one-shot game. In contrast, my model shows how limit order 51 submission involves a multi-period strategy. My application of dynamic programming to a limit order market is most closely related to the work of Cohen, Maier, Schwartz, and Withcomb (1981), Bertsimas and Lo (1998), and Harris (1998). The first paper considers only single-period limit orders and focuses on the limit-market decision rather than the dynamic management of orders. Bertsimas and Lo (1998) derive optimal dynamic trading strategies that minimize the cost of trading a large block of equity. In contrast to that paper, I ignore the volume issue, but focus on how an individual limit order can be managed to obtain maximum profitability. Harris (1998) solves a similar dynamic strategy problem. I use a less stylized pricing and order flow process and focus on the management of an existing order, rather than the submission strategy. Recent work on the empirical analysis of limit order submission strategies includes Biais, Hillion, and Spatt (1995), Hamao and Hasbrouck (1995), Al-Suhaibani and Kryzanowski (2000), and similar papers that study the provision of order flow on individual exchanges. Harris and Hasbrouck (1996) and Lo, MacKinlay and Zhang (2001) examine the performance of different order submission strategies. Sandas (2001) tests the zero-profit condition for liquidity suppliers in a limit order market. My work is also related to Hollifield, Miller, Sandas and Slive (2001) and Chapter 3 of this dissertation, where we model the timing and profitability of liquidity supply and demand on the Vancouver Stock Exchange. The current work explicitly incorporates the cancellation decision into the trader's optimization problem and is therefore able to make more structural predictions about the trading strategies and resulting market characteristics. My work contributes to the literature by presenting an analysis of the dynamic limit order submission problem that brings together the theoretical modelling of the optimal strategy with a comprehensive data set on limit orders. While Harris (1998) uses a similar theoretical model where traders may dynamically manage their orders, he focuses his analysis on the original submission strategy. I, on the other hand, focus on the choices of traders who have an order outstanding on the limit order book, and wish to optimally manage that order to mitigate execution risk and adverse selection costs. This work, therefore, provides guidance on what optimal dynamic strategies should look like. In addition, I have married the theoretical modelling with empirical data that shows the complete limit order book and complete order history of stocks on the Vancouver Stock Exchange. Using this data, I am 52 able to calibrate the theoretical model closely to a real order flow, and I am able to compare the predictions of the model with observations on a real stock. This is the only work to bring together a theoretical model and data to analyze the dynamic submission problem. The next section describes my theoretical model of dynamic order management. Section 4.3 solves the model numerically using dynamic programming. In section 4.4 I assess how accurately the predictions of the theoretical model match what I observer in the Vancouver Stock Exchange data. Section 4.5 concludes the paper and discusses directions for future research. 4.2 Model I consider afinite-horizonmodel of a fixed order size market with a single optimizing trader and an exogenous order flow. This is a highly simplified model which deliberately abstracts away the game-theoretic nature of the order placement choice. Yet this model is able to capture the execution risk and adverse selection costs suffered by the trader. It also includes both direct costs of managing orders in the market, and time-priority costs of moving orders. While the full specification of the model is quite complex with many state variables and a detailed transition matrix, the underlying idea is very simple. The model describes a limit order market with a set of trading and priority very similar to the rules on a modern limit order exchange such as the Vancouver Stock Exchange or Paris Bourse. Two different types of traders appear at the exchange and make decisions that affect the limit order book. Exogenous traders arrive who submit orders according to a pre-specified distribution. A single optimizing trader arrives who submits his order to maximize his expected profit. Together, the actions of these traders control the evolution of a limit order book subject to the rules of priority on the exchange. 4.2.1 A M a r k e t W i t h Exogenous O r d e r Flow I model a value trader who manages his portfolio to maximize his trading profit: the difference between the price he receives for the security and the underlying common value of the security. By appearing repeatedly at the market, he is able to update his information and make new trading decisions throughout the trading day. In this way, he can manage the risks of limit 53 order trading to obtain the best execution price. A risky security is traded in a limit order market which is open for T periods, indexed by t € {0... T — 1}. The terminal, liquidating value of the security is VT- Each period prior to the terminal date, the common value of the security is vt, which can be interpreted as the common knowledge expectation of the terminal value of the security conditional on the time t information set. This value evolves according to an equal probability binomial process with volatility parameter a. So the law of motion for the common value process is Vt+\ = < vt+cr v —a t with probability 1/2 (4.1) with probability 1/2. Each period t that the market is open, an exogenous trader arrives. I refer to this trader as exogenous because her order choice does not depend on market conditions or expectations of order payoffs. She submits an order of a pre-specified, fixed size which I normalize to one. She chooses a buy or sell order with equal probability, and her desired trading position is given by her private trading shock u\ for a buyer or uf for a seller, both of which are drawn from a distribution with cumulative distribution function G - The G distribution t t has the same shape in each period t € T , but the mean of the distribution is an increasing function of Vt, so the location of the distribution moves over time in the same direction as the common value. The final price of the order, designated as e\ and ef for the buyer or seller respectively, must fall on the pricing grid defined by P = {L4>, (L + 1)0,...., U<f>} where <fi is the minimum price variation or tick size, L(f> is the minimum allowable price, and U<p is the maximum allowable price. The translation from trading shock to price is given by e\ = min{/ € P : / > u\) (4.2) ef max{Z € P : I < u } (4.3) = s t If a buy order arrives, I set ef = oo while if a sell order arrives I set e\ = 0 for notational simplicity. The flow of exogenous orders drives the transition of a limit order book, which I will designate by B at each time t. The book consists of a record of the number of buy and t sell orders available for trading at each price on the pricing grid P. I use b[ to designate the number of buy orders available at price I € P in period t and a\ similarly for the sell side. 54 Then the book can be written as B = (4.4) {{a },{b }{orleP}. l t l t t For convenience of reference, I also define the best ask and bid price on the book as if a\ = 0VZ oo (3t = min{7 6 P : a[ > 0} otherwise 0 if 6' = OVZ max{7 € P : b[ > 0} (4.5) (4.6) otherwise. When a new order arrives, it impacts the state of the book according to a set of trading rules similar to those of an actual limit order market. A buy order e transacts against an b order on the book if e\ > a . I refer to this as a market buy order, and it has the effect of t removing one limit sell order from the book: af^ = af* — 1. Similarly, a sell order ef transacts if el < p and updates the book such that bf t = 6^ — 1. If the new order is not a market +1 order, it is added to the book itself, so b t+1 = b * + 1 for a buy order or a t t+1 =a l t + 1 for a sell order. In summary, the evolution of the book is given by the following if the exogenous trader is a buyer a[ — 1 if e > a and I = a b t H+i t t (4.7) otherwise b[ + l if e < a and I = e\ b t b t+i l b\ t (4.8) otherwise for each / € P. And similarly if the exogenous trader is a seller, H+l = < = a[ + 1 if ef > /3 and / = e\ t a\ otherwise b\ - 1 if ef < fa and I = (3 t < bl otherwise (4.9) (4.10) for each I € P. Under this trading process, I can derive a few simple properties of the limit order book, 55 Lemma 2 If the book starts empty—a = 0 Vi and b = OVJ— then there can never be both l l Q 0 buy orders and sell orders at the same point on the pricing grid: b|o| = 0 V ! 6 ? , V t 6 r . (4.11) In addition, the best bid price is always strictly less than the best ask price: Pt < atVt G T. (4.12) Starting from any book where the lemma holds (including the empty book) and applying equations (4.7)-(4.10) leads to another book where the lemma holds, which proves the result by induction. 4.2.2 The Optimizing Trader There is a single, risk-neutral, optimizing trader who arrives at the market at period t = 0 and can submit an order of size one. I assume that this trader wishes to purchase the security, and has a valuation for the security exactly equal to the common value v . The trader's t valuation will change over time as the common value evolves. The solution for a seller is exactly analogous but is not presented here. An individual liquidity demand component of the valuation could be added at the expense of added notational complexity. State Variables The optimizing trader may submit an order to the limit order book in period 0. Then in some subsequent periods he pays a waiting cost and returns to the market with the ability to manage his order. The trader's frequency of returning to the market is exogenously specified: he returns to the market once every N periods. This is tracked by the state variable n , 1 t which is equal to zero when the optimizing trader appears at the market and evolves according to (4.13) This could be just as easily modeled by giving the trader a certain probability of being able to manage his order in each period. The results would be very similar to the existing specification. A more interesting model would allow the optimizing trader to endogenize the choice of returning to the market. This extension should be feasible in the current framework and will be a focus of future research. 1 56 When arriving at the market, the optimizing trader can choose to leave his order exactly as it is, he can change his order to any other price on the pricing grid by paying the move cost A , m or he can remove his order from the book and leave the market by paying the cancellation cost A . c The optimizing trader's order will transact either if he changes the order to a price that exceeds the best ask order on the book, or if his order is first in the buy limit order queue and a market sell order arrives. The buy limit order queue is sorted first by price priority, then by time priority. So higher priced buy orders transact first, but if two buy orders are at the same price, then the first one entered at that price will also be the first one to trade. If his order transacts, he earns a profit equal to the difference between the common value of the security at the transaction time and the price he must pay to purchase the security. The position of the optimizing trader's order in the market in each period t is tracked by three state variables. First, 7 is an indicator variable that tracks if the optimizing trader t is still participating in the market (j = 1) or if he has traded or canceled his order and has t therefore left the market (-y = 0). Second, the price of the optimizing trader's order at the t beginning of the period (before the trader manages his order) is given by p . Third, the time t priority position of the optimizing trader's order is given by m . This variable indicates how t many orders on the book Bt at the price pt were placed before the exogenous trader's order. A value of mt = 0 indicates that the exogenous trader's order is first in time priority. Then the state for the problem consists of the optimizing trader's position in the market, the number of periods since the exogenous trader had an opportunity to manage his order, the exogenous limit order book, and the common value of the security: St = {'yt^Pttf^t^t, Bt,vt}. Action On observing the market in each period t G T, the optimizing trader observes the current state S . t He then chooses an action 6 G 6(St). The action can either be a cancellation, t which I denote C, or an order price 0t E P. If the trader does not have an opportunity to manage his order this period, then the only available action is to maintain the same price. Then the action space can be described as 6(5 ) = { {p } if n ^ 0 {Pl){C}} otherwise. t t t 57 (4.14) If Ot = C, then the cancellation cost is paid. If 6 ^ pt then the move cost is paid and the t optimizing trader's order goes to the back of the time priority queue at the new price. If the optimizing trader moves his order to a new price such that 0 > a , then he has converted t t his order to a market order that will transact immediately and earn the profit Vt — at — X m minus accumulated costs. Otherwise he remains on the book as a limit order. After the optimizing trader makes his submission decision, an exogenous trader arrives at the market and submits her order e£ or e\. If she submits a buy order or a limit sell order, then the book is updated and the market proceeds to the next period. If she submits a sell market order, then the optimizing trader's order will trade if it is first in the limit order queue. The optimizing trader's order is first in the queue if it has the highest bid price (0 > Pt) or if it is tied for the highest bid price (6 = Pt) and it is first in the time priority t t queue (Ot — Pt and mt = 0). If his order transacts, the optimizing trader earns a profit of Vt — Ot minus accumulated waiting and change costs. After the exogenous trader submits her order, the common value shock arrives and the market proceeds to the next period. The exogenous order book Bt+i and the common value vt+i are determined as discussed in the last section. The evolution of the indicator variable that tracks the presence of the optimizing trader in the market is given by ' 0 iiO = C t 0 if 0 > t 7m =< 0 a t if 6> > p and ef < 0 t t 0 ii 0 = 7t otherwise. t Pt (4.15) t and ef < 0 and 6 = pt and m = 0 t t t In other words, the trader remains in the market as long as he does not cancel his order, convert to a market order, or get executed against an incoming market sell order. The time priority position of the trader is given by if 0 ^ pt and either ef > 0 or 0 < Pt t mt+i &£' - 1 HOt^pt t t and e < 0 and 0 = Pt s t t t m — 1 if Qt = Pt and ef < Ot and 0 = t m t t (4.16) Pt otherwise, so the optimizing trader goes behind other orders in the book if he moves, but he gains in the queue when a market sell order removes an order ahead of him. Finally, the price of the 58 optimizing trader's order is given by Pt+i = Of (4.17) The evolution of the state space St over time is then characterized by equations (4.1), (4.7)-(4.10), (4.13), and (4.15)-(4.17). In the initial period 70 = 1, p = 0, m = 0, n = 0, 0 a = 0 VZ, b = 0 VZ, and VQ = v. l l 0 2 Q 0 0 As a special exception, the optimizing trader may set #o 7 ^ Po i the initial period without paying the move cost. The time-line of the problem n 3 looks for each period t _ {0... T — 1} is drawn in Figure 4.1. Optimizing trader chooses 8 t e Order \ or e\ arrives Value shock ±cr arrives t+1 Figure 4.1: Time-line for the dynamic limit order market problem It is worthwhile to note some characteristics of this market. The uncertainty in the market comes entirely from two sources: the common value shock that changes Vt each period and the exogenous order shock uf or u\ that determines the exogenous order e£ or e\. The exogenous shock, in turn, depends on vt so it moves with the common value. As the order flow then moves with the common value, buy limit orders will be more likely to transact after a common value decrease, and less likely to transact after a common value increase. This is what I call the adverse-selection cost of limit order submission. Even if the optimizing trader has the opportunity to manage his order each period after learning the new common value and before the exogenous order arrives, this will not entirely eliminate the adverse selection cost in this market. Because order management is costly, the optimizing trader may still suffer some adverse selection in order to avoid the cost of moving or canceling his order. This aligns nicely with the situation on real limit order markets, where an actively involved trader In some solutions in the next section I posit that the book is not empty in the initial period. This could be easily incorporated into the explicit model by assuming that a negative time horizon exists where the exogenous traders come to the market but the optimizing trader does not. I do not focus on the initial submission problem for the optimizing trader because I wish to highlight the dynamic order management problem. It would be a trivial change to add a submission cost in period 0. 2 3 59 can eliminate adverse selection by actively monitoring his order, but may choose not to if the costs are too high. Since the optimizing trader is endowed with no particular reason to trade—there is no liquidity component to his valuation—he can be thought of as a supplier of liquidity to the market. While a pure limit order market has no specialist or dealer designated to supply liquidity, limit order traders may act as market makers by supplying liquidity when it is profitable to do so. The optimizing trader is just such a market maker. The model can be used to reveal how an informal market maker optimally manages his liquidity supply. 4.3 Optimal Strategy The purpose of the model is to provide an optimal strategy to a trader arriving at a limit order market who desires to purchase securities. The solution can provide practical guidance to a trader wishing to obtain the most favorable terms of execution. Further, it allows me to assess how certain characteristics of the market, such as the frequency of changes and cancellations, react to exogenous changes in information and order flow. 4.3.1 One-Period Strategy To provide intuition on the basic trade-offs in the model, I begin by analyzing the strategy of a trader arriving at the market one period before closing. This trader must solve a relatively simple problem. Any order that remains on the book after the exogenous order arrives in period T will expire untraded. Any costs incurred prior to this period are sunk. Hence the trader is concerned only with maximizing the immediate return from his action this period. To choose the optimal action, the trader assesses the expected trading profit minus transaction costs for three different scenarios. His one-period profit at t = T — 1 assuming that he still has a limit order on the book is obtained from —X — X w if 6t — C c max TT (S ,6t) = I (vt-et)x(st,et) t t — I{n =0}^v> ' t I {Bitot} \ m ife <a t t (4-18) where I{Q ^ } is the indicator 7{ =o} is (vt - at)variable - Xw - for \m moving the order to a different if 0t > aprice, t T PT nt the indicator variable for having an opportunity to manage the order, and X(St, Qt) is the 60 one-period execution probability of a limit order at price 9tThe first row gives the payoff from cancellation. The last row gives the payoff from conversion to a market order at the best ask price at- The second row gives the payoff from leaving a limit order on the book. This payoff depends through X(S ,O ) t on the exogenous t order that will arrive later in this period. The function X(-) gives the probability that the period T — 1 exogenous order will transact against the optimizing trader's order, so this is the transaction probability and fill ratio for a limit order. The following describes some basic properties of the single period problem. Lemma 3 The one-period limit order execution probability X(St,9t) is increasing in the chosen price, 6 . Consequently, increasing #x-i leads to a higher execution probability and a t lower profit conditional on execution. The proof of this lemma relies on the fact that the one-period execution probability X(-) is zero for 9t < fit, one for 9t> at, and increasing in between due to the structure of the exogenous trading shocks that define ef or e\. Then since there is only one opportunity to trade in the last period, the execution probability as a whole must be equal to X(ST-I, 9T-I), and the profit conditional on execution is VT-\ — min(0r_i, a r - i ) , which is weakly decreasing in 9T-II can provide additional structure on the single-period problem by analyzing how the transaction costs X , X , and A influence the decision of the optimizing trader. First, it is w c m clear the waiting cost X has no influence on the problem in the last period, since the trader w must pay this cost regardless of his decision. So without loss of generality, I temporarily consider = 0 for this part of the discussion. Next, I consider the problem in the case where cancellation is prohibitively costly (A = oo) but moving an order is costless (A = 0). c m Then the profit that the optimizing trader will obtain in the last period can be written as TV*(S ) = max(vt - 9 )X(St,6 ). t T 4 (4.19) t In contrast, if both cancellation and moving are prohibitively costly (A = A = oo), then c m Although I do not explicitly deal with the market order case here, it can be considered a special case of the same formula where 9 = a and X{St,a ) — 1. The trader will never optimally choose 6 > a because he would decrease his profit without increasing his execution probability. 4 t t t t t 61 the profit available in the last period is simply the profit from not managing the order: n'(S ) = (v - )X(S ,pt). t t Pt (4.20) t In the standard.case with positive, finite values for the cancellation and moving cost (0 < X , A < oo), the optimizing trader will choose to move his order in the last period when c m n*(S )-7r'(S )>\ t t (4.21) m where I can note that IT* (St) — ir'(S ) > 0 regardless of the state. The optimizing trader will t choose to cancel his order when max( r*(5 ),7r (,S )) < - A . 7 t , , f (4.22) c With these results, I can reveal more about the relationship between costs and decisions in this model. If I assume that the pricing grid P is wide enough such that 30 € P : t X(St, 0 ) = 0 VSf, then if A < A , the optimizing trader will never choose to cancel his order. t m c The first assumption is not strong, since a limit order market traditionally allows traders to place orders at very low prices where they have practically no chance of execution, even in the absence of competition. In this case, order moves can be used as a perfect replacement for cancellations. This implies that in the last period, cancellations will only be used if there are no profitable trading opportunities and they offer a saving on transaction costs. In previous periods, cancellations have the added advantage that they avoid future waiting costs so the choice between cancellations and moves will depend both on their relative costs, and on the waiting time and expected time in the book. Expanding on this result, when can are there no profitable trading opportunities? This depends on the distribution of exogenous orders, Gt, and on the current state of the book Bt. If there exists a profitable market order (at < vt), then the optimizing trader will choose this in preference to cancellation if the market order profit minus the move cost exceeds the negative cancellation cost: vt — at — \ m > —A . In looking for profitable limit c orders, the optimizing trader must consider the competition from the buy side of the book. If there are high price buy limit orders on the book (3 > Vt), then there is no opportunity t for a limit order to make a profit. 62 4.3.2 Multi-period Solution I have now identified all the parts of a dynamic programming problem. The state space S t summarizes the information that the trader has available at each time t G T . Using this information, the trader chooses the control 9 G 6(5(). Section 4.2 outlines the law of motion t of the state variable conditional on the control and the two sources of randomness in the market. Finally, equation (4.18) gives the one-period reward function. The optimizing trader chooses his action at each time to maximize his expected profit over the horizon of the market. The problem can be formulated as a dynamic programming problem with a finite horizon, a discrete state space, and a finite action space. As such, there exists a deterministic Markovian policy which is the optimal solution to the dynamic order management problem. In solving the multi-period problem, the strategy is obtained 5 by recursively applying the one-period reward function, starting from the last period. The Bellman equation is given by u (St) = t max {7 7r (5t, 0 ) + E u (S i)} t t t t t+1 t+ (4.23) 0t60(St) subject to the boundary condition UT(ST) = 0 because all orders expire untraded when they reach period T. The problem lends itself quite naturally to discrete backward induction because both the action space and the state space are easily formulated as discrete variables. To make the problem manageable with finite computer resources, it is necessary to restrict the state space by allowing the book to have only a small number of possible prices, and limiting the number of exogenous orders that can be present at each price. In addition, one state variable can be eliminated by tracking both the book and the optimizing trader's order as values relative to the common value vt in each period. In this way, the orders available on the book change each period with the common value. This has the advantage of assuring that there are always orders available up to several ticks away from the common value on each side. The disadvantage is that some orders that are representable on the book in period t may not be representable in period t + 1. These orders will, by construction, be far away from the common value and unlikely to affect order outcome, so they are simply dropped from the See Puterman, 1994 or Bertsekas, 1987 5 63 analysis. 6 To calibrate the model parameters, I chose a stock from the Vancouver Stock Exchange data set at random from among the 30 most active companies on the exchange after eliminating those trading under $3. The chosen stock is Clearly Canadian Beverage (ticker symbol CLV) over the time span January 1, 1991 to November 30, 1992. This stock trades at a tick size of 1/8 during this time interval. Averaging over the entire time-period of the sample, and counting both limit orders and market orders, I found that a new order arrives every 2 minutes. I then set the common value volatility parameter a equal to the mean absolute value of the midquote return over a 2 minute period — a value of approximately one half tick or 0.0625. To choose the exogenous trader order placements, I examined a histogram of order placement relative to the common value for all buy orders on the exchange as presented in Figure E . l . This histogram is heavily right-skewed. This represents, at least in part, the fact that the observed order placements are truncated for market orders. When a market order is submitted, I see the price at which it transacted: the price of the lowest ask order on the book. To calibrate the distribution Gt, what I need is the highest price that the trader was willing to trade at, which is unobservable. To eliminate this truncation bias, I discard the right half of the distribution and reflect the left half around the midquote to create a symmetric distribution. This is obviously a coarse approximation, since we may expect value traders to shade their submission prices below the midquote in order to earn profits. Yet for a first approximation, it is sufficient. There is no easy way to calibrate the three different cost parameters in the problem or the time interval between optimizing trader management opportunities. Waiting costs, cancellation costs, and moving costs all include both direct and indirect costs and are all difficult to measure. I chose to set these values at a level that seems reasonable, but further experimentation and empirical work will be needed to determine the correct values. I set the time between management opportunities for the optimizing trader to N = 5. Since an exogenous trader arrives every two minutes, this implies the optimizing trader may manage his order every 10 minutes. The base case parameters with the exception of the distribution Gt are summarized in Table E . l . Table E.2 shows the solution of the dynamic order management problem under the The dynamic programming solution for this problem is implemented using C code on a Pentium IV 1500Mhz processor. It requires approximately 512MB of R A M and one hour of processing time to solve using a single set of parameter values with 13 price levels and up to one order per price level. 6 64 base case parameter values when the book is empty in the initial period. In my discussion, I focus on the order management problem, rather than the order submission problem; I assume that an order already exists on the book at a given price and the optimizing trader must manage it. The model is fully capable of answering questions about market versus limit decisions and initial limit order placement decisions, but can yield its most interesting results in contrast to existing models when focusing on the dynamic management problem. Hence the table shows the solution of the dynamic order management problem conditional on the optimizing order being present on the book at each possible price level. The solution gives the optimal immediate action and the value function under the optimal strategy at each price level. The prices are measured relative to the common value of the security. For example, the top value is obtained by a limit order 0.375 below the common value. The optimal immediate action for an order at this level is obviously to leave it unchanged at —0.375, and following the optimal strategy to the end of the market will yield a value of 0.08928 meaning a profit of approximately 9 cents. A trader should be willing to submit a new limit order at this price level as long as his submission costs are less that 9 cents. Following this optimal strategy, the optimizing trader will execute his order with 81.88% probability. With 0.1997 probability, the order will execute after the trader converts it to a market order, so with 0.8188 — 0.1997 = 0.6191 probability, the order will execute when a sell market order matches against it. If the order does not execute, it must either expire at the end of the market, or be canceled by the trader. The latter happens with probability 0.1401, so the former must happen with probability 1 - 0.8188 - 0.1401 = 0.0411. Overall, the order will remain in the book for an expected value of just over 10 periods (20 minutes). Finally, the optimal strategy involves an expected number of price changes (not counting conversions to market orders) of 0.0396. Although the optimal limit order position is at —0.375, a trader with a limit order on the book at any price from —0.750 to —0.125 will choose to leave the order untouched at its current position. The extra value obtained by having an order at the optimal position cannot outweigh the cost of moving the order. For prices below —0.375, the value is increasing with the price as the increased execution probability outweighs the decreased profit conditional on execution. For prices above —0.375, the value decreases with the price since the increased execution probability cannot compensate for decreased profit conditional on execution. An 65 optimizing trader with an order on the book at a price greater than —0.125 finds that it is optimal to pay the moving cost in order to change to a more favorable price. Given that the cost of moving does not depend on either the original price or the new price, any trader who chooses to move will move to the optimal limit order price: —0.375. I now contrast this solution with the optimal strategy that holds when the optimizing trader is prevented from managing his order. Table E.3 shows a solution in a case identical to the base case, except with the cancellation cost A and move cost \ c m set to very high values. Obviously this trader must always choose to maintain his current limit order price, and will therefore have a market execution probability, cancellation probability, and expected number of moves equal to zero. The trader who is prevented from managing optimally chooses a less aggressive limit order with a price of —0.250. He does this in order to gain better execution probability in the absence of the ability to use market orders to guarantee execution. Yet he will also face increased adverse selection risk, both because his order is placed closer to the common value, and because he cannot use cancellations or moves to prevent execution after the common value moves down. The execution risk outweighs the adverse selection costs in this case, causing the optimizing trader to place a less aggressive limit order. The value he obtains from this limit order is .060369. Contrasting this with the maximum value in Table E.2, I find that the ability to manage increases the value of submitting a limit order by 48%. This comparison can be summarized as follows. Numerical Result 1 The ability to implement a dynamic limit order management strategy constitutes a substantial part of the value of limit order submission. In addition, the ability to manage an order in the future can change the optimal initial submission strategy. An optimizing trader who leaves his order on the book is subject to changes in the common value that occur during each time period. Hence an order submitted at a distance of —0.375 from the common value will most likely be at a different distance from the common value the next time that the trader has the opportunity to manage his order. In particular, as the common value moves up, the optimizing trader's order will move down the rows of Table E.2 and vice versa. I can therefore analyze the effect of common value changes by analyzing the changes from moving up or down the rows of the table. After an increase in the common value, the execution probability of a buy limit order 66 will decrease. Since the exogenous order distribution tracks the common value, there will be less chance of exogenous sell orders at prices below the optimizing trader's price. Further there will be a greater chance of exogenous buy orders being submitted above the optimizing trader's price, thereby taking priority in the limit order queue. These factors cause a decrease in the limit order execution probability. The optimizing trader can partially offset this effect by increasing his use of market orders, but the net effect is still a decrease in overall probability of execution. Cancellation probability, in contrast, can either increase or decrease as the common value increases. Cancellations are used either to prevent execution at unfavorable price, or to leave an unprofitable market and avoid future moving and waiting costs. The first factor will cause cancellations to decrease as the common value increases, while the second factor will cause cancellations to increase. The net effect is mostly an increase in cancellation probability as the common value increases. The overall time in the book incorporates the decreased execution probability and the increased cancellation probability. The decreased execution probability predominates and the time in book increases with the common value. These results are summarized as follows. Numerical Result 2 Optimizing traders who do not move or cancel their order after a common value increase face a decreased limit order execution probability and an increased market order probability. In net, the probability of execution decreases and the expected time in the book increases. The effect of a common value decrease is the reverse. Of course, a trader in a limit order market is unlikely to face an empty limit order book. Tables E.4 and E.5 show the optimal strategy with a number of different limit order books. Each row shows the value and strategy for a particular limit order when the book has one limit order at every price except for a specific spread around the common value. In Table E.4, I show the limit order with the highest value for each different book. When there is a small spread on the book, the optimizing trader chooses to submit a less aggressive limit order in order to get ahead of other orders in the limit order queue. Table E.5 shows how the strategy for a limit order at a particular price (—0.375) would change depending on the spread in the book. Here, if the spread is very small, there are no profitable trading opportunities in the market and the trader cancels his order. With larger spreads, the trader chooses to leave his order unchanged. As the spread increases, the value of the position first increases 67 and then decreases. The increase happens when the number of limit buy orders ahead of the optimizing trader in the queue decreases. When the spread is smaller, the optimizing trader more often uses market orders to gain immediate transactions and makes more use of cancellations to exit unprofitable markets. Once the optimizing trader becomes first in the queue, increasing the spread decreases the value since the available ask orders become more expensive. The same effect causes cancellation probability to first decrease and then increase. The following result summarizes the effect of varying the spread on the book. Numerical Result 3 An optimizing trader will submit less aggressive limit orders when there is competition on the book. Table E.6 returns to the case of an empty book, but shows the strategy that would be used with only 10 periods remaining in the market. Here I see that the trader will cancel any limit order priced above the common value. In the initial period, these limit orders where simply moved to another price. As the end of the market nears, the value available to the trader at the optimal limit order price decreases. Hence it is not worthwhile to pay the extra cost of moving the order, and the trader simply cancels. The short-horizon trader also chooses to submit a less aggressive limit order. This is necessary because the risk of non-execution is much higher as the trader nears the end of the market. He therefore places an order that is closer to executing, but he still winds up with a significantly lower execution probability than the initial-period trader. Numerical Result 4 Limit order traders subject to shorter trading horizons will shade their order price upwards to decrease the risk of non-execution. The ratio of moves to cancellations will also decrease as the expected limit order profit decreases with the shorter horizon. Table E.7 shows the solution of the order management problem with an empty book in the first period when the volatility of the common value is double the value from the base case. Here the common value changes by one tick each period rather than 1/2 tick. This is a purely comparative static analysis, since the exogenous order flow is fixed at the original values, which would not be the case in a real market when the volatility doubles. The limit order with the highest value in this case is far more aggressive than in the base case, with a price of —0.625. There is also a significantly higher cancellation probability (0.2687 compared 68 to 0.2208 at the same price or 0.1401 at the optimal price). Both of these changes are the result of the optimizing trader trying to mitigate the increased adverse selection costs brought on by the higher common value volatility. The more aggressive limit order and higher use of cancellations results in a lower execution probability and a higher overall time in the book. Numerical Result 5 If the volatility of the common value increases—with all else held constant—the trader will submit more aggressive limit orders and make increased use of cancellations to reduce adverse selection costs. 4.4 Empirical Analysis Table E.8 shows some of the basic characteristics of the data used in the remainder of the analysis. A total of ten stocks are analyzed over a six month period. The stocks were selected randomly from a pool of thirty of the most active stocks on the Vancouver Stock Exchange during 1991. The first column gives the stock ticker symbol, while the second column shows the number of trader-initiated events occurring for each stock over the time period. Trader initiated changes include market orders, limit orders, order changes, and cancels, but exclude changes to the book not directly initiated by a trader such as order expirations. Since the sample was selected to be relatively active, there is only a moderate variation in the overall amount of activity, with 35,826 events on average and a standard deviation of 12,974 events. Overall in the sample, approximately 50 percent of the events are new limit order submissions, where a limit order is defined as an order submitted to the market that does not trade immediately. Approximately 21 percent of events are market orders, defined as an order that trades immediately, in its entirety, and therefore never enters the limit order book. A further 7 percent of orders are partial-fill orders, that see some of their volume immediately executed, while the remainder is added to the limit order book. In total, approximately 12 percent of events consist of changes to existing orders. That is further broken down based on what happens to the order immediately after it is changed. For 9 percent of events, the change is to another limit order, which remains on the book. For 2 percent of events, the change is to a market order, which executes immediately and completely, and for 1 percent of events the change is to a partial-fill order. Comparing new orders to changed orders, a slightly higher proportion of new orders trade immediately, while 69 changed orders tend to remain on the book. Finally, an average of 10 percent of events on the exchange consist of trader-initiated order cancellations. Across stocks, cancellations range from a low of 6.5 percent of events to a high of 16.3 percent of events. Table E.9 looks at the entire sample of ten stocks, and breaks down the events into four types. Here, limit orders are orders where no volume trades on submission, and market orders are orders where at least some volume does trade on submission. In other words, the partial-fill category has been included with market orders. The top panel shows only events related to buy orders, while the bottom panel shows sell-order events. The rows break the events down by the bid-ask spread at the time of the event. The bid-ask spread is calculated simply as the best ask price on the board-lot book minus the best bid price on the board lot book. The spread is then reported as a percentage of the midquote, in order to control for the magnitude of prices. The first row lists events at times with negative or zero bid-ask spreads. By the rules of limit-order exchange, a negative bid-ask spread should never be possible, since the existence of a buy order at a price greater than the best ask price will trigger a sale. Looking at real-time data, however, there can certainly be short periods of time where non-positive spreads can exist. This happens most often when a large amount of order activity is produced in a short period of time, and the market computers must take a few seconds to catch-up and execute pending transactions. Alternatively, a non-positive spread can occasionally be the result of a weakness in the book-building algorithm used to analyze the data. For example, a market order that walks up the book executing against several limit orders may be wrongly added to the book for a brief period of time — thereby creating a negative spread — if it takes a particularly long time to execute or if other orders execute between its trades. In my sample, the event types when there is a non-positive spread are more weighted towards limit orders in comparison to the case of a slightly positive spread. This is, perhaps, an artifact of the data problems just mentioned. The slowness of the exchange in carrying out transactions that results in a negative spread, also makes orders that intended to trade immediately appear as limit orders, since they are briefly entered in the book. For both buy and sell orders, the table shows that the proportion of limit order declines as the spread increases. This is consistent with previously reported results and with theory 70 that predicts that traders enter the market to supply liquidity at times when it is profitable to do so. As the spread increase, the opportunity for profitable limit order submissions increases because the competition in supplying liquidity has decreased. In keeping with this result, the frequency of the market order event decreases as the spread increases. Demanding liquidity by submitting market orders becomes comparatively expensive as the spread increases. The relative frequency of order management declines substantially as the spread increases. For example, while almost 14 percent of all buy events are cancellations when the spread is less than 3 percent of the midquote, this declines to slightly more than 5 percent of events for spreads over 18 percent of the midquote. This holds for both buy and sell orders, and it also holds for both order changes and cancellations. Again, this results from the fact that, as the spread increases, it becomes more profitable to supply liquidity to the book. This discourages traders from cancelling or moving existing limit orders. Table E.10 shows the proportion of different event types conditional on the volatility of the midquote over the previous hour. Here I do not see find substantial change in the relative proportion of limit orders, but I do find a slight decrease in the proportion of market orders. In terms of order management, the relative frequency of cancellations increases substantially as the midquote volatility increases. This is consistent with traders using cancellations to avoid adverse selection risk from common value changes. Order changes, on the other hand, increase in frequency when comparing the zero volatility case to the very small volatility case, but then decrease in frequency as the volatility rises further. Table E . l l shows how the proportion of different order types vary depending on the amount of transaction volume in the previous hour. Again, these results show an increase in the proportion of cancellations, and a decrease in the proportion of changes as activity increases. Next, instead of looking at submission events, I examine all submitted orders and determines their outcomes. An order submitted to the market can have one of four outcomes: it can be completely executed against other orders, it can be changed by assigning a new price or volume, it can be canceled by the trader and removed from the book, or it can expire either at the end of the trading day (for day orders) or on a specific date (for good 'til orders). Table E.12 looks at the outcome of orders conditional on their distance from execution at the time of submission, where distance is the difference between the order price and the best price on the other side of the book, divided by the midquote. It is not at all 71 surprising to see that the proportion of orders that are completely filled declines and the proportion of expired orders increases as the order is submitted further away from the quote. Not all orders submitted at or past the quote are completely filled, because there may not be enough volume at the quote to offset the entire order. The fraction of orders that end in changes increases and then decreases as the order is submitted further from the quote. The proportion of orders canceled increases and then decreases for buy orders, but only increases for sell orders as the order is submitted further from the quote. Table E.13 shows the outcome of orders conditional on the relative spread (spread divided by midquote) at the time of submission. Ignoring the case where the spread is nonpositive, the fill probability decreases, while the change and cancellation probability increase with a greater spread. Table E.14 shows the outcome of orders conditional on the volume traded over the sixty minutes prior to submission. Here, the fill and cancellation probability increase, while the change probability decreases with the level of recent trading activity. 4.5 Conclusion I have modeled the decision problem of a trader in a limit order market who has the opportunity to manage his order over the trading day using cancellations and order changes. This trader manages his order to optimally supply liquidity to the market; thus he plays the role of a market maker in a market where no formal dealership structure exists. I calibrate the model to a particular stock on the Vancouver Stock Exchange. I find that the ability to dynamically manage a limit order dramatically alters both the initial submission strategy and the value of submitting limit orders to the market. Further, I can characterize the optimal strategy after changes in the common value, changes in the trading horizon, and changes in the volatility of the underlying asset value. 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Wang, 1998, "Market architecture: Limit-order books versus dealership markets" Working paper, Duke University. 75 Appendix A Market order frequency Define Mt as a random variable that equals one in the event a market order is submitted on the buy side of the book at time t and 0 otherwise. Then the market order frequency can be defined as (A.l) Unfortunately, the probability of a market order in any particular period depends on the history of past events. If a sell market order was submitted last period, then the sell side of the book is empty this period and a limit order will be submitted for sure. To reduce notation, the event M = 1 will be referred to as M while the event M = 0 will be referred to as Lt- Then this idea can be expressed as t Pr(M ) 4 t t = Pr(M |L _ )Pr(L _ ) + Pr(M |L _ )Pr(M _ ) = Pr(M |L _i)Pr(L _i) t t i 1 t t t 1 t t 1 f 1 (A.2) since Pr(M |L _i) = 0. Unfortunately, neither of these numbers can be calculated from the optimization process. Therefore, a different technique is required. For simplicity of notation, r is henceforth used as an index of trader type at time t and the event r< = i will be referred to as T\. Here, trader type refers to the liquidity shock, and knowledge of how the common asset value will evolve over the next period. The total number of types is N . Then the conditional probability of seeing a market order in period t t t T 76 t, given that the type is r\ can be written Pv(M \rl) = t Pr(M ,r») t Pr(M ,r , M _ i ) + Pr(M , T ^ , L - i ) = t t i t Pr(r^) t t Pr(r<) Pr(M ,r^,L -i) t t (A.3) (A.4) Pr(r*) (A.5) = ^E{^|r^TiL ,2^i)Pr(r«)Pr(r«L 1 1 > L*-i)} (A.6) fc=i {Pr(M |T*,r/L ,L _ )Pr(T 'L L _ )} T = 1 ( 1 t 1) t 1 ^{PrCMtlr^T^.L^OPrCr^JPr^lr/L!)} (A.7) (A.8) fc=i (A.9) Equation (A.4) uses the fact that the probability of seeing two market orders in a row is zero, regardless of type. Equation (A.6) uses the fact that the draw of type is independent of past events. Equation (A.9) defines a recursion in terms of values which can be obtained in the optimization process: the conditional limit order fill rate, and the unconditional probability of facing each trader type. In fact, in this case, the conditional limit order fill rate is either one or zero, since knowing both the period t and t — 1 types gives us the bid and cutoff values with certainty. The unconditional limit order probability at time t can be easily obtained as (A.10) From here, the unconditional limit order frequency can be obtained in two ways. Under the assumption that a limit in (A.l) exists, the stationarity of the problem can be used to set Pr(M |r ) = P r ( M _ i | T _ ) and the system of equations implied by (A.9) can be solved. Alternatively, the recursion can be used with the initial condition that Pr(Mi|rf) = 0 for all i given that the book is empty at the opening of the market. For now, the latter technique is used in the text, although the former should be feasible after a few dozen pages of algebra. t t l t 1 T 1 77 Appendix B Proofs for Liquidity Supply and Demand in Limit Order Markets Proof of Lemma 1 If it is optimal for a trader with valuation v to submit a buy order, then it is also optimal for traders with private values v' > v to submit buy orders. Let s be an arbitrary sell order, and suppose that d^(v) = 1. Then, <J K W " PM) + # V + V(l - <0 - c> <f (v - P M ) + * ! ? > Vsf >rf $f + V(l - (Ps,t -v)- s + V(l ~ (Ps,t - v') ~ C) " c r f) ~ c s tit + V(l ~ Vsf) ~ c (B.l) The first line follows because v' > v; the second line follows because it is optimal for a trader with value v to submit a buy order at b; the third line follows because v' > v. Let pb',t be the optimal buy submission for the trader with valuation v'. By optimality, - P >,t) + + V(l - b - c, <,v - w,t)+#y+vu - <r) - > (B.2) c <v - PM)+&+m - <?) - c (-) B 3 Subtracting equation (B.3) from equation (B.2) and rearranging (v-v>)(^y-^y)>o. (B.4) The result follows from equations (B.4) and (3.16). Symmetric arguments hold on the sell side. • Proof of Proposition 1 The threshold characterization follows from the monotonicity in Lemma 1. • 78 B.l The conditional log-likelihood function Let U denote the time of the i order submission and I the total number of order submissions. Conditioning on the common value, order size, xt and zt the conditional log-likelihood function is th { £ j4 { In (G ( O U j - i f c ) A ) t i tj i=l fi s s \ t t KS=1 J In ( [ G ti (^(Marginal) - y ) - G (^(0,1) - y )] X ) ti + d Jln(l-G b Bt 0 + ( £ < u ) - £ l n u u ti (6^(0,1)-y )\ ) ti ti ti ([Gu (< (0.1). -Vu) ~ G (^(Marginal) - y )] X ) y ti ti [G (^ (Marginal) - y ) - G (9 UJ/ t t t s el1 t ti (Marginal) - y )] X dt^. (B.5) t t The first line is contribution from the instantaneous probability of a sell market order at time U; the second line is the contribution from the instantaneous probability of a sell limit order; the third line is the contribution from the instantaneous probability of a buy market order; fourth line is the contribution from the instantaneous probability of a buy limit order; and the final line is the integrated hazard rate. In our estimation, we assume that the common value yt only changes when an order is submitted at tj. B.2 Execution probabilities in the Weibull competing risks model Suppose a limit order is submitted at time U. Let t be the hypothetical execution time for the order and t the hypothetical cancellation time for an order. Assume that the times are independent Weibull random variables, with cdf's F and F : e c e c F (r) = l-exp(-exp( ;a F (r) = l-exp(-exp( ^ e c 7 7 ; t j t < )(T-t ) i Q e )(T-t ) i ). a c ( -) B ), 6 (B.7) The execution probability is defined as the probability that the order executes between U and U + T, P r ( f e < f c , f e < i i + r ) = r + (1 - Fc(t - U)) dFe(t - ti) Jti = I* exp(-exp(7;x )(t-ti) )ti Jti Qc exp( ;x )a (t - U) <- exp (- exp( ^ )(t - U)^- ) dt. (B.8) 7 ti a e 1 7 t< 1 We compute equation (B.8) numerically with T equal to two trading days (48,600 seconds). 79 B.3 The picking off risk Assuming that the order either fully executes or fully cancels, so that Qt+T G {0,1}, the picking off risk is (buy = E t / (yt+r - yt)dQt+T jbuy _ JT=Q = E = E j t t [ (yt+ .JT=0 Vt)dQt+T (yt+r - yt)dQt+r dly (yt+r - yt)dQt+ d J = b b l , Q t + T = 1,Qt+r = l] Pr t {Qt+T = 11^ = l) .JT=0 = E t / T d ^ = l,Q t + r = l (B.9) JT=0 The second line follows from the law of iterated expectations, the third line follows because dQt+r > 0 for some t + r if and only if Qt+r = 1 and the final line follows from the definition of the execution probability. 80 Appendix C Tables and Figures for Asymmetric Information in Limit Order Markets 81 T — 1 Type E(e \9 -i) y r - 2 +a +L 0 +L +<7 +L —a +L 0 +L —a +L +a -L 0 -L +a -L —a -L 0 -L —a -L T T er-i +a +o—a +<r —a —a +a +a —a +o—a —a Prob 1/16 1/8 1/16 1/16 1/8 1/16 1/16 1/8 1/16 1/16 1/8 1/16 CumProb 1 15/16 13/16 12/16 11/16 9/16 8/16 7/16 5/16 4/16 3/16 1/16 Cutoff 10.4 10.3 10.2 10.2 10.1 10 10 9.9 9.8 9.8 9.7 9.6 AE{v ) T 0 -.0133 -.0308 -.0333 -.0364 -.0222 0 -.0286 -.08 -.1 -.133 -.2 Profit .0125 .025 .05 .0625 .075 .1 .1125 .1 .075 .0625 .05 .025 Table C l : Calculation of optimal bid for the uninformed trader with high liquidity shock at time T — 2 in a finite horizon game. Parameter values are a = .1, L = .2, <> / = .5, VT-2 = 10. Prob is the probability in period T — 2 of facing the trader of the specified type in period T — 1. CumProb is the Cumulative probability of seeing this trader and all traders with lower cutoffs. CumProb can also be interpreted as the execution probability of a bid order placed at that cutoff price in T — 2. Cutoff is the price at or above which the T — 1 trader will accept any bid on the book for immediate execution. AE(VT) is the change in the T — 2 traders expectation of the final payoff conditional on executing a limit bid order equal to that cutoff price. Profit from a limit order placed at that cutoff price is given in the last column. When two or more T — 1 traders have the same Cutoff, the T — 2 trader must trade against both of them or neither of them. For this reason, it is always the higher of the two that is relevant. For example, when the submitted price is 10, both the (—a, +L, —a) trader and the —L, +a) trader will take the order, so the relevant profit is .1, not .1125. Info-O l!qukfflyrf.2 ligmtaO.I U 0 . 2 rho-0.90 lnk»0.1 Hquk*ty.O2 l i g m a ^ . l 1^0.2 rtio.0.99 (a) Uninformed (b) Informed High Realization Figure C l : Bidding Strategies for High Liquidity Shock Types 82 (a) Info -0.1 -0.1 -0.1 -0.1 +0.1 +0.1 +0.1 +0.1 +0.0 +0.0 +0.0 +0.0 Information Liquidity Liquid Val -0.2 -0.1 -0.2 +0.1 +0.2 -0.1 +0.2 +0.1 -0.2 -0.1 -0.2 +0.1 +0.2 -0.1 +0.2 +0.1 -0.2 -0.1 -0.2 +0.1 +0.2 -0.1 +0.2 +0.1 Info -0.1 -0.1 -0.1 -0.1 +0.1 +0.1 +0.1 +0.1 +0.0 +0.0 +0.0 +0.0 = .65 -0.1 -0.2 +0.2 - 0.16 - - +0.1 -0.2 - 0.16 0.17 (b) Information Liquidity Liquid Val -0.2 -0.1 -0.2 +0.1 +0.2 -0.1 +0.2 +0.1 -0.2 -0.1 -0.2 +0.1 +0.2 -0.1 +0.2 +0.1 -0.2 -0.1 -0.2 +0.1 +0.2 -0.1 +0.2 +0.1 cj) 4> = -0.1 -0.2 +0.2 - 0.17 - 0.17 - - 0.15 +0.0 -0.2 +0.2 - - 0.16 - 0.16 - 0.08 0.08 - 0.08 - - - 0.17 - +0.2 +0.0 -0.2 - - 0.09 0.09 - .70 +0.1 -0.2 - - 0.17 - 0.17 - 0.15 - - Table C.2: Cross-Type Limit Fill Probabilities 83 +0.2 - +0.2 0.09 0.09 - 0.09 - 0.07 - lnfo-0 BqukJty-0,2 dgma-0.1 L-0.2 rtio-O.Bfl bito-0.1 flquklty-0.2 iigmn-0.1 LJ),2 rho^},99 (a) Uninformed (b) Informed High Realization Figure C.2: Limit Fill and Market Order Probabilities for High Liquidity Shock Types Figure C.3: Limit Fill and Market Order Probabilities for the Market as a Whole 84 Appendix D Tables and Figures for Liquidity Supply and Demand in Limit Order Markets Table D . l : Summary Statistics Barkhor Res. Ticker BHO Number of order submissions Percent of submissions that are: Sell limit orders Sell market orders Buy limit orders Buy market orders Mean percentage spread Standard deviation of percentage spread Mean depth Standard deviation of depth Eurus Res. ERR War Eagle Mining WEM 55,444 56,599 47,578 31.7 21.6 28.5 18.2 31.5 23.7 27.9 16.9 31.3 19.9 32.0 16.8 3.9 4.5 21.0 18.8 2.8 4.3 10.0 8.7 3.4 4.9 10.0 9.0 The sample period is May 1, 1990, to November 30, 1993. The depth is equal to the average of the bid and ask depth in the limit order book within 2.5% of the mid-quote, measured in units of 1,000's of shares. 85 Table D.2: Order Size Sell orders Limit Market Mean T-test for equal means P-value Mean T-test for equal means P-value Mean T-test for equal means P-value Buy orders Limit Market BHO 5.689 5.951 6.613 6.268 (0.036) (0.053) (0.058) (0.068) -4.118 3.841 (0.000) (0.000) ERR 2.627 2.830 3.413 3.414 (0.023) (0.026) (0.032) (0.047) -5.875 -0.029 (0.000) (0.977) WEM 2.848 3.132 3.711 3.197 (0.020) (0.035) (0.029) (0.038) -7.034 10.849 (0.000) (0.000) Buy and sell orders Limit Market 6.127 (0.034) 6.096 (0.042) 0.560 (0.575) 2.996 3.073 (0.019) (0.025) -2.427 (0.015) 3.284 (0.018) 3.162 (0.026) 3.918 (0.000) The mean order size in 1,000's of shares is reported for sell limit and market orders, buy limit and market orders, and for buy and sell limit and market orders. Standard errors are reported in parentheses. A test statistic with p-values in parentheses for a t-test of equal means is reported for each pair of means. 86 Table D.3: Conditioning Variables Name Spread Close ask depth Far ask depth Close bid depth Far bid depth Recent trades Lagged duration Mid-quote volatility TSE market index TSE mining index Overnight interest rate Canadian dollar Lagged return Distance to mid-quote First hour Second hour Third hour Fourth hour Fifth hour Sixth hour Description Book variables Bid-ask spread Log of ask depth at best ask quote Log of ask depth within 9 cents of best ask quote Log of bid depth at best bid quote Log of bid depth within 9 cents of best bid quote Activity variables Number of traders in the last ten minutes Sum of last ten durations of order book changes Volatility of the mid-quote over the last ten minutes Market-wide variables Absolute value of the lagged close to close return on T S E market index Absolute value of the lagged close to close return on T S E mining index Absolute value of the lagged change in the Canadian interest rate Absolute value of the lagged change in the Canadian/US exchange rate Value proxies Absolute value of the lagged open to open mid-quote return Percentage deviation between mid-quote and moving-average mid-quote Time dummies dummy variable for 6:30-7:30 dummy variable for 7:30-8:30 dummy variable for 8:30-9:30 dummy variable for 9:30-10:30 dummy variable for 10:30-11:30 dummy variable for 11:30-12:30 87 Table D.4: Weibull Hazard Rate Model for Order Submissions Power: a Constant Spread Close ask depth Far ask depth Close bid depth Far bid depth X§: All parameters = 0 P-value Recent trades Duration (xlOOO) Mid-quote volatility X§: All parameters = 0 P-value BHO 0.586 (0.003) -3.486 (0.157) -2.287 (0.418) -0.029 (0.005) 0.119 (0.009) -0.011 (0.005) -0.034 (0.008) 266.690 (0.000) 0.040 (0.001) -0.124 (0.002) 0.526 (0.086) 8856.980 (0.000) 88 ERR 0.567 (0.002) -1.667 (0.130) -0.359 (0.092) 0.015 (0.006) 0.018 (0.007) 0.034 (0.005) 0.023 (0.006) 155.180 (0.000) 0.052 (0.001) -0.109 (0.002) -0.550 (0.074) 10326.160 (0.000) WEM 0.561 (0.002) -2.031 (0.216) -1.196 (0.222) 0.001 (0.006) -0.002 (0.008) 0.011 (0.006) -0.026 (0.007) 37.830 (0.000) 0.068 (0.001) -0.102 (0.001) -0.152 (0.127) 12647.250 (0.000) Table D.4: Weibull Hazard Rate Model for Order Submissions (continued) BHO T S E market index T S E m i n i n g index Overnight interest rate C a n a d i a n dollar X 4 : A l l parameters = 0 P-value Lagged return Distance to mid-quote x| A l l parameters = 0 P-value F i r s t hour : Second hour T h i r d hour F o u r t h hour F i f t h hour S i x t h hour X§: A l l parameters = 0 P-value X21'. Constant hazard rate P-value N -0.002 (0.005) -0.011 (0.005) 0.001 (0.005) 0.000 (0.005) 6.230 (0.183) 0.088 (0.005) 6.141 (0.428) 547.000 (0.000) -0.249 (0.017) -0.371 (0.017) -0.202 (0.016) -0.169 (0.016) -0.137 (0.016) -0.113 (0.020) 521.680 (0.000) 17399.970 (0.000) 56316 ERR WEM 0.032 (0.004) -0.005 (0.004) 0.031 (0.005) -0.026 (0.004) 135.340 (0.000) 0.025 (0.005) 0.834 (0.099) 98.880 (0.000) -0.084 (0.016) -0.154 (0.016) -0.178 (0.015) -0.080 (0.016) -0.070 (0.016) -0.041 (0.016) 189.480 (0.000) 12955.990 (0.000) 57441 0.003 (0.005) -0.004 (0.004) 0.002 (0.005) -0.033 (0.005) 52.080 (0.000) 0.043 (0.005) 2.816 (0.367) 145.310 (0.000) -0.295 (0.017) -0.331 (0.016) -0.213 (0.016) -0.150 (0.016) -0.141 (0.016) -0.090 (0.016) 561.290 (0.000) 14301.930 (0.000) 48446 The table reports parameters estimates with asymptotic standard errors in parentheses for a Weibull model for the hazard for order submissions, and x -tests of the null hypothesis of all parameters jointly equal to zero for each subset of conditioning variables and for all conditioning variables jointly with p-values in parentheses. The time until the first order each day is measured from the opening. The time elapsed from the last order each day until the close is included and treated as a censored observation. 2 89 Table D.5: Ordered Probit Models for Order Submissions Buy Spread Close ask depth Far ask d e p t h Close b i d depth Far b i d depth O r d e r size X§: A l l parameters = 0 P-value Recent trades D u r a t i o n (xlOOO) Mid-quote volatility X§: A l l parameters = 0 P-value T S E market index T S E m i n i n g index Overnight interest rate C a n a d i a n dollar x\- A l l parameters = 0 P-value Lagged return ( x l O ) Distance to mid-quote X_'- A l l parameters = 0 P-value F i r s t hour Second hour T h i r d hour F o u r t h hour F i f t h hour S i x t h hour X§: A l l parameters = 0 P-value X 2 o A l l parameters = 0 P-value N u m b e r of observations : BHO 40.600 (2.126) 0.061 (0.009) - -0.145 (0.010) 0.032 (0.018) 0.168 (0.010) 888.860 (0.000) -0.007 (0.001) 0.003 (0.002) -1.405 (0.139) 224.660 (0.000) -0.001 (0.083) 0.077 (0.078) 0.004 (0.079) 0.008 (0.078) 1.050 (0.902) 0.093 (0.081) -16.597 (0.858) 377.170 (0.000) 0.212 (0.030) 0.219 (0.029) 0.153 (0.029) 0.081 (0.030) 0.102 (0.029) 0.083 (0.030) 82.940 (0.000) 1504.130 (0.000) 25,904 ERR W E M Buy Sell Sell Buy Sell 34.276 (1.359) -0.142 (0.009) 0.183 (0.013) 0.049 (0.008) 15.698 (1.356) -0.031 (0.016) 8.153 (0.713) -0.106 (0.011) 0.020 (0.015) 0.006 (0.009) - -0.206 (0.012) 0.078 (0.017) 0.064 0.145 (0.010) (0.011) 1039.590 559.910 (0.000) (0.000) -0.005 -0.015 (0.001) (0.002) -0.010 0.014 (0.002) (0.002) 1.686 ' -0.447 (0.136) (0.379) 213.250 214.600 (0.000) (0.000) -0.315 -0.473 (0.078) (0.081) -0.032 0.136 (0.071) (0.085) 0.061 0.089 (0.073) (0.083) 0.098 0.121 (0.072) (0.085) 17.600 37.210 (0.002) (0.000) 0.106 -0.057 (0.076) (0.080) 14.291 -19.215 (1.317) (2.822) 118.660 48.240 (0.000) (0.000) 0.223 0.229 (0.028) (0.031) 0.187 0.201 (0.027) (0.030) 0.122 0.210 (0.027) (0.031) 0.141 0.121 (0.028) (0.031) 0.172 0.081 (0.032) (0.028) 0.094 0.045 (0.028) (0.032) 81.410 93.370 (0.000) (0.000) 1498.950 1319.090 (0.000) (0.000) 29,540 25,345 - - 0.071 (0.008) 308.700 (0.000) -0.019 (0.002) -0.011 (0.002) 2.160 (0.164) 212.110 (0.000) -0.228 (0.070) 0.036 (0.071) 0.189 (0.070) -0.002 (0.072) 17.950 (0.001) 0.126 (0.070) 31.432 (1.225) 661.670 (0.000) 0.268 (0.028) 0.199 (0.027) 0.199 (0.027) 0.187 (0.027) 0.158 (0.028) 0.131 (0.028) 106.130 (0.000) 1352.180 (0.000) 31,254 19.569 (2.546) 0.018 (0.014) - -0.140 (0.014) 0.059 (0.024) 0.213 (0.012) 482.030 (0.000) -0.020 (0.002) -0.005 (0.002) -1.155 (0.463) 151.390 (0.000) -0.041 (0.087) -0.051 (0.089) -0.153 (0.088) 0.021 (0.087) 3.950 (0.413) -0.533 (0.088) -16.888 (3.278) 54.250 (0.000) 0.320 (0.033) 0.329 (0.032) 0.199 (0.034) 0.203 (0.031) 0.165 (0.032) 0.143 (0.030) 141.510 (0.000) 918.470 (0.000) 23,199 16.334 (1.512) -0.166 (0.013) 0.127 (0.019) -0.003 (0.011) - 0.018 (0.011) 239.540 (0.000) -0.018 (0.002) -0.018 (0.002) 2.262 (0.394) 172.830 (0.000) -0.093 (0.083) 0.024 (0.081) 0.050 (0.083) -0.164 (0.083) 6.190 (0.186) -0.156 (0.081) 21.768 (1.243) 309.110 (0.000) 0.197 (0.032) 0.203 (0.029) 0.119 (0.030) 0.056 (0.029) 0.042 (0.030) 0.108 (0.029) 72.440 (0.000) 861.090 (0.000) 24,379 T h e table reports parameter estimates for an ordered probit model of the choice between a market order, a one tick away l i m i t order, a n d l i m i t order more than one tick away from the best quotes. S t a n d a r d errors are i n parentheses. For each stock, a model is estimated for b u y orders and for sell orders. T h e dependent variable is equal to zero for a market order, one for a one tick l i m i t order a n d two for a l l other l i m i t orders. A x -test for the null hypothesis of a l l parameters j o i n t l y being equal to zero is reported for each subset of regressors a n d for the overall m o d e l . 2 90 Table D.6: Competing Risks Models for Execution and Cancellation Hazard Rates for B H O Variable Power: a Constant Spread Close ask d e p t h Far ask d e p t h Close b i d depth Far b i d depth O r d e r size M a r g i n a l sell Execute C a n c e l 0.655 0.625 (0.031) (0.015) -15.689 -5.075 (1.513) (0.890) -1.530 -0.538 (2.690) (1.477) - -0.276 (0.081) -0.225 (0.066) -0.186 - -0.154 (0.040) -0.007 (0.030) - - 1 Tick Execute 0.580 (0.009) -10.420 (0.420) 56.836 (2.360) -0.057 (0.023) - sell Marginal buy Cancel Execute C a n c e l 0.512 0.700 0.615 (0.017) (0.044) (0.016) -1.721 -4.310 -6.029 (0.854) (1.851) (0.630) -9.189 16.289 3.457 (4.214) (2.783) (1.685) -0.008 0.000 0.007 (0.025) (0.098) (0.037) - 0.138 (0.021) 0.002 (0.022) -0.221 -0.049 0.039 (0.075) (0.039) (0.027) (0.030) Recent trades 0.026 0.013 0.036 0.032 (0.005) (0.003) (0.002) (0.003) D u r a t i o n (xlOOO) -0.144 -0.108 -0.084 -0.022 (0.026) (0.010) (0.006) (0.004) Mid-quote volatility 5.611 0.095 3.325 -1.090 (0.796) (0.487) (0.235) (0.496) T S E market index 0.065 0.013 -0.001 0.022 (0.054) (0.028) (0.023) (0.025) T S E m i n i n g index -0.030 0.056 -0.058 0.032 (0.065) (0.028) (0.021) (0.021) Overnight interest rate -0.027 0.003 -0.021 0.010 (0.072) (0.031) (0.020) (0.021) C a n a d i a n dollar 0.095 -0.020 0.015 0.022 (0.059) (0.030) (0.022) (0.022) Lagged r e t u r n 0.146 0.004 0.103 0.071 (0.058) (0.030) (0.020) (0.022) Distance t o mid-quote 26.973 0.224 38.506 21.246 (2.755) (2.082) (1.825) (2.186) F i r s t hour -0.679 -0.672 -0.569 -1.124 (2.755) (2.082) (1.825) (2.186) Second hour -0.201 -0.664 -0.634 -0.963 (0.265) (0.115) (0.087) (0.084) T h i r d hour -0.020 -0.368 -0.438 -0.902 (0.273) (0.118) (0.085) (0.082) F o u r t h hour 0.057 -0.347 -0.405 -0.855 (0.277) (0.122) (0.087) (0.085) F i f t h hour 0.279 -0.049 -0.341 -0.644 (0.277) (0.121) (0.088) (0.082) S i x t h hour 0.396 0.133 -0.208 -0.463 (0.290) (0.123) (0.089) (0.083) X i : Constant hazard rate 302.610 290.010 1731.080 576.140 P-value (0.000) (0.000) (0.000) (0.000) N u m b e r of observations 1,748 4,498 q 0.049 -- 1 Tick Execute 0.613 (0.009) 2.043 (0.473) 68.731 (2.670) 0.201 (0.023) - -0.082 (0.022) buy Cancel 0.524 (0.010) -6.928 (0.733) 2.203 (4.327) 0.053 (0.025) - -0.013 (0.025) -0.075 (0.101) (0.043) -0.498 0.025 -0.289 -0.115 (0.101) (0.041) (0.027) (0.031) 0.031 0.017 0.035 0.030 (0.006) (0.003) (0.002) (0.002) -0.069 -0.081 -0.064 -0.048 (0.026) (0.009) (0.006) (0.006) -2.170 0.705 -4.309 1.985 (1.084) (0.364) (0.273) (0.430) 0.053 -0.005 0.020 -0.003 (0.077) (0.031) (0.023) (0.027) 0.010 0.009 -0.039 0.004 (0.084) (0.032) (0.022) (0.025) 0.062 0.023 0.020 -0.031 (0.087) (0.034) (0.021) (0.023) -0.375 -0.058 -0.042 -0.048 (0.101) (0.032) (0.023) (0.025) 0.090 0.067 0.055 0.086 (0.084) (0.033) (0.021) (0.025) -21.657 -2.826 -45.174 -11.919 (3.453) (1.730) (1.977) (2.120) 0.081 -1.041 -0.212 -1.006 (3.453) (1.730) (1.977) (2.120) -0.228 -0.870 -0.105 -1.111 (0.393) (0.129) (0.089) (0.093) -0.513 -0.764 -0.262 -1.002 (0.425) (0.136) (0.089) (0.086) -0.278 -0.767 -0.189 -0.771 (0.436) (0.139) (0.091) (0.084) -0.458 -0.429 -0.174 -0.891 (0.478) (0.143) (0.090) (0.084) 0.003 -0.249 0.012 -0.653 (0.485) (0.153) (0.093) (0.088) 188.960 247.370 1710.490 648.100 (0.000) (0.000) (0.000) (0.000) 1,353 4,105 - -- T h e table reports parameters estimates w i t h asymptotic standard errors i n parentheses for a competing risks model of hazard rates for executions a n d cancellations for marginal l i m i t orders a n d one tick away l i m i t orders. A x -test for the null hypothesis of a l l parameters j o i n t l y being equal to zero is reported for each model w i t h the p-value i n parenthesis. 2 91 Table D.7: Competing Risks Models for Execution and Cancellation Hazard Rates for E R R Variable Power: a Constant Spread Close ask depth Far ask d e p t h Close b i d depth Far b i d depth O r d e r size Recent trades D u r a t i o n (XlOOO) Mid-quote volatility T S E market index T S E m i n i n g index Overnight interest rate C a n a d i a n dollar Lagged return M a r g i n a l sell 1 T i c k sell Marginal buy Execute C a n c e l Execute Cancel Execute C a n c e l 0.652 0.615 0.544 0.499 0.817 0.585 (0.022) (0.012) (0.011) (0.011) (0.029) (0.012) -12.974 -6.642 -10.345 -5.602 -3.862 -4.232 (0.886) (0.979) (0.584) (1.210) (1.229) (0.657) 3.463 2.205 22.510 -5.585 3.650 2.710 (1.287) (0.706) (1.565) (2.220) (1.195) (0.630) -0.284 0.000 0.105 -0.050 (0.036) (0.036) (0.061) (0.034) -0.210 -0.060 (0.055) (0.032) 0.091 -0.014 0.195 0.019 (0.054) (0.030) (0.033) (0.033) -0.261 -0.074 (0.056) (0.031) -0.115 0.144 -0.017 -0.025 -0.248 0.005 (0.048) (0.029) (0.031) (0.032) (0.057) (0.033) 0.011 0.016 0.018 0.014 0.055 0.030 (0.006) (0.004) (0.005) (0.006) (0.007) (0.005) -0.150 -0.056 -0.088 -0.007 -0.099 -0.058 (0.022) (0.008) (0.015) (0.008) (0.018) (0.007) 4.047 1.041 3.682 1.530 -2.525 -0.078 (0.466) (0.558) (0.322) (0.705) (0.700) (0.381) 0.175 0.050 0.025 0.004 0.001 0.044 (0.040) (0.024) (0.030) (0.030) (0.047) (0.027) -0.156 -0.003 0.007 0.005 0.010 -0.049 (0.051) (0.026) (0.033) (0.033) (0.048) (0.026) -0.089 0.049 -0.080 0.047 0.081 -0.002 (0.046) (0.023) (0.036) (0.032) (0.042) (0.026) -0.102 -0.031 -0.066 -0.044 -0.072 -0.019 (0.049) (0.025) (0.035) (0.034) (0.049) (0.026) - - 0.010 -0.003 (0.040) Distance t o mid-quote - (0.022) 36.361 9.413 - - - - 0.002 -0.060 (0.035) (0.041) - 0.074 (0.034) -0.030 (0.029) buy Cancel 0.538 (0.015) -2.764 (0.807) 0.209 (2.702) -0.050 (0.044) -0.079 (0.037) -0.023 (0.040) - - -0.188 (0.036) 0.014 (0.007) -0.079 (0.014) -4.958 (1.084) -0.027 (0.032) -0.017 (0.031) -0.024 (0.045) -0.006 (0.036) -0.018 (0.041) 0.038 (0.006) -0.043 (0.011) -0.242 (0.464) 0.046 (0.036) -0.124 (0.039) 0.046 (0.047) -0.075 (0.044) 0.026 0.007 (0.029) (0.030) 49.488 25.914 (4.648) (2.932) (4.639) (5.050) (4.455) (2.469) F i r s t hour -0.548 -1.070 -0.347 -0.822 -0.287 -1.092 (4.648) (2.932) (4.639) (5.050) (4.455) (2.469) (6.043) (6.374) Second hour -0.553 -0.966 -0.475 -1.030 -0.196 -1.020 -0.168 -0.805 (0.198) (0.095) (0.124) (0.113) (0.242) (0.102) (0.131) (0.127) T h i r d hour -0.415 -0.914 -0.337 -0.973 -0.294 -0.943 -0.130 -1.088 (0.200) (0.096) (0.127) (0.118) (0.245) (0.104) (0.130) (0.136) F o u r t h hour -0.098 -0.611 -0.105 -0.817 -0.120 -0.769 -0.328 -1.023 F i f t h hour S i x t h hour Constant hazard rate P-value N u m b e r of obervations XIQ: -45.869 - 1 Tick Execute 0.638 (0.015) 3.927 (1.855) 25.817 (1.810) 0.071 (0.038) -9.096 -101.778 -31.003 (6.043) (6.374) 0.061 -0.719 (0.202) (0.097) (0.121) (0.113) (0.252) (0.108) (0.132) (0.130) -0.112 -0.770 -0.110 -0.631 -0.100 -0.635 -0.224 -0.771 (0.203) (0.102) (0.125) (0.111) (0.262) (0.110) (0.140) (0.134) -0.190 -0.381 -0.225 -0.486 -0.167 -0.366 -0.051 -0.770 (0.217) (0.098) (0.127) (0.107) (0.276) (0.105) (0.134) 290.140 (0.000) 258.470 (0.000) 531.010 (0.000) 2,458 211.250 (0.000) 2,368 403.440 (0.000) 304.150 (0.000) 2,138 539.690 (0.130) 218.130 (0.000) (0.000) 1,732 T h e table reports parameter estimates w i t h asymptotic standard errors i n parentheses for a competing risks model of hazards for executions a n d cancellations for marginal l i m i t orders a n d one tick away l i m i t orders. A x -test for the null hypothesis of a l l parameters j o i n t l y being equal to zero is reported for each model w i t h the p-value i n parenthesis. 2 92 Table D.8: Competing Risks Models for Execution and Cancellation Hazard Rates for W E M Marginal sell 1 Tick sell Marginal buy Execute Cancel Execute Cancel Execute Cancel 0.644 0.616 0.580 0.494 0.786 0.652 (0.028) (0.014) (0.014) (0.013) (0.038) (0.014) Constant -5.933 -4.505 -26.407 1.111 1.221 -3.551 (1.603) (0.910) (1.942) (1.764) (2.012) (1.087) Spread 3.982 3.535 19.346 3.637 3.219 2.109 (2.133) (1.084) (1.426) (3.024) (1.681) (0.826) Close ask depth 0.020 -0.015 - -0.018 -0.077 - (0.044) (0.045) (0.081) (0.037) Far ask depth -0.429 -0.145 (0.080) (0.044) Close bid depth 0.017 -0.051 0.168 0.080 (0.067) (0.036) (0.040) (0.040) Far bid depth - -0.488 -0.092 - (0.072) (0.036) Order size -0.196 0.057 -0.130 -0.059 -0.303 -0.045 (0.079) (0.041) (0.045) (0.045) (0.074) (0.034) Recent trades 0.040 0.039 0.044 0.044 0.064 0.033 (0.010) (0.005) (0.007) (0.008) (0.008) (0.005) Duration (XlOOO) -0.078 -0.054 -0.062 -0.025 -0.070 -0.045 (0.018) (0.008) (0.010) (0.009) (0.020) (0.007) Mid-quote volatility 0.279 -0.173 12.892 -2.736 -5.368 -0.947 (0.914) (0.518) (1.137) (1.031) (1.182) (0.635) TSE market index -0.040 -0.014 0.041 0.007 0.074 0.010 (0.058) (0.029) (0.035) (0.035) (0.058) (0.027) TSE mining index -0.260 0.012 -0.036 0.065 0.102 0.000 (0.069) (0.028) (0.039) (0.035) (0.059) (0.028) Overnight interest rate 0.020 0.029 0.005 -0.002 0.064 -0.027 (0.054) (0.027) (0.037) (0.036) (0.055) (0.029) Canadian dollar -0.005 -0.069 -0.047 -0.007 -0.142 -0.063 (0.057) (0.031) (0.046) (0.042) (0.067) (0.028) Lagged return -0.046 0.006 0.102 0.002 0.152 0.038 (0.060) (0.027) (0.033) (0.036) (0.061) (0.029) Distance to mid-quote 25.431 7.155 42.530 31.780 -31.161 -5.006 (2.616) (2.543) (3.719) (3.914) (5.153) (2.731) First hour -0.916 -1.019 -0.807 -0.957 0.051 -1.029 (2.616) (2.543) (3.719) (3.914) (5.153) (2.731) Second hour -1.025 -1.013 -0.778 -0.802 0.021 -0.685 (0.224) (0.105) (0.140) (0.127) (0.316) (0.104) Third hour -0.484 -0.730 -0.477 -0.873 0.387 -0.712 (0.222) (0.108) (0.130) (0.126) (0.319) (0.112) Fourth hour -0.059 -0.742 -0.441 -0.726 -0.204 -0.483 (0.214) (0.113) (0.133) (0.127) (0.342) (0.108) Fifth hour -0.141 -0.503 -0.459 -0.562 0.629 -0.330 (0.236) (0.117) (0.139) (0.127) (0.332) (0.116) Sixth hour -0.083 -0.328 -0.130 -0.407 0.451 -0.118 (0.223) (0.108) (0.127) (0.119) (0.330) (0.107) X j : Constant hazard rate 218.200 273.520 501.500 188.440 244.620 273.480 P-value (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) Number of observations 1,679 1,793 1,792 Variable Power: a 9 1 Tick buy Execute Cancel 0.586 0.468 (0.014) (0.013) 12.912 -9.605 (2.005) (1.960) 35.598 -7.160 (2.528) (3.948) 0.050 0.043 (0.046) (0.046) - -0.197 (0.041) - - -0.079 (0.041) - -0.201 -0.101 (0.040) (0.044) 0.039 0.038 (0.007) (0.007) -0.035 -0.029 (0.010) (0.010) -10.441 3.840 (1.181) (1.150) 0.000 0.026 (0.036) (0.036) 0.046 -0.013 (0.038) (0.041) 0.088 0.001 (0.034) (0.036) 0.027 -0.030 (0.041) (0.045) 0.102 0.072 (0.037) (0.038) -81.301 -12.414 (5.880) (5.566) -0.197 -0.624 (5.880) (5.566) -0.102 -1.073 (0.136) (0.151) -0.266 -0.683 (0.132) (0.119) -0.207 -0.554 (0.133) (0.121) -0.191 -0.847 (0.136) (0.132) -0.072 -0.487 (0.130) (0.118) 493.200 201.660 (0.000) (0.000) 1,689 The table reports parameter estimates with asymptotic standard errors in parentheses for a competing risks model of the hazards for executions and cancellations for marginal limit orders and one tick away limit orders. A x -test for the null hypothesis of all parameters jointly being equal to zero is reported for each model with the p-value in parenthesis. 2 93 to CO CM oi o Ol f00 rH rH 00 r-; 05 in CM d l TP co lO rlO td rH O CO CN CO O O in O o ci CM i 1 o d r H CO r H CO CM to o o in o o O O O O to c i o o o m o t o o m rH O C M C M T P C M C O C M r H r H d d <3 d d d q q q o TP t to =* SCM 00 rH rH — s Ol !D O H DO S I D N CO N to r H r H r HO ci rH rA H CO to N CM r H CO TP O O O 00 o o o o o o o o i n t O C M i n i n O C l t O O l O l O C O O t - r C O r H O r H O r H r H r H O r H d d d d d d d d i 1 TP TP rco CM TP to q rH C- CM in rH to TP o d to rH O d rH CO o d o o o d rH CO o d m o CM o rH O d_ d CM O 00 CM Ol OO CM CM r H d d CO o rH d CM TP rH d o d o ot? o o d tc? 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CO o* in to CO o CO CO CO m CN in CM rH OJ rH m CO CO TP 2S CN CN o O o o rH CN rH . 00 r~ O d d dCN in d d di *d— d d d• d d o_ d• d_ o CO 1 1 -* 1 1 1 rH f- o to to in TP CO o> CO o tto to to CN to to rH CO TP in o CN o to in TP f- TP t~ TP m TP CO TP CN TP d d lO rH d _l d d , d d d d d d d d CO T f CN CN .—1 00 T f TP 1 Ol c<? to in in* in CO CN co" CN 0C? 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C O C O r H r H C M l - - O J r H r H C O ^ O O rH m ( O r f H ^ T l ' ^ O n H f f l D H OS O O d o CO O J >6 co o rH co in TP m CN CM rH o CN d d o o o o o o o o o o o o di ciciddciddcidciddi ° > rH o S V CN rH £5 ototocN-tfrHiomcocoioco H O I O H N O O O O O l l O C f r - C O floor-qoooooo^ooq O r H O r H O d d r H O r H O r H o OJ o co i n o i r ^ c o t o o o o o o c o o j o o j o o O CN r H C O C N T P C N C O C N C O O C O r H C O o o m co r H O O O O O O O O O O O d d l> rH m CN CM t o c o r H i n O J C N t o o o o i n c - r ~ TP in CM r H o o i n - ^ t o c N c o t o m i n o o rH rH O t^- q t ^ c N t o q t o t q t o o t o i n t o dddddddicDdd'Zid rH o 00 CO OJ o <^ O 3 a3 O -c •8 bO bO C3 •a a o 3 O J3 3 o EH in "ft! Z fe H 95 «2 Table D.10: Estimation Results for the Discrete Choice Model BHO Trader arrival rate 0.450 (0.002) Constant -2.146 (0.018) T S E market index 0.029 (0.005) T S E mining index 0.032 (0.005) Overnight interest rate -0.114 (0.006) Canadian dollar 0.105 (0.005) Lagged return 0.271 (0.005) Xs- Constant arrival rate 3871.950 (0.000) Private value distribution 0.849 Mixing probability: p (0.040) 0.058 (0.001) a_ — ai 0.472 (0.035) Time-varying variance T S E market index -0.000 (0.007) T S E mining index -0.032 (0.006) Overnight interest rate 0.072 (0.006) Canadian dollar -0.041 (0.007) Lagged return -0.104 (0.006) X%'- Constant private value variance 434.550 (0.000) Continuation value 0.021 (0.001) Power: a a ERR WEM 0.465 (0.002) -2.077 (0.019) 0.018 (0.005) -0.000 (0.005) -0.000 (0.005) 0.061 (0.005) 0.152 (0.007) 680.940 (0.000) 0.453 (0.002) -2.554 (0.024) 0.032 (0.005) 0.011 (0.005) 0.066 (0.005) -0.038 (0.005) 0.185 (0.006) 985.280 (0.000) 0.891 (0.034) 0.040 (0.001) 0.682 (0.064) 0.842 (0.048) 0.037 (0.001) 0.483 (0.045) 0.025 (0.006) -0.018 (0.007) -0.051 (0.006) -0.022 (0.006) -0.135 (0.005) 814.080 (0.000) 0.024 (0.001) -0.047 (0.007) -0.045 (0.007) -0.032 (0.007) 0.026 (0.007) -0.116 (0.008) 369.700 (0.000) 0.011 (0.001) The table reports parameter estimates for the discrete choice model. Asymptotic standard errors are reported in parentheses. 96 rH LO CM rH rH rH CM tO rH rH rH CO rH LO Tf rH CM CM O d „ 10 r§ <^ o> 9 W r-< rH Tf tO CM LO OJ Tj< Ol tO LO OJ r-i r-i LO Tf LO LO r-i rH t-- rH O tO CO CO o OJ r- CM rH CN O rH CM O Tf Tf rH O Tf rH CM rH rH O CM t— CM rH rH rH rH O CM c~ T f rH tO CO LO CN O r - i t-i rH rH 00 tO rH Tf r-i rH rH t - q O r~ Tf tO rH 00 rH TP q rH f LO to O rH CN LO CO O CO CN r-i Tf LO r-i CN O s Ob •5 *-.\co O 0 3 rH CN CO Tf CO CN f X Tf tlO T f O t ~ CN LO tO O r - i r - i ' •S r CN Ol rH a to Tf' LO LO O XI O f- 00 xT d Tf CO to o> Tf CO Tf ' 00 CN o o d to OJ r - LO Tf Tf CO CM LO LO LO CO n s to q r-, LO Tf Tf O r-i I q CN LO r H CN O OJ OJ tO tO LO CN OJ Tf ^ i Tf rH 00 LO <D LO > H O I 2 rH f- Tf Tf ^ CN tO CO Ou, ^ CC LO CN CM Tf Hnri (3 CO tO r H & CO r H r H tO " tO CO T f T j i CO LO tO Ol CO OJ t- ta "* °° 9 _ Tf •S T f LO 00 O CM CM d Tf CN tO CN Tf CO Tf J3 CO 00 O 00 t-~ r H [3 1— CN CO tO r-i r - i CO co CO Tf Tf O Tf CN LO LO CN CO r H r H CO tO COJ CO GJ t* OJ r-i LO Tf Tf CO Tf Tf Tf CN rH O CO t - rH rH rH rH CO rH O rH t~ to to CO Tf _ rH LO O Tf CN t~ 00 d 60 c CN rH O Tf tO CN CN tO CM O CO O rH _ rH rH O Tf rH rH O CN LO CO rH rH CO n m w K ^ °- n -r -r Tf •rf Tf O O t-; O i-i r-i CM Tf CN rH rH tO TP rH rH c3 S CM CM O rH rH rH Tf rH rH CN CN c3 C C3 'So E s 5 s -g S rH O * s 3 3 CQ 03 to S3 S >> 3 a 97 Tf rH O CO OJ q r-i a> o a> d d co LO CM r - i Ji ^ bO c3 * £ cci rH t- r-i C8 c S Tf LO s OJ e -E rn >>>>>> 3 13 3 3 0 O a '5b c3 S earaca z ^ £ rH c3 g Table D.12: Order Submission Probabilities by Trader Valuation (-oo,-5%] Mean Standard deviation Sell market Sell limit No order Buy limit Buy market Sell market Sell limit No order Buy limit Buy market Mean Standard deviation Sell market Sell limit No order Buy limit Buy market Sell market Sell limit No order Buy limit Buy market (-5%,-2.5%] Private value (-2.5%,+2.5%) [2.5%,+5%) BHO Probability of value in interval 0.212 0.127 0.322 0.127 0.026 0.007 0.039 0.007 Mean order submission probabilities 0.735 0.247 0.014 0.000 0.244 0.642 0.247 0.003 0.019 0.089 0.364 0.068 0.002 0.022 0.367 0.743 0.000 0.000 0.008 0.187 Standard deviation of order submission probabilities 0.300 0.357 0.070 0.010 0.286 0.393 0.201 0.042 0.093 0.251 0.282 0.218 0.029 0.128 0.271 0.356 0.000 0.004 0.050 0.316 ERR Probability of value in interval 0.115 0.143 0.484 0.143 0.018 0.009 0.049 0.009 Mean order submission probabilities 0.851 0.316 0.011 0.001 0.147 0.656 0.164 0.001 0.002 0.027 0.608 0.009 0.000 0.001 0.210 0.752 0.000 0.000 0.007 0.238 Standard deviation of order submission probabilities 0.233 0.351 0.059 0.026 0.231 0.355 0.116 0.030 0.027 0.120 0.154 0.069 0.003 0.023 0.116 0.334 0.000 0.000 0.039 0.330 98 [+5%,+oo) 0.212 0.026 0.000 0.000 0.010 0.301 0.689 0.000 0.006 0.065 0.305 0.309 0.115 0.018 0.000 0.000 0.001 0.218 0.781 0.008 0.011 0.015 0.263 0.264 Table D.12: Order Submission Probabilities by Trader Valuation (continued) Private value (-oo,-5%] Mean Standard deviation Sell market Sell limit No order Buy limit Buy market Sell market Sell limit No order Buy limit Buy market (-5%,-2.5%] (-2.5%,+2.5%) [2.5%,+5%) WEM Probability of value in interval 0.111 0.130 0.519 0.130 0.017 0.012 0.055 0.012 Mean order submission probabilities 0.921 0.483 0.036 0.000 0.079 0.512 0.413 0.003 0.000 0.004 0.118 0.011 0.000 0.001 0.403 0.616 0.000 0.000 0.030 0.370 Standard deviation of order submission probabilities 0.157 0.413 0.090 0.001 0.157 0.411 0.171 0.047 0.007 0.053 0.176 0.090 0.003 0.021 0.155 0.415 0.002 0.007 0.089 0.414 [+5% +oo) 1 0.111 0.017 0.000 0.000 0.001 0.164 0.835 0.000 0.008 0.021 0.237 0.238 The table reports the in-sample mean and the standard deviation of the probability of drawing a private value (u) from five different intervals. For each interval and stock the table reports the insample mean and standard deviation of the probability of a trader optimally submitting a sell market order, a sell limit order, no order, a buy limit order, or a buy market order. 99 e""(o,i) e' "(i,2) 8 efteNO) e^.NO) ej^ti^) eJ" (o,i) y e,(2,2) Valuation v =y +u ( t Figure D . l : The graph provides an example of the traders optimal order submission strategy. The horizontal axis is the trader's valuation, and the vertical axis is the expected utility from various order submissions. The horizontal axis and the vertial axis have different scale. Sell orders are plotted with dashed lines ( ) and buy orders are plotted with dashed-dotted lines (-.-.). The maximized utility function is plotted with a dark solid line (—). The continuation value is equal to zero. 100 Figure D.2: Comparative statics for BHO. The top left picture plots the probability of a sell market order (-) and the probability of a sell limit order (- -) conditional on an order submission as a function of the spread. The top middle picture plots the probability of a buy market order (-) and the probability of a buy limit order (- -) conditional on an order submission as a function of the spread. The top right picture plots the expected time to the next order submission as a function of the spread. The middle row and the bottom row of pictures plots the corresponding comparative statics for the choice probabilities and time to the next order for the bid side depth and the distance between the common value proxy and the mid-quote. 101 Figure D.3: Comparative statics for ERR. The top left picture plots the probability of a sell market order (-) and the probability of a sell limit order (- -) conditional on an order submission as a function of the spread. The top middle picture plots the probability of a buy market order (-) and the probability of a buy limit order (- -) conditional on an order submission as a function of the spread. The top right picture plots the expected time to the next order submission as a function of the spread. The middle row and the bottom row of pictures plots the corresponding comparative statics for the choice probabilities and time to the next order for the bid side depth and the distance between the common value proxy and the mid-quote. 102 Ot -2.5 . . . . J -1.5 -0.5 0.5 1.5 2.5 Common Value - Mid-Quote [%) l 0 -2.5 .—.—. • , J -1.5 -0.5 0.5 1.5 2.5 Common Value - Mid-Quote [%] L -2.5 . , . . J -1.5 -0.5 0.5 1.5 2.5 Common Value - Mid-Quote [%] Figure D.4: Comparative statics for W E M . The top left picture plots the probability of a sell market order (-) and the probability of a sell limit order (- -) conditional on an order submission as a function of the spread. The top middle picture plots the probability of a buy market order (-) and the probability of a buy limit order (- -) conditional on an order submission as a function of the spread. The top right picture plots the expected time to the next order submission as a function of the spread. The middle row and the bottom row of pictures plots the corresponding comparative statics for the choice probabilities and time to the next order for the bid side depth and the distance between the common value proxy and the mid-quote. 103 oi0.1 • • • 0.35 0.75 1.15 Lagged Return 1 1.4 0 L 0.1 ' • • -I 0.35 0.75 1.15 1.4 Lagged Return L 0.1 , . 0.35 0.75 . J 1.15 1.4 Lagged Return Figure D.5: Comparative statics for BHO. The top left picture plots the sell market (-) and sell limit (- -) choice probabilities as a function of the absolute value of changes in the stock's lagged return. The top middle picture plots the buy market (-) and buy limit (- -) choice probabilities as a function of the absolute value of changes in the stock's lagged return. The top right picture plots the expected time to the next order submission as a function of the absolute value of changes in the stock's lagged return. The middle and the bottom row of pictures also plot the choice probabilities and time to the next order submission. In the middle row of pictures the valuation distribution and the arrival rate of traders are held constant at their mean values and in the bottom row of pictures the threshold valuations are held constant at their mean values. 104 0.1 0.35 0.75 1.15 1.4 0.1 0.35 Lagged Return 0.1 0.35 0.75 1.15 1.4 0.1 0.35 Lagged Return 0.1 0.35 0.75 Lagged Return 0.75 1.15 1.4 0.1 0.35 Lagged Return 0.75 1.15 1.4 0.1 0.35 Lagged Return 1.15 1.4 0.1 0.35 0.75 Lagged Return 0.75 1.15 1.4 1.15 1.4 1.15 1.4 Lagged Return 0.75 Lagged Return 1.15 1.4 0.1 0.35 0.75 Lagged Return Figure D.6: Comparative statics for ERR. The top left picture plots the sell market (-) and sell limit (- -) choice probabilities as a function of the absolute value of changes in the stock's lagged return. The top middle picture plots the buy market (-) and buy limit (- -) choice probabilities as a function of the absolute value of changes in the stock's lagged return. The top right picture plots the expected time to the next order submission as a function of the absolute value of changes in the stock's lagged return. The middle and the bottom row of pictures also plot the choice probabilities and time to the next order submission. In the middle row of pictures the valuation distribution and the arrival rate of traders are held constant at their mean values and in the bottom row of pictures the threshold valuations are held constant at their mean values. 105 Lagged Return Lagged Return Lagged Return Figure D.7: Comparative statics for WEM. The top left picture plots the sell market (-) and sell limit (- -) choice probabilities as a function of the absolute value of changes in the stock's lagged return. The top middle picture plots the buy market (-) and buy limit (- -) choice probabilities as a function of the absolute value of changes in the stock's lagged return. The top right picture plots the expected time to the next order submission as a function of the absolute value of changes in the stock's lagged return. The middle and the bottom row of pictures also plot the choice probabilities and time to the next order submission. In the middle row of pictures the valuation distribution and the arrival rate of traders are held constant at their mean values and in the bottom row of pictures the threshold valuations are held constant at their mean values. 106 Appendix E Tables and Figures for Dynamic Strategies in Limit Order Markets 107 35001 r Figure E . l : Histogram of the buy order prices relative to the midquote for the stock "Clearly Canadian Beverages" (CLV) on the Vancouver Stock Exchange over the time period from January 1, 1991 to November 30, 1992. The width of each bar is one tick (12.5 cents). Variable T N a Value 60 5 0.02 0.01 0.10 0.125 0.0625 Table E . l : Base case parameter values. The parameter values used to solve the numerical model in the base care are presented. T is the time horizon, N is the number of time intervals between management opportunities for the optimizing trader, C is the monitoring or waiting cost, C is the cancellation cost, and C is the cost to move an order. The tick size is <j> and the parameters that measures the volatility of the common value is a. The value N gives the maximum number of ticks away from the common value that exogenous (liquidity) orders will fall. w c m 108 Price Value Action 0.750 0.625 0.500 0.375 0.250 0.125 0.000 -0.125 -0.250 -0.375 -0.500 -0.625 -0.750 -0 010720 -0 010720 -0 010720 -0 010720 -0 010720 -0 010720 -0 010720 0 054497 0 087641 0 089280 0 078762 0 062386 0 048012 -0.375 -0.375 -0.375 -0.375 -0.375 -0.375 -0.375 -0.125 -0.250 -0.375 -0.500 -0.625 -0.750 Execution Probability Total Limit Market 0.8188 0.6191 0.1997 0.8188 0.6191 0.1997 0.8188 0.6191 0.1997 0.8188 0.6191 0.1997 0.8188 0.6191 0.1997 0.8188 0.6191 0.1997 0.8188 0.6191 0.1997 0.8779 0.8427 0.0351 0.8640 0.7610 0.1030 0.8188 0.6191 0.1997 0.7654 0.4717 0.2936 0.7184 0.3501 0.3683 0.6889 0.2761 0.4128 Cancel Prob. 0.1401 0.1401 0.1401 0.1401 0.1401 0.1401 0.1401 0.1117 0.1110 0.1401 0.1812 0.2208 0.2470 Number Moves 1.0396 1.0396 1.0396 1.0396 1.0396 1.0396 1.0396 0.0148 0.0239 0.0396 0.0583 0.0745 0.0842 Time in Book 10.248 10.248 10.248 10.248 10.248 10.248 10.248 2.956 6.283 10.248 13.661 16.129 17.470 Table E.2: Base case solution. This table presents the solution for the order management problem under the parameter values in Table E . l in period 0 when the book is empty. Optimal strategies are computed assuming that a limit order has already been submitted to the book at the given price. The Price itself is relative to the common value. The Action is the new price chosen by the trader or C for a cancellation. The Value is the optimal value function for the problem calculated using equation (4.23). Three execution probabilities are given. The Limit execution probability is the probability that a future market sell order will transact against the optimizing trader's order. The Market execution probability is the probability that the optimizing trader will convert to a market order. The total is the sum of the market and limit execution probabilities. The Cancel Prob. is the probability that the optimizing trader will find it optimal to cancel the order prior to it executing and prior to the close of the market. The Number Moves is the expectation under the optimal strategy of the number of times the optimal trader will change the price of his order. Time in Book is the expected number of periods before the order is transacted, canceled, or expired. 109 Price Value Action 0.750 0.625 0.500 0.375 0.250 0.125 0.000 -0.125 -0.250 -0.375 -0.500 -0.625 -0.750 -0 734574 -0 643091 -0 520708 -0 396286 -0 272204 -0 150731 -0 041087 0 038028 0 060369 0 040975 0 004424 -0 038105 -0 070659 0.750 0.625 0.500 0.375 0.250 0.125 0.000 -0.125 -0.250 -0.375 -0.500 -0.625 -0.750 Execution ProbabilityTotal Limit Market 1.0000 1.0000 0.0000 1.0000 1.0000 0.0000 1.0000 1.0000 0.0000 1.0000 1.0000 0.0000 0.9997 0.9997 0.0000 0.9979 0.9979 0.0000 0.9871 0.9871 0.0000 0.9487 0.9487 0.0000 0.8572 0.8572 0.0000 0.7275 0.7275 0.0000 0.5926 0.5926 0.0000 0.4768 0.4768 0.0000 0.4063 0.4063 0.0000 Cancel Prob. 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 Number Moves 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 Time in Book 0.000 0.000 0.000 0.000 0.000 0.000 1.743 4.424 10.053 17.680 25.430 32.042 36.075 Table E.3: N o management solution. This table presents the solution for the order management problem under the parameter values in Table E . l with the exception that the cancellation and move costs A and A are set to very high values. The strategy is shown for period 0 when the book is empty. All other values are as in Table E.2. c Spread Price Value Action 0.250 0.500 0.750 1.000 1.250 1.500 -0.125 -0.125 -0.250 -0.375 -0.375 -0.375 0.023379 0.062739 0.095158 0.093923 0.091382 0.089804 -0.125 -0.125 -0.250 -0.375 -0.375 -0.375 m Execution Probability Total Limit Market 0.7741 0.6991 0.0751 0.8994 0.8675 0.0319 0.8871 0.7814 0.1058 0.8413 0.6343 0.2070 0.8301 0.6267 0.2034 0.8221 0.6212 0.2009 Cancel Prob. 0.2132 0.0918 0.0871 0.1149 0.1274 0.1364 Number Moves 0.0330 0.0062 0.0229 0.0440 0.0406 0.0406 Time in Book 4.546 2.359 5.918 10.221 10.258 10.255 Table E.4: Optimal limit order across spreads. This table presents the solution for the order management problem under the parameter values in Table E . l with a variety of different limit order books. Each row considers a different book and selects the limit order with the highest value in that book. Each book contains exogenous limit orders on each price point from —0.750 to 0.750 with the exception that there are no orders within the given spread around the common value. All other values are as in Table E.2. 110 Spread Price Value Action 0.125 0.250 0.500 0.750 1.000 1.250 1.500 -0.375 -0.375 -0.375 -0.375 -0.375 -0.375 -0.375 -0.030000 -0.020356 -0.011035 0.009884 0.093923 0.091382 0.089804 -0.375 -0.375 -0.375 -0.375 -0.375 -0.375 C Execution Probability Total Limit Market 0.0000 0.0000 0.0000 0.6520 0.4435 0.2085 0.6779 0.4432 0.2347 0.7116 0.4434 0.2682 0.8413 0.6343 0.2070 0.8301 0.6267 0.2034 0.8221 0.6212 0.2009 Cancel Prob. 1.0000 0.3244 0.2971 0.2590 0.1149 0.1274 0.1364 Number Moves 0.0000 0.3114 0.2837 0.1850 0.0440 0.0406 0.0406 Time in Book 0.000 9.511 9.707 10.526 10.221 10.258 10.255 Table E.5: Fixed limit order across spreads. This table presents the solution for the order management problem under the parameter values in Table E . l with a variety of different limit order books. Each row presents the solution for the optimizing limit order trader with a price of —0.375 but with a different book. The book has one exogenous limit order at each price point from —0.750 to 0.750 except for the orders within the given spread around the common value. All other values are as in Table E.2. Price Value Action 0.750 0.625 0.500 0.375 0.250 0.125 0.000 -0.125 -0.250 -0.375 -0.500 -0.625 -0.750 -0.030000 -0.030000 -0.030000 -0.030000 -0.030000 -0.030000 -0.028177 0.044807 0.066612 0.059523 0.046259 0.030598 0.017215 C C C C C C 0.000 -0.125 -0.250 -0.375 -0.500 -0.625 -0.750 Execution Probability Total Limit Market 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.8052 0.8052 0.0000 0.6886 0.6846 0.0040 0.5086 0.4815 0.0271 0.3557 0.2848 0.0709 0.2697 0.1575 0.1121 0.2222 0.0825 0.1397 0.1970 0.0417 0.1553 Cancel Prob. 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.1098 0.0651 0.0156 0.0000 0.0000 0.0000 0.0000 Number Moves 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 Table E.6: Short horizon solution. This table presents the solution for the order management problem under the parameter values in Table E . l . The strategy is shown for period 50 when the book is empty. All other values are as in Table E.2. Ill Price Value Action 0.750 0.625 0.500 0.375 0.250 0.125 0.000 -0.125 -0.250 -0.375 -0.500 -0.625 -0.750 -0.016603 -0.016603 -0.016603 -0.016603 -0.016603 -0.016603 -0.016603 0.023396 0.058165 0.074550 0.082494 0.083397 0.081061 -0.625 -0.625 -0.625 -0.625 -0.625 -0.625 -0.625 -0.125 -0.250 -0.375 -0.500 -0.625 -0.750 Execution Probability Total Limit Market 0.6990 0.3088 0.3902 0.6990 0.3088 0.3902 0.6990 0.3088 0.3902 0.6990 0.3088 0.3902 0.6990 0.3088 0.3902 0.6990 0.3088 0.3902 0.6990 0.3088 0.3902 0.7339 0.6298 0.1041 0.7203 0.5329 0.1874 0.6822 0.4142 0.2681 0.7005 0.3630 0.3375 0.6990 0.3088 0.3902 0.7022 0.2819 0.4203 Cancel Prob. 0.2687 0.2687 0.2687 0.2687 0.2687 0.2687 0.2687 0.2573 0.2644 0.2956 0.2717 0.2687 0.2631 Number Moves 1.0239 1.0239 1.0239 1.0239 1.0239 1.0239 1.0239 0.0309 0.0255 0.0414 0.0243 0.0239 0.0185 Time in Book 15.664 15.664 15.664 15.664 15.664 15.664 15.664 5.370 8.541 11.587 13.881 15.664 16.587 Table E.7: High volatility solution. This table presents the solution for the order management problem under the parameter values in Table E . l with the exception that the volatility parameter a is equal to one full tick: 0.125. The strategy is shown for period 0 when the book is empty. All other values are as in Table E.2. Stock AWR CLV DDD ERR ONE PRW SBU SPR TKO WEM Mean Median Std # 27400 35603 55385 49491 25927 33772 14264 32822 31140 52455 35826 33297 12974 Limit 0.486 0.501 0.495 0.475 0.584 0.506 0.485 0.473 0.499 0.509 0.501 0.497 0.032 New Mkt 0.244 0.169 0.219 0.196 0.209 0.206 0.202 0.210 0.215 0.194 0.206 0.208 0.019 Percent of Activity Changes P F i l l Limit M k t 0.057 0.105 0.017 0.042 0.107 0.013 0.082 0.079 0.024 0.090 0.086 0.027 0.055 0.063 0.018 0.083 0.086 0.030 0.085 0.091 0.035 0.091 0.067 0.020 0.084 0.029 0.073 0.069 0.088 0.026 0.073 0.086 0.024 0.077 0.086 0.025 0.017 0.014 0.007 Table E.8: Basic Summary Statistics 112 PfiU 0.007 0.005 0.010 0.011 0.006 0.012 0.013 0.006 0.010 0.011 0.009 0.010 0.003 Cancel 0.085 0.163 0.092 0.115 0.065 0.076 0.088 0.134 0.091 0.103 0.101 0.091 0.029 Spread # -Inf - 0.00 0.00 - 0.02 0.02 - 0.04 0.04 - 0.06 0.06 - 0.08 0.08 - 0.10 0.10 - 0.12 0.12 - 0.14 0.14 - 0.16 0.16 - 0.18 0.18 - Inf 4350 71959 54206 20775 12210 5392 3813 2525 1638 1026 3196 -Inf - 0.00 0.00 - 0.02 0.02 - 0.04 0.04 - 0.06 0.06 - 0.08 0.08 - 0.10 0.10 - 0.12 0.12 - 0.14 0.14 - 0.16 0.16 - 0.18 0.18 - Inf 4028 73966 52862 19745 11392 5037 3519 2525 1571 938 3028 Limit Market Buy Orders 0.594 0.282 0.434 0.427 0.624 0.269 0.708 0.204 0.727 0.195 0.800 0.131 0.814 0.123 0.804 0.130 0.833 0.096 0.856 0.080 0.885 0.063 Sell Orders 0.615 0.256 0.465 0.425 0.611 0.285 0.680 0.227 0.704 0.204 0.765 0.152 0.778 0.142 0.768 0.133 0.792 0.129 0.800 0.114 0.817 0.097 Change Cancel 0.129 0.123 0.110 0.106 0.102 0.094 0.084 0.078 0.066 0.053 0.057 0.124 0.139 0.107 0.087 0.078 0.070 0.063 0.067 0.070 0.064 0.052 0.091 0.131 0.125 0.123 0.130 0.120 0.120 0.105 0.094 0.079 0.067 0.128 0.110 0.104 0.093 0.092 0.083 0.080 0.098 0.079 0.086 0.086 Table E.9: Submissions Conditional on Spread 113 Midquote-volatility -Inf- 0.005 0.005 - 0.010 0.010 - 0.015 0.015 - 0.020 0.020 - 0.025 0.025 - 0.030 0.030 - 0.035 0.035 - 0.040 0.040 - 0.045 0.045 - 0.050 0.050 - Inf -Inf- 0.005 0.005 - 0.010 0.010 - 0.015 0.015 - 0.020 0.020 - 0.025 0.025 - 0.030 0.030 - 0.035 0.035 - 0.040 0.040 - 0.045 0.045 - 0.050 0.050 - Inf # Limit Market B u y Orders 34260 0.556 0.354 16280 0.352 0.545 15540 0.551 0.338 12312 0.541 0.343 11745 0.540 0.342 8137 0.542 0.335 6619 0.544 0.330 5773 0.526 0.337 4558 0.541 0.322 3887 0.536 0.335 33639 0.302 0.548 Sell Orders 34482 0.525 0.386 16797 0.384 0.515 16113 0.526 0.371 0.542 12493 0.349 12146 0.538 0.358 8397 0.564 0.323 6834 0.569 0.327 5770 0.557 0.340 4407 0.579 0.310 3980 0.570 0.314 0.584 32165 0.295 Change Cancel 0.111 0.138 0.131 0.129 0.119 0.118 0.120 0.104 0.120 0.112 0.118 0.089 0.103 0.111 0.116 0.118 0.123 0.126 0.137 0.137 0.128 0.150 0.141 0.159 0.141 0.139 0.132 0.120 0.132 0.125 0.123 0.128 0.122 0.089 0.101 0.103 0.109 0.104 0.113 0.105 0.103 0.111 0.117 0.121 Table E.10: Submissions Conditional on Midquote Volatility 114 Recent Volume # 0 - 1000 1000 - 5000 5000 - 10000 10000 - 15000 15000 - 20000 20000 - 25000 25000 - 30000 30000 - 35000 35000 - 40000 40000 - Inf 26024 28859 21866 13933 9120 6612 4873 4133 3066 34264 0 - 1000 1000 - 5000 5000 - 10000 10000 - 15000 15000 - 20000 20000 - 25000 25000 - 30000 30000 - 35000 35000 - 40000 40000 - Inf 27009 28675 21036 13293 9013 6319 4984 4092 3092 36071 Limit Market Buy Orders 0.577 0.316 0.554 0.345 0.564 0.333 0.554 0.334 0.543 0.347 0.334 0.551 0.524 0.349 0.532 0.338 0.538 0.333 0.510 0.336 Sell Orders 0.530 0.373 0.537 0.373 0.551 0.356 0.560 0.336 0.565 0.328 0.554 0.334 0.567 0.330 0.574 0.314 0.334 0.549 0.547 0.324 Change Cancel 0.126 0.132 0.126 0.128 0.124 0.123 0.123 0.124 0.114 0.098 0.107 0.101 0.103 0.112 0.111 0.115 0.127 0.130 0.129 0.154 0.150 0.157 0.147 0.141 0.144 0.130 0.123 0.129 0.117 0.100 0.097 0.090 0.092 0.104 0.107 0.113 0.103 0.112 0.117 0.129 Table E . l l : Submissions Conditional on Recent Volume 115 Distance # -Inf - 0.00 0.00 - 0.01 0.01 - 0.02 0.02 - 0.03 0.03 - 0.04 0.04 - 0.05 0.05 - 0.06 0.06 - 0.07 0.07 - 0.08 0.08 - 0.09 0.09 - Inf 46363 6051 26604 17931 13368 9865 7033 5958 4761 3300 19653 -Inf - 0.00 0.00 - 0.01 0.01 - 0.02 0.02 - 0.03 0.03 - 0.04 0.04 - 0.05 0.05 - 0.06 0.06 - 0.07 0.07 - 0.08 0.08 - 0.09 0.09 - Inf 49271 6641 25294 17233 13237 9519 6674 5755 4593 3179 18798 Filled Changed Buy Orders 0.825 0.096 0.633 0.135 0.588 0.126 0.491 0.136 0.428 0.141 0.390 0.146 0.354 0.140 0.347 0.139 0.307 0.144 0.267 0.148 0.214 0.130 Sell Orders 0.828 0.102 0.660 0.131 0.608 0.130 0.509 0.148 0.461 0.155 0.406 0.163 0.358 0.173 0.359 0.164 0.325 0.175 0.174 0.299 0.225 0.166 Canceled Expired 0.041 0.171 0.172 0.191 0.211 0.203 0.190 0.186 0.182 0.183 0.175 0.046 0.083 0.136 0.207 0.250 0.292 0.345 0.362 0.398 0.441 0.511 0.026 0.138 0.145 0.169 0.178 0.188 0.188 0.189 0.191 0.206 0.215 0.047 0.088 0.135 0.198 0.233 0.274 0.316 0.321 0.342 0.357 0.436 Table E.12: Outcome Conditional on Distance from Execution at Submission 116 Midquote # -Inf - 0.00 0.00 - 0.01 0.01 - 0.02 0.02 - 0.03 0.03 - 0.04 0.04 - 0.05 0.05 - 0.06 0.06 - 0.07 0.07 - 0.08 0.08 - 0.09 0.09 - Inf 3812 11792 50199 29927 18479 11763 7196 6398 4864 2610 13847 -Inf - 0.00 0.00 - 0.01 0.01 - 0.02 0.02 - 0.03 0.03 - 0.04 0.04 - 0.05 0.05 - 0.06 0.06 - 0.07 0.07 - 0.08 0.08 - 0.09 0.09 - Inf 3512 13527 52311 29301 18078 11144 6773 5894 4450 2377 12827 Filled Changed B u y Orders 0.467 0.168 0.689 0.100 0.624 0.105 0.556 0.119 0.503 0.136 0.482 0.137 0.458 0.137 0.471 0.141 0.429 0.144 0.394 0.171 0.358 0.153 Sell Orders 0.521 0.138 0.692 0.102 0.634 0.118 0.577 0.133 0.144 0.549 0.506 0.153 0.482 0.160 0.472 0.166 0.439 0.173 0.422 0.180 0.358 0.191 Canceled Expired 0.249 0.124 0.136 0.142 0.156 0.145 0.144 0.139 0.140 0.149 0.146 0.168 0.104 0.154 0.203 0.228 0.259 0.283 0.272 0.312 0.314 0.368 0.183 0.109 0.122 0.129 0.134 0.142 0.129 0.139 0.140 0.143 0.149 0.185 0.112 0.143 0.180 0.195 0.223 0.252 0.252 0.276 0.280 0.331 Table E.13: Outcome Conditional on Spread at Submission 117 Past Vol # 0 - 1000 1000 - 5000 5000 - 10000 10000 - 15000 15000 - 20000 20000 - 25000 25000 - 30000 30000 - 35000 35000 - 40000 40000 - 45000 45000 - 50000 50000 - Inf 39876 28168 20951 13224 8751 6272 4577 3774 2970 2614 2368 27342 0 - 1000 1000 - 5000 5000 - 10000 10000 - 15000 15000 - 20000 20000 - 25000 25000 - 30000 30000 - 35000 35000 - 40000 40000 - 45000 45000 - 50000 50000 - Inf 38173 27784 20195 12687 8596 5969 4750 3879 3044 2623 2427 30067 Filled Changed Buy Orders 0.430 0.131 0.558 0.127 0.561 0.128 0.560 0.128 0.572 0.125 0.573 0.126 0.122 0.589 0.580 0.123 0.591 0.121 0.604 0.111 0.582 0.125 0.622 0.107 Sell Orders 0.452 0.148 0.574 0.143 0.589 0.142 0.596 0.140 0.142 0.590 0.605 0.131 0.605 0.124 0.597 0.134 0.612 0.128 0.618 0.124 0.609 0.129 0.630 0.117 Canceled Expired 0.129 0.118 0.124 0.136 0.144 0.139 0.152 0.166 0.175 0.171 0.183 0.196 0.331 0.217 0.208 0.198 0.180 0.183 0.159 0.161 0.140 0.141 0.132 0.100 0.121 0.107 0.109 0.120 0.126 0.132 0.143 0.157 0.149 0.141 0.167 0.170 0.300 0.194 0.179 0.165 0.163 0.152 0.148 0.139 0.133 0.138 0.120 0.105 Table E.14: Outcome Conditional on Recent Volume at Submission 118
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Analysis of limit order markets Slive, Joshua 2002
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Title | Analysis of limit order markets |
Creator |
Slive, Joshua |
Date Issued | 2002 |
Description | This dissertation consists of three essays analyzing the strategies of traders who participate in limit order markets from both theoretical and empirical perspectives. In the first essay, I examine the implications of asymmetric information on the strategies of traders facing a multi-period limit order market. In the second essay, we examine liquidity provision on a limit order market using a simple order-choice model where traders with extreme liquidity needs place market orders and traders with less extreme liquidity needs place limit orders or stay out of the market. In the third essay, I solve the dynamic problem of an investor in a limit order market who manages his order over the trading day. |
Extent | 6320036 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-11-11 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0099720 |
URI | http://hdl.handle.net/2429/14739 |
Degree |
Doctor of Philosophy - PhD |
Program |
Business Administration - Finance |
Affiliation |
Business, Sauder School of Finance, Division of |
Degree Grantor | University of British Columbia |
Graduation Date | 2002-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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