UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Tests of the black scholes option pricing model Chew, Boon Yong 1978

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1978_A45 C49.pdf [ 3.08MB ]
Metadata
JSON: 831-1.0094470.json
JSON-LD: 831-1.0094470-ld.json
RDF/XML (Pretty): 831-1.0094470-rdf.xml
RDF/JSON: 831-1.0094470-rdf.json
Turtle: 831-1.0094470-turtle.txt
N-Triples: 831-1.0094470-rdf-ntriples.txt
Original Record: 831-1.0094470-source.json
Full Text
831-1.0094470-fulltext.txt
Citation
831-1.0094470.ris

Full Text

TESTS OF THE BLACK SCHOLES OPTION PRICING MODEL by BOON YONG jlCHEW B.Acc.(Hons.), U n i v e r s i t y of Singapore, 1975 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF BUSINESS ADMINISTRATION i n THE FACULTY OF GRADUATE STUDIES (Faculty of Commerce and Business A d m i n i s t r a t i o n ) We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1978 |c) Boon Yong Chew, 1978 In p r e s e n t i n g t h i s t h e s i s in 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 at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e that 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 ag ree 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 p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that 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 o f Commerce and Business A d m i n i s t r a t i o n The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date A p r i l 24, 1978 i i To my Mother i i i ABSTRACT Black and Scholes developed the f i r s t Option P r i c i n g Model based on observable v a r i a b l e s . This model was subsequently extended by Merton, Cox and Ross, Schwartz and others. In the p a s t , e m p i r i c a l s t u d i e s using the Black-Scholes option p r i c i n g model have obtained f a i r l y s a t i s f a c t o r y r e s u l t s . However, these t e s t s have e i t h e r assumed that d i s c r e t e hedging w i l l not s i g n i f i c a n t l y a f f e c t the r e s u l t s i n any way or that i t causes un-c e r t a i n returns which could be d i v e r s i f i e d away. This paper shows that the use of d i s c r e t e hedging w i l l r e s u l t i n a s i g n i f i c a n t b i a s i n the excess re t u r n s . As such a b i a s i s shown to be a f u n c t i o n of the d i s t r i b u t i o n of the r a t e of r e t u r n on the stock, there i s a p o s s i b i l i t y that the covariance between the excess r e t u r n on a hedged p o s i t i o n and that of the market are not zero. This i m p l i e s the existence of systematic r i s k which could not be d i v e r s i f i e d away. Tests of the Montreal Stock Exchange's option market were al s o c a r r i e d out. These t e s t s were subjected to c e r t a i n s t a t i s t i c a l problems as assumptions of the r e g r e s s i o n model used were v i o l a t e d . Despite these v i o l a t i o n s , the r e s u l t s i n d i c a t e that p r o f i t o p p o r t u n i t i e s do e x i s t i n the market. However, i t i s d o u b t f u l that such p r o f i t o p p o r t u n i t i e s would s t i l l e x i s t i f t r a n s a c t i o n costs e t c . are taken i n t o c o n s i d e r a t i o n . i v i CONTENTS Page 1. I n t r o d u c t i o n • 1 2. Theory of Option P r i c i n g 2.1 The Black-Scholes Model 2 2.2 Extensions of the Black-Scholes Model 5 3. P r e v i o u s E m p i r i c a l Tests and R e s u l t s 9 4. E f f e c t of D i s c r e t e Hedging on Excess Returns 21 5. The Canadian Option Market 29 6. Data 34 7. Tests and R e s u l t s 7.1 Performance of the Black-Scholes Model 36 7.2 E f f i c i e n c y of the Montreal Option Market 42 8. Conclusions 49 Appendix V APPENDIX Page I. Histograms of the Rates of Return f o r S h e l l Canada 51 I I . L i s t i n g of Underlying Stocks Whose Options are Traded on the Montreal Options Market as at 31st December 1976 . . . 55 I I I . Inadequacy of the I n t e r e s t Rate Adjustment Factor 56 v i LIST OF TABLES Table Page I Returns on the Hedged P o s i t i o n s Using a B u y a l l Strategy (Constant Variance) 38 I I Kolmogorov-Smirnoff Goodness of F i t Test on Excess Return. . 40 I I I Returns on the P o r t f o l i o s Using a B u y a l l Strategy (Constant Variance) 38 IV Returns on the Hedged P o s i t i o n s Using a B u y a l l Strategy ( D a i l y Adjusted Variances) 41 V Returns on the P o r t f o l i o s Using a B u y a l l Strategy ( D a i l y Adjusted Variance) 41 VI Returns on the Hedged P o s i t i o n s Using a Buy and S e l l Strategy ( D a i l y Adjusted Variance) . . . 45 VI I Returns on the P o r t f o l i o s Using a Buy and S e l l Strategy (Dai l y Adjusted Variance) 45 LIST OF FIGURES I R e l a t i o n s h i p Between Stock and Option P r i c e 22 I I Histogram of Excess Returns f o r a Log-Normal D i s t r i b u t i o n . . 27 v i i ACKNOWLEDGEMENT This t h e s i s would not have been p o s s i b l e without the a s s i s t a n c e of v a r i o u s members of the F a c u l t y of Commerce and Business A d m i n i s t r a t i o n , U n i v e r s i t y of B r i t i s h Columbia. In p a r t i c u l a r , the candidate i s g r a t e -f u l to Pr o f e s s o r Phelim P. Boyle f o r h i s c o n s i s t e n t encouragement. As chairman of the committee, h i s guidance has been i n d i s p e n s a b l e . The candidate expresses h i s g r a t i t u d e to Professo r Eduardo S. Schwartz f o r v a l u a b l e d i s c u s s i o n and suggestions. P r o f e s s o r Kent Brothers' a s s i s t a n c e i n the area of s t a t i s t i c a l techniques i s very much appreciated. As members of the committee, Profes s o r Derek A t k i n s and Professo r James C.T. Mao read the t h e s i s and o f f e r e d h e l p f u l suggestions towards i t s improvement. The candidate extends h i s a p p r e c i a t i o n to Pro f e s s o r A l a n Kraus, P r o f e s s o r David Emanuel, Profess o r Rolf Banz and Mr. A.L. Ananthanarayanan f o r t h e i r h e l p f u l comments and opinions. L a s t l y , the candidate i s g r a t e f u l to Colleen Colclough f o r p a t i e n t l y t y p i n g t h i s paper d e s p i t e numerous a l t e r a t i o n s . 1. INTRODUCTION An o p t i o n i s a contract to buy ( c a l l options) or s e l l (put option) a s p e c i f i e d number of shares i n a given time p e r i o d at an agreed p r i c e . An 'American' op t i o n i s one that can be e x e r c i s e d at any time up to the date the o p t i o n e x p i r e s . An 'European' op t i o n on the other hand i s one wi t h the e x e r c i s e r i g h t l i m i t e d to a s p e c i f i e d f u t u r e date. As put and c a l l options are the simplest forms of contingent c l a i m s , i t i s g e n e r a l l y b e l i e v e d that a study of these w i l l provide i n s i g h t s i n t o more complex con-tingent claims s i t u a t i o n s . In 1973, Black and Scholes [2] published a paper which represented a major breakthrough i n t h i s area. Since then, a s u b s t a n t i a l amount of r e -search has been done i n t h i s f i e l d . Most of t h i s research has been of a t h e o r e t i c a l nature. The few stu d i e s d e a l i n g w i t h e m p i r i c a l t e s t s have a l l used American Data. The purpose of t h i s paper i s twofold; F i r s t , to t e s t the p e r f o r -mance of the Black-Scholes option p r i c i n g model using data from the Mon-t r e a l Option Market and second to t e s t the e f f i c i e n c y of that market. E f f i c i e n c y r e f e r s to the p o s s i b i l i t i e s of earning higher than normal r e -turns a f t e r t a k i n g i n t o c o n s i d e r a t i o n the r i s k taken. Section 2 introduces the Black-Scholes model and discusses some sub-sequent c r i t i c i s m s and extensions. S e c t i o n 3 presents some of the major e m p i r i c a l t e s t s . Section 4 analyses the e f f e c t of d i s c r e t e hedging on excess r e t u r n s . Section 5 describes the main features of the Montreal Option Market. A d e s c r i p t i o n of the data used i s given i n Section 6. Sec-t i o n 7 discusses the t e s t s performed and the r e s u l t s and Secti o n 8 presents some conclusions and suggestions f o r f u r t h e r research. - 2 -2. THEORY 2.1 The Black-Scholes Model: The e a r l i e s t o p t i o n ^ v a l u a t i o n formula was derived by B a c h e l i e r (1900). Since then, there have been papers by Sprenkle (1962), Boness (1964) and Sameulson (1965) and others. A good survey of these e a r l i e r r e s u l t s o u t l i n i n g t h e i r strengths and l i m i t a t i o n s i s given i n Smith [11]. Black and Scholes [ 2 ] present the f i r s t complete op t i o n p r i c i n g mod-e l ' which depends only on observable v a r i a b l e s . Their model i s developed on the p r i n c i p l e that i n e q u i l i b r i u m c o n d i t i o n s assets which are s i m i l a r would y i e l d the same r a t e of r e t u r n . In d e r i v i n g t h e i r formula f o r the value at time t,, of a European op t i o n which matures at time t * , the f o l l o w -in g assumptions were made: (1) The stock p r i c e f o l l o w s a Geometric Brownian motion through time, i . e . ~ = Pdt + adz (2.1) where S i s the value of the stock, ydt i s the d r i f t or d e t e r m i n i s t i c term and adz i s the s t o c h a s t i c term w i t h dz f o l l o w i n g a normal d i s t r i b u t i o n w i t h mean zero and vari a n c e dt. (2) The short term r i s k f r e e i n t e r e s t r a t e i s known and i s con-stant through time. (3) The stock pays no dividends or other d i s t r i b u t i o n s during the l i f e of the options. (4) The opt i o n i s European, i . e . i t can only be ex e r c i s e d at date of matur i t y . (5) There are no t r a n s a c t i o n costs or taxes. Borrowing and lending i n t e r e s t r a t e s are equal and that there are no ^ Options r e f e r r e d to i n t h i s paper are c a l l options. - 3 -r e s t r i c t i o n s on s h o r t - s e l l i n g . With these assumptions, the valu e of the options can be expressed as a f u n c t i o n of S, the stock p r i c e and T,, the time to m a t u r i t y . (T = t * - t ) . I t i s then p o s s i b l e to create a long (short) p o s i t i o n i n the stock and a short (long) p o s i t i o n i n the op t i o n i n such a way that p r o f i t s ( l o s ses) as a r e s u l t of a r i s e ( f a l l ) i n stock p r i c e s w i l l be o f f -set by lo s s e s ( p r o f i t s ) as a r e s u l t of the short (long) p o s i t i o n i n options. Consequently, the value of t h i s hedged p o r t f o l i o w i l l not depend on the p r i c e of the stock but on T, the time to ma t u r i t y and other con-s t a n t s . I f we w r i t e the v a l u e of the op t i o n as C(,S,T), then changes i n the stock p r i c e dS w i l l change the value of the o p t i o n by approximately V s where Cg i s the p a r t i a l d e r i v a t i v e of C(S.T) w i t h respect to S. I f we were to e s t a b l i s h a hedged p o r t f o l i o such that f o r one share of stock long, we s e l l number of options short then the change i n the value of the hedged p o r t f o l i o as a r e s u l t of a change i n stock p r i c e of dS w i l l be dS - - J - C Q dS - 0 (2.3) By a d j u s t i n g the hedged p o r t f o l i o c o n t i n u o u s l y , the r e t u r n on the hedged p o r t f o l i o w i l l then be non-stochastic and t h e r e f o r e r i s k l e s s . Consequent-l y the p o r t f o l i o must earn the r i s k l e s s r a t e of r e t u r n . In g e n e r a l , i f the value of a hedged p o r t f o l i o c o n s i s t i n g of one share of stock long and jr~ options short i s V = S - -C (2.4) where C i s the value of one option. The change i n the value of the p o r t -f o l i o i n a short i n t e r v a l dt i s dV = dS - ^- dC (2.5) - 4 -In e q u i l i b r i u m c o n d i t i o n s , the r a t e of r e t u r n on V should be equal to the r i s k f r e e r a t e . Therefore |^ = r d t C2 .6) S u b s t i t u t i n g ( 2 . 4 ) and (.2.5) i n t o ( 2 . 6 ) we have dS - ±~ dC = (S - C) r d t (2.7) °S CS Using s t o c h a s t i c c a l c u l u s to expand dC and s i m p l i f y i n g , Black and Scholes found that C T = rC - r S C s - h C g s a2 S 2 (2.8) which defines a d i f f e r e n t i a l equation f o r the values of the o p t i o n subject to the boundary c o n d i t i o n that C(S , 0 ) = Max [0,S-'E] (2.9) This c o n d i t i o n s t a t e s that at m a t u r i t y date the o p t i o n p r i c e must be equal to the maximum of e i t h e r the d i f f e r e n c e between the stock p r i c e and the e x e r c i s e p r i c e or zero. Transforming the d i f f e r e n t i a l equation (2.8) i n t o the heat exchange equation from physics and s o l v i n g , Black and Scholes a r r i v e d at the o p t i o n v a l u a t i o n formula: C(S,t) = S.N(d 1) - X e ~ r T N(d 2) where N (d ) = ln(S/X) + (r + q 2 / 2)T ( 2 . 1 0 ) cr/T - . d 2 = d l " C = p r i c e of the o p t i o n f o r a s i n g l e share of stock S = the current p r i c e of the stock X = the s t r i k i n g or e x e r c i s e p r i c e of the o p t i o n r = short term i n t e r e s t r a t e ( r i s k l e s s ) - 5 -= v a r i a n c e of the r a t e of r e t u r n on stock T = time to m a t u r i t y N(d^) = cumulative nominal d e n s i t y f u n c t i o n 2 Equation (2.8) i s a f u n c t i o n of the v a r i a b l e s , r , S, T, a , X. Cox and Ross [sf ] have noted that i n d e r i v i n g i t , no assumptions were made w i t h regard to the preferences of the i n v e s t o r s except that two assets which are p e r f e c t s u b s t i t u t e s should earn the same r a t e of r e t u r n . This i m p l i e s that the s o l u t i o n obtained by s o l v i n g f o r equation (2.8) w i l l hold f o r any preference s t r u c t u r e . Therefore, i f we assume that the market i s composed of r i s k - n e u t r a l i n v e s t o r s , then the e q u i l i b r i u m r a t e of r e t u r n f o r the stock p r i c e and the o p t i o n should be equal to the r i s k l e s s r a t e of r e -t u r n , r . The o p t i o n p r i c e w i l l then be the discounted expected op t i o n p r i c e at the date of m a t u r i t y . The expected op t i o n value at m a t u r i t y can be obtained by i n t e g r a t i n g over the d i s t r i b u t i o n of stock p r i c e s at mat-u r i t y to o b t a i n S e r T N(d x) - XN(d 2) (2.11) -rT M u l t i p l y i n g (2.10) by the discount f a c t o r of e , we o b t a i n equation (2.9). 2.2 Extension of the Black-Scholes Model The b a s i c Black-Scholes model has been extended i n a number of ways by modifying the u n d e r l y i n g assumptions. (2) With no dividend payment, Merton [ 8] has shown that p r i o r to e x p i r a t i o n , the holder of an American op t i o n w i l l r a t h e r s e l l than e x e r c i s e the o p t i o n . This i m p l i e s that an American c a l l w i l l never be Refer to Merton [ 8 ] f o r a more d e t a i l e d d i s c u s s i o n . - 6 -ex e r c i s e d before i t s maturity date and thus w i l l be valued at the same p r i c e as a European c a l l . Therefore, the Black-Scholes model could be used to value American options on non-dividend paying stocks. When dividends are paid and assuming that D i s the continuous dividend per share per u n i t time, Merton [8] a r r i v e d at the f o l l o w i n g d i f f e r e n t i a l equation f o r the value of an op t i o n : where s u b s c r i p t s denote p a r t i a l d e r i v a t i v e s . I f the o p t i o n i s European, equation (2.12) i s subject to the boundary c o n d i t i o n A general c l o s e d formed s o l u t i o n f o r equation (2.12) subject to boundary c o n d i t i o n s (2.9) has not been found. Only i n one p a r t i c u l a r case has a s o l u t i o n been found f o r the value of a European c a l l w i t h a f i n i t e time to e x p i r a t i o n . This i s the case when un d e r l y i n g stock i s assumed to pay a constant dividend y i e l d of p. i . e . D = pS. S u b s t i t u t i n g D = pS i n equation (2.12) and s o l v i n g Merton a r r i v e d at the f o l l o w i n g formula f o r the European option or dividend paying However, w i t h d i v i d e n d payments, the value of the American o p t i o n i s no longer equal to the value of the European op t i o n because of the p o s s i b i -l i t y of premature e x e r c i s e . Therefore equation (2.13) i s not a p p l i c a b l e to the American o p t i o n . (2.12) C(S,0) = Max[0,S - E] (2.9) stock; (2.13) - 7 -To a r r i v e at the equation f o r the value of the American o p t i o n , i t i s necessary to account f o r the p o s s i b i l i t y of premature e x e r c i s e . Equation (2.12) w i l l be subject to the f o l l o w i n g a d d i t i o n a l boundary c o n d i t i o n : Assuming that the stock p r i c e w i l l drop by the amount of the d i v i -dend immediately a f t e r the stock goes ex-dividend, the value of the o p t i o n f o r the ex-dividend stock p r i c e must be greater than the e x e r c i s e p r i c e cum-dividend, because, otherwise i t w i l l be e x e r c i s e d . No c l o s e d form s o l u t i o n has been found f o r t h i s case. However, Schwartz [10] s u c c e s s f u l l y employed numerical i n t e g r a t i o n to s o l v e t h i s equation and a r r i v e d at an optimal e x e r c i s e s t r a t e g y . Boyle [ 4 ] adopted a d i f f e r e n t approach whereby the d i s t r i b u t i o n of the stock p r i c e at m a t u r i t y date can be generated u s i n g Monto Carlo s i m u l a t i o n techniques. The advantage of such a technique i s the ease at which dividends can be taken i n c o n s i d e r a t i o n . Furthermore, a change i n the u n d e r l y i n g d i s t r i b u t i o n of stock p r i c e s can be e f f e c t e d by changing a d i f f e r e n t process of generating the random v a r i a b l e s employed. The method was used to evaluate a h y p o t h e t i c a l options using 5,000 t r i a l s . These r e s u l t s were compared to that which were a r r i v e d at using numerical i n t e g r a t i o n and i n a l l cases the 95%. confidence i n t e r v a l contains the true answer. So f a r , stock p r i c e s are assumed to f o l l o w a log-normal d i f f u s i o n process. However, Cox and Ross [ 5 ] argued that new i n f o r m a t i o n tends to a r r i v e i n the market i n d i s c r e t e lumps r a t h e r than a smooth flow. Therefore, stock p r i c e s can be more a c c u r a t e l y described as f o l l o w i n g a jump r a t h e r than a d i f f u s i o n process. Assuming that the percentage change i n the stock p r i c e from t to t + dt i s Xdt k-1 0 (2.14) 1-Xdt where k-1 = the jump amplitude p = d r i f t term Adt = p r o b a b i l i t y of a jump i n the time i n t e r v a l dt they were able to form a hedged p o s i t i o n w i t h the stock, the option and the r i s k f r e e bond.and a r r i v e at a complicated formula f o r the option p r i c e independent of A. - 9 -3. EMPIRICAL TESTS To date, the only major e m p i r i c a l t e s t s of the Black-Scholes o p t i o n p r i c i n g model have been performed by Black and Scholes i n 1971 and G a l a i i n 1975. As i t i s intended to r e p l i c a t e some of these t e s t s , the procedures employed and the r e s u l t s obtained are discussed i n con s i d e r a b l e d e t a i l . Black-Scholes Tests The f i r s t e m p i r i c a l t e s t s of the model were performed by Black and Scholes [ 1 ] , who obtained t h e i r data from the d i a r i e s of an op t i o n broker from 1966 to 1969. At the w r i t i n g date of the c a l l o p t i o n , the model was used to compute i t s t h e o r e t i c a l v a l u e as w e l l as the number of shares to (3) balance against each o p t i o n , i . e . N(d^). I f the market p r i c e was greater ( l e s s ) than the model p r i c e the option was considered overvalued (undervalued). Four p o r t f o l i o s were constructed based on the f o l l o w i n g s t r a t e g i e s . a) Buying a l l c a l l s at model p r i c e s b) Buying a l l c a l l s at market p r i c e s c) Buying undervalued c a l l s and s e l l i n g overvalued c a l l s at model p r i c e s d) Buying undervalued c a l l s and s e l l i n g overvalued c a l l s at market p r i c e s . On every purchase (sale) of a c a l l o p t i o n a hedge was e s t a b l i s h e d by s e l l i n g (buying) N(d^) number of shares. Each p o r t f o l i o con-s i s t e d of a number of hedges. However, as the p r o p o r t i o n N(d^) was not adjusted continuously, the hedge w i l l generate an u n c e r t a i n r e t u r n . Black NCd-^ ) i s the d e r i v a t i v e of the Black Scholes o p t i o n p r i c i n g formula w i t h respect to the stock p r i c e . - 10 -and Scholes [2] however have shown that t h i s r e t u r n i s t h e o r e t i c a l l y u n c o r r e l a t e d w i t h the market and could t h e r e f o r e be d i v e r s i f i e d away by h o l d i n g a p o r t f o l i o of hedges. As the market w i l l not compensate the i n v e s t o r s f o r d i v e r s i f i a b l e r i s k , , the r e t u r n on the hedge should be equal to the r i s k l e s s r a t e r . Therefore dC - C c dS = (C - C S) r A t (.3.1) The r e a l i z e d excess d o l l a r r e t u r n i s t h e r e f o r e defined as dC - CgdS - (C - C gS) rAt (3.2) The p e r i o d from May 1966 to J u l y 1969 was d i v i d e d i n t o ten sub-periods. For each p o r t f o l i o the r e a l i z e d excess d o l l a r r e t u r n s were aggregated d a i l y to form a t o t a l d a i l y p o r t f o l i o r e t u r n . A r e g r e s s i o n model of the form R = a + B R + e (3.3) pt p p mt . t where R = Return on Standard and Poor Composite Index on mt c day t R = T o t a l p o r t f o l i o r e t u r n on day t Pt a = The i n t e r c e p t a. I t s s i g n i f i c a n c e i s taken to be a measure of the performance of the model f o r the r e s p e c t i v e p o r t f o l i o s , g = Slope c o e f f i c i e n t ez = R e s i d u a l . was used f o r each of the ten sub periods and the t o t a l p e r i o d f o r each p o r t f o l i o . The r e s u l t s obtained could be summarized as f o l l o w s : - 11 -1) As expected the 3 c o e f f i c i e n t s were i n s i g i n f i c a n t l y d i f f e r e n t from zero i i ) For the f i r s t two s t r a t e g i e s where the cont r a c t s were purchased at model and again at market p r i c e s , the excess p o r t f o l i o returns were i n s i g n i f i c a n t l y d i f f e r -ent from zero i i i ) When undervalued c o n t r a c t s were bought and overvalued c o n t r a c t s were s o l d at model p r i c e s , the a's were found to be s i g n i f i c a n t l y negative. However, when t h i s s t r a t e g y was performed w i t h market p r i c e s , the a's were s i g n i f i c a n t l y p o s i t i v e . In other words, buy-ing and s e l l i n g at model p r i c e s w i l l i n c u r s i g n i f i -cant l o s s e s while buying and s e l l i n g at market p r i c e s r e s u l t s i n s i g n i f i c a n t gains. Black and Scholes explained the f i n d i n g s i n terms of the measure-ment e r r o r s of past v a r i a n c e s . Estimates of true variances using past variances r e s u l t i n a wider spread i n the d i s t r i b u t i o n of the estimated variance than that of the true v a r i a n c e . Consequently, high variances w i l l be over-estimated w h i l e low variances w i l l be under-estimated. There-f o r e the model tends to o v e r p r i c e options on high variance stock and underprice options on low variance stock. However, the reverse i s true i n the case of the options t r a d e r , i . e . the e s t i m a t i o n of the spread of the d i s t r i b u t i o n of variances i s smaller than what i t should be. Therefore, the market w i l l tend to underprice options of high variance stock and ov e r p r i c e options on low variance stock. I f the true o p t i o n p r i c e i s between the model and the market p r i c e , then buying and s e l l i n g at model p r i c e s r e s u l t s i n buying high and s e l l i n g low thus e x p l a i n i n g the s i g n i f i -- 12 -c a n t l y negative r e t u r n s . Using market p r i c e s r e s u l t s i n the opposite e f f e c t . To t e s t t h e i r hypothesis Black and Scholes d i v i d e d a l l the stocks i n t o 4 p o r t f o l i o s . The f i r s t p o r t f o l i o c o n s i s t e d of stocks which have the lowest 25% of the v a r i a n c e and so on. Contracts were bought and s o l d as before using model and market p r i c e s . R e a l i z e d excess p o r t f o l i o r e t u rns were c a l c u l a t e d and regressed against the market. The r e s u l t s c l e a r l y showed that the model over-estimates the values of options on h i g h v a r i a n c e s e c u r i t i e s and under-estimates the values of options on low v a r i a n c e s e c u r i t i e s . The reverse was t r u e f o r the market. Thus i t would be p o s s i b l e to make money by f o l l o w i n g a s t r a t e g y of s e l l i n g c o n t r a c t s on low v a r i a n c e s e c u r i t y and by buying c o n t r a c t s on high v a r i a n c e s e c u r i t i e s . However, Black-Scholes found that t h i s p r o f i t would be e l i m i n a t e d as a r e s u l t of t r a n s a c t i o n costs of t r a d i n g i n the current o p t i o n market. To demonstrate that the model could p r i c e c o r r e c t l y i f i t has the proper v a r i a n c e , Black and Scholes repeated the above t e s t s using variances c a l -c u l a t e d w i t h the a c t u a l stock p r i c e s over the l i f e of the o p t i o n . The r e s u l t s f o r the p o r t f o l i o using the f i r s t two s t r a t e g i e s were as before i n s i g n i f i c a n t . However, i n c o n t r a s t to previous t e s t s , the r e s u l t s when co n t r a c t s were bought and s o l d at model p r i c e s were i n s i g n i f i c a n t w h i l e t r a d i n g at market p r i c e s r e s u l t e d i n a very s i g n i f i c a n t p o s t i v e r e t u r n . These t e s t s are i n a way biased against the market but nevertheless show that the model w i l l perform w e l l given the c o r r e c t v a r i a n c e . 3.2 G a l a i ' s Tests The second set of e m p i r i c a l t e s t s on the Black-Scholes o p t i o n p r i c -i n g model was performed by G a l a i [6] on the Chicago Board Options Exchange. Using data from A p r i l 20, 1973 to November 30, 1973, G a l a i d i v i d e d h i s - 13 -t e s t s i n t o ex post and ex ante t e s t s . Ex post t e s t s assume that i t i s p o s s i b l e to trade on a day's c l o s i n g p r i c e s a f t e r the t r a d i n g r u l e has been based on that p r i c e . Ex ante t e s t s on the other hand assume that t r a n s a c t i o n s can only take place on the next day's c l o s i n g p r i c e s . Ex Post Tests: I n i t i a l l y G a l a i d u p l i c a t e d the Black-Scholes t e s t s by maintaining the i n i t i a l o ption p o s i t i o n and a d j u s t i n g the hedge through changes i n stock p o s i t i o n . Returns on the hedge are c a l c u l a t e d i n two ways. The f i r s t method i s c a l c u l a t e d as f o l l o w s ; on the day a f t e r the hedged p o s i t i o n was e s t a b l i s h e d (3.4) *Ht = < c t M ~ Ct-1> " N«l>t-1 ( S t " Vl> where = Return on the hedge i n p e r i o d t C " = Model opt i o n p r i c e on per i o d t C^_^ = A c t u a l o p t i o n p r i c e on p e r i o d t - 1 , i . e . the day the opti o n was f i r s t traded S = Stock p r i c e at per i o d t . For subsequent days, the model p r i c e s are assumed to be the a c t u a l o p t i o n p r i c e s and *Ht *X " C t - 1 } " N < d l > t - 1 < S t - S t - l > ( 3 ' 5 ) The second method uses a c t u a l o p t i o n p r i c e s f o r the f i r s t as w e l l as subsequent days to c a l c u l a t e R^t, so v - ( c t - <£-i> - N ( d i ) t - i ( s t - s t - i } ( 3 - 6 ) The r e t u r n s on the hedged p o s i t i o n s were t e s t e d and G a l a i reported that they appeared to f i t a normal d i s t r i b u t i o n . The average d a i l y r e t u r n of each hedged p o s i t i o n was then t e s t e d f o r s i g n i f i c a n c e u s i n g a t t e s t . - 14 -The above equations f o r c a l c u l a t i n g r e t u r n s d i f f e r from those used by Black and Scholes (equation (3.2)) by a term (C - C gS) r A t . This i n t e r e s t r a t e f a c t o r was omitted because the e f f e c t was found by G a l a i to be . . -t (4) mmxmal. Using the above t e s t procedure, the average r e t u r n s f o r almost a l l of the hedged p o s i t i o n s using the second method of computation were not s i g n i f i c a n t l y d i f f e r e n t from zero. This i s i n con t r a s t w i t h r e s u l t s ob-tained using the f i r s t method where almost a l l the hedged p o s i t i o n s were found to be s i g n i f i c a n t l y d i f f e r e n t from zero. G a l a i explained these r e s u l t s as f o l l o w s : A c t u a l o p t i o n p r i c e s d i f f e r from the tr u e model p r i c e s by an e r r o r term. This can be w r i t t e n as: C A = St + e ( 3 ' 7 ) = A c t u a l o p t i o n p r i c e C., = Model p r i c e M r e = E r r o r term where E(e) = 0 The v a r i a n c e of r e t u r n s c a l c u l a t e d using the a c t u a l p r i c e s w i l l t h e r e f o r e be greater than those using the model p r i c e . Consequently, most of the t s t a t i s t i c s of the c o e f f i c i e n t s using the model p r i c e s w i l l be d i v i d e d by a smaller v a r i a n c e thus r e s u l t i n g i n t h e i r s i g n i f i c a n c e . The o r i g i n a l B-S t e s t s were performed on the over-the-counter options f o r which there were no secondary markets. Therefore, i t i s necessary to i ) assume model p r i c e s to be a c t u a l p r i c e s a f t e r the f i r s t day i i ) maintain the hedge p o s i t i o n by a d j u s t i n g the stock A 10% r i s k f r e e r a t e compounded d a i l y i s 0.00027401 i f the e q u i t y p o s i -t i o n i s $10. This i n t e r e s t r a t e f a c t o r i s only $0.0027401. - 15 -p o s i t i o n d a i l y However, G a l a i ' s t e s t s were done on the Chicago Board Option Exchange i n which there i s a very a c t i v e secondary market. The adjustment of the hedge by changing stock p o s i t i o n s w i l l ignore the in f o r m a t i o n a v a i l a b l e i n the d a i l y d e v i a t i o n s of a c t u a l o p t i o n p r i c e s from model p r i c e s . There-f o r e the second ex post t e s t was performed by a d j u s t i n g the hedge p o s i t i o n s through changes i n the op t i o n p o s i t i o n . For each day, the model p r i c e C^ was compared to the a c t u a l o p t i o n t A M A p r i c e Cj.. I f Cfc > C t, then the op t i o n would be purchased at the day s a c t u a l c l o s i n g p r i c e s and immediately an amount N(d^) of und e r l y i n g stock s o l d . I f Cfc > C^, then the op t i o n would be s o l d and N(d^) stocks purchased. This hedged p o s i t i o n at t i s assumed to be l i q u i d a t e d at t+1 and the re t u r n i s V - ( c t + i - ct> - N < d i > t < s t + i - st> ( 3 - 8 ) When the hedge was l i q u i d a t e d at t+1, a new hedge was immediately M A set up based on the r e l a t i o n s h i p between C f c +^ and This procedure was repeated and the average r e t u r n s at the end of the pe r i o d was c a l c u -l a t e d . Comparing the r e s u l t s w i t h the f i r s t t e s t , shows that the model i s able to d i f f e r e n t i a t e over and underpriced options very w e l l . The average returns was $10 per option per day versus about $1 i n the case of the f i r s t t e s t s . As a v a r i a t i o n of the above t e s t s , the hedge was not l i q u i d a t e d u n t i l the second, t h i r d or f i f t h days, i . e . i f a p o s i t i o n was taken at day t , i t w i l l not be l i q u i d a t e d u n t i l t=2. Meanwhile at t = l another p o s i t i o n was taken using t+1 p r i c e s . The average r e t u r n s f o r m a i n t a i n i n g the hedge f o r two days i s s t i l l s i g n i f i c a n t though the amount was l e s s than that derived f o r the 1 day ho l d i n g p e r i o d . No attempts were made by - 16 -G a l a i to adjust f o r the c o r r e c t t r a n s a c t i o n costs i n each case. Instead, an a r b i t r a r y percentage of 1% was s e l e c t e d and i t s i n c l u s i o n e l i m i n a t e d the s i g n i f i c a n t r e t u r n s a r r i v e d at i n e a r l i e r t e s t s . The Black-Scholes o p t i o n p r i c i n g model assumes zero d i v i d e n d payment during the l i f e of the options and G a l a i i n performing the above t e s t s ignored the dividend payments. He however test e d the e f f e c t of divide n d payments i n the f o l l o w i n g ways: i ) Four p o r t f o l i o s were constructed w i t h the f i r s t p o r t f o l i o c o n s i s t i n g of a l l options on stocks w i t h the lowest d i v i d e n d y i e l d and the l a s t p o r t f o l i o con-s i s t i n g of a l l options on stocks w i t h the highest dividend y i e l d . The d a i l y average returns on each p o r t f o l i o were regressed against the r e t u r n on Standard and Poor's Index. As the d a i l y average r e -t u r n on p o r t f o l i o i s based on a d i f f e r e n t number of options traded, a problem of h e t e r o s k e d a s t i c i t y was suspected. To overcome t h i s problem a weighted l e a s t square was performed. As i n the B-S case, the i n t e r -cept i n d i c a t e s the average p r o f i t s on a p o r t f o l i o w i t h zero systematic r i s k . The dp f o r the four p o r t f o l i o s were found to be s i g n i f i c a n t l y d i f f e r e n t from zero. However as he moved from the highest dividends p o r t f o l i o to the lowest, the increases. i i ) The returns f o r hedged p o s i t i o n s using equation (3.8) were c a l c u l a t e d f o r a l l hedges where no ex-dividend dates were expected during the period remaining to - 17 -e x p i r a t i o n . The o v e r a l l d a i l y average r e t u r n s was $10.7 per contract and the m a j o r i t y of the hedged p o s i t i o n s had s i g n i f i c a n t average r e t u r n s , i i i ) S o l u t i o n s of the dividend adjusted d i f f e r e n t i a l equa-t i o n using numerical methods are expensive w i t h more advanced computers. G a l a i t h e r e f o r e used a simple adjustment to the Black-Scholes formula. The adjustment i s c a r r i e d out by s u b t r a c t i n g the present value of the expected dividends from the p r i c e of the underlying s e c u r i t y . The o p t i o n p r i c i n g equations become C (V,D . .... D ) = (V - E e: r T D .) N(d ) - X r t N(d ) (3.9) x=l where d l = t W V - ^ e " r t D t.)/X] + (r + |^)T}/o ft d 2 = d x - a ft Unpublished simulated a n a l y s i s by Black shows that t h i s adjustment produces very s i m i l a r r e s u l t s to that a r r i v e d at using numerical i n t e g r a t i o n . Model p r i c e s and N(d^) c a l c u l a t e d using the above equations were then used i n equation (3.8) to a r r i v e at the average r e t u r n s f o r each hedged p o s i t i o n . The r e s u l t s showed that the o v e r a l l d a i l y average r e -turns were higher than those a r r i v e d at using the model without a d j u s t i n g f o r d i vidends. Furthermore, the number of hedged p o s i t i o n s w i t h s i g n i f i -- 18 -cant average returns were greater. Generally the r e s u l t s of the t e s t s support the hypothesis that the model performs b e t t e r i n s i t u a t i o n s i n which the assumptions are b e t t e r met. Ex Ante Tests: Instead of using the c l o s i n g p r i c e s of the same day f o r determining and e s t a b l i s h i n g the hedge, i t i s assumed i n these t e s t s that the tr a d e r had to wait 24 hours before executing the d e s i r e d t r a n s -a c t i o n . On day t the Black-Scholes o p t i o n p r i c i n g formula was used, as before, to determine i f an option i s over or under p r i c e d . However, the hedge i s e s t a b l i s h e d based on t+1 c l o s i n g p r i c e s and l i q u i d a t e d at t+2 c l o s i n g p r i c e s . The r e t u r n on the hedge i s c a l c u l a t e d as ^ + 2 " ( Ct+2 " " N ( V t ( Vt+2 ' Vl> ( 3' 1 0 ) The returns f o r the ex ante t e s t s were s i g n i f i c a n t . However, the averages were lower than those obtained on the ex post t e s t s . Repeating the t e s t s w i t h a h o l d i n g p e r i o d of 2 days r e s u l t s i n s u b s t a n t i a l l y lower r e t u r n s . E f f e c t s of dividends were test e d by repeating the procedures i n the ex post t e s t s . D i v i d i n g the options i n t o 4 p o r t f o l i o s ranged according to the div i d e n d y i e l d s of the un d e r l y i n g s t o c k , provided r e s u l t s s i m i l a r to that of the ex post t e s t s . However, i f a divide n d adjustment was made to the Black-Scholes formula, the r e s u l t was s i g n i f i c a n t l y d i f f e r e n t . In the ex post t e s t adjustment of B-S formula r e s u l t s i n s i g n i f i c a n t l y higher average r e t u r n s , but w i t h the ex ante t e s t s , the average returns were s i m i l a r to those when no adjustments were made. G a l a i explained that "the added accuracy i n determining the hedge r a t i o and the p o s i t i o n i s washed out by del a y i n g execution by one day". Options spreading i s a technique commonly used by t r a d e r s . I t r e f e r s - 19 -to the p o s i t i o n of being long and short on d i f f e r e n t options of the same underlying stock. G a l a i t e s t e d the performance of the model based on the spreading s t r a t e g y by using a spreading r a t i o derived from Black-Scholes Formula. The r a t i o i s derived as f o l l o w s : I f we buy 1 share of stock i , hedging i s done by s e l l i n g 1/N(d^) of opt i o n k. However, we could a l s o e s t a b l i s h another hedge by s e l l i n g 1 share of stock i and buying 1/N(d^_.) of o p t i o n j . Note that o p t i o n k and j have the same und e r l y i n g stock but d i f f e r i n s t r i k i n g p r i c e or m a t u r i t y date. As the one share of stock i i s common i n both hedges, i t i s p o s s i b l e f o r us to a r r i v e at the same r i s k l e s s p o s i t i o n by s e l l i n g 1/N(d^) of o p t i o n k and buying 1/N(d ) of options j or by buying one o p t i o n j and s e l l i n g N(d )/N(d ) l j l j l k of o p t i o n k. On day t , the d e c i s i o n as to whether to buy or s e l l o ption k (or M M M M A A op t i o n j ) w i l l depend on the r a t i o of C. /C. . I f C. /C, > C. /C,. , then we j t k t j t k t 31 kt w i l l buy 1 o p t i o n j and s e l l N(d ,)/N(d ) of opt i o n k. At t+1 t h i s p o s i t i o n l j l k i s l i q u i d a t e d and a new p o s i t i o n based on c l o s i n g p r i c e s at t+1 w i l l be Rs t + i = [ c j t + i - V - i^fy [ c k t + i - c k t ] ( 3 - n ) l k t M M A A In cases where C. /C, < C. /C, , 1 opt i o n i w i l l be s o l d and j t k t j t k t N(d^)/N(d^k) of o p t i o n k purchased. The r e t u r n w i l l then be R s t + i - ?£r4 f c k t + i " \ t ] " [ c j t + i " V (3-12) l k t J The r e s u l t s showed t h a t : i ) the average r e t u r n s f o r the spreading s t r a t e g y were greater than the average r e t u r n s f o r hedging s t r a t e g y i i ) the options w i t h high average r e t u r n s were u s u a l l y on high p r i c e d stock - 20 -The above spreading s t r a t e g y i s not o p e r a t i o n a l as i t i s assumed that c l o s i n g p r i c e s at which the d e c i s i o n r u l e i s based upon i s known before hand so that t r a d i n g can be done at that c l o s i n g p r i c e . To make i t o p e r a t i o n a l , G a l a i used day t c l o s i n g p r i c e s to determine the p o s i t i o n to take. Transactions were done using t+1 p r i c e s and the hedged p o s i t i o n l i q u i d a t e d w i t h t+2 c l o s i n g p r i c e s (as i n the ex ante t e s t s ) . The average returns were c a l c u l a t e d and found to be l e s s than h a l f of those obtained using the ex post spreading s t r a t e g y . However, they were s t i l l s i g n i f i c a n t and above normal p r o f i t s could s t i l l be made. - 21 -4. EFFECT OF DISCRETE HEDGING .ON EXCESS RETURN The e m p i r i c a l t e s t s of the Black Scholes o p t i o n p r i c i n g model per-formed so f a r produced f a i r l y s a t i s f a c t o r y r e s u l t s . However, these t e s t s were c a r r i e d out on the assumption that the use of d i s c r e t e i n s t e a d of continuous hedging' w i l l not i n any way b i a s the excess r e t u r n s . Black and Scholes [2, pp. 642] wrote: "Note a l s o that the d i r e c t i o n of the change i n the e q u i t y v a l u e i s independent of the d i r e c t i o n of change i n the stock p r i c e . This means that the covariance between the r e t u r n on the e q u i t y and the r e t u r n on the stock w i l l be zero. I f the stock p r i c e and the value of the "market p o r t f o l i o " f o l l o w a j o i n t continuous random walk w i t h constant covariance r a t e , i t means that the covariance between the r e t u r n on the equity and the r e t u r n on the market w i l l be zero." Since the covariance between the r e t u r n on the equity and the r e t u r n on the market w i l l be zero they j u s t i f i e d the use of d i s c r e t e hedging on the grounds that " i f the p o s i t i o n i s not adjusted c o n t i n u o u s l y , the r i s k i s s m a l l and can be d i v e r s i f i e d away by forming a p o r t f o l i o . " In t h i s s e c t i o n we w i l l attempt to show that w i t h d i s c r e t e hedging, although the d i r e c t i o n of the change i n the e q u i t y value i s independent of the d i r e c t i o n of change i n the stock p r i c e , i t i s however not indepen-dent of the magnitude of the change i n the stock p r i c e . Consequently i t i s p o s s i b l e that the covariance between the r e t u r n on the market and the r e t u r n on the equity w i l l not be zero r e s u l t i n g i n systematic r i s k that i s not d i v e r s i f i a b l e by the formation of the p o r t f o l i o s . F i g ure I i l l u s t r a t e s the r e l a t i o n s h i p between the o p t i o n value and the stock p r i c e at a p o i n t i n time. The curves W and represent the value of the option at T and T - l days to m a t u r i t y r e s p e c t i v e l y . On day F i g u r e I: R e l a t i o n s h i p Between Stock and Option P r i c e -- 23 -T, assuming the stock p r i c e i s S^, which gives an option v a l u e of a hedged p o s i t i o n i s e s t a b l i s h e d by buying one o p t i o n at and s e l l i n g N(d^) stocks at S^. Since N(d^) i s the d e r i v a t i v e of the o p t i o n p r i c i n g formula w i t h respect to the stock p r i c e , i t i s t h e r e f o r e the slope of the curve W at p o i n t P. This slope can be b e t t e r represented by drawing a l i n e AM tangent to the curve W at p o i n t P. Assuming that the hedged p o s i t i o n i s adjusted d a i l y , the e f f e c t s on the hedged p o s i t i o n i s two-fold. F i r s t , the N(d^) used w i l l be repre-sented by the slope of the l i n e AM i n s t e a d of the slope at v a r i o u s p o i n t s of the curve W. Second, the value of the o p t i o n w i l l reduce by a d i s c r e t e amount represented by a downward s h i f t i n the curve W to W-^  as we approach ma t u r i t y . As we move from T to T - l days to m a t u r i t y , l e t us assume that the stock p r i c e moves from S^ to S 2 . The option p r i c e w i l l move from Ci to C^. The excess r e t u r n on the hedged p o s i t i o n i s c a l c u l a t e d as: (5) ( C 2 - C±) - N(d x) ( S 2 - S n) (4.1) With the stock p r i c e at S 2 , the slope of the l i n e AM has a lower gradient than the slope of W at the p o i n t Z. This i m p l i e s that the N(d^) used i n e s t a b l i s h i n g the hedged p o s i t i o n i s lower than what i t should be i f continuous hedging i s p o s s i b l e . The use of a lower N(d^) w i l l b i a s the excess r e t u r n upward. This b i a s i s represented by AZ. The downward s h i f t of the curve from W to W^  w i l l r e s u l t i n a neg-a t i v e b i a s (represented by ZB) as i s lower than what i t would be i f At i s very s m a l l . Note that t h i s b i a s i s a f u n c t i o n of At and not the To s i m p l i f y the a n a l y s i s the i n t e r e s t r a t e adjustment f a c t o r of (C^ - N(d^.)S^)rAt i s ignored at t h i s stage. I t w i l l be considered subsequently. - 24 -change i n stock p r i c e . Since we assume At to be 1 day, t h i s e f f e c t , causes a small negative b i a s independent of the change i n the stock p r i c e . The net r e s u l t of these two e f f e c t s (represented by AB) i s p o s i t i v e as ZA i s greater than ZB. AB i s a l s o the excess r e t u r n as c a l c u l a t e d by equation (4-1). This can be shown as f o l l o w s : The term (C^ - C-^ ) i n equation (4-1) i s represented by BR and N(d^) ( S 2 - S^) i s the d i s t a n c e AR. The excess r e t u r n i s t h e r e f o r e BR - AR = AB. ^ Using the same a n a l y s i s , i f the stock p r i c e moves downward from S^ to S^jthe N(d^) used i n the establishment of the hedged p o s i t i o n i s higher than what i t should be. This w i l l s t i l l cause a p o s i t i v e b i a s as the excess r e t u r n i s now c a l c u l a t e d as N(d x) (S]_ - S 3) - (C± - C 3) (4-2) For any movement of stock p r i c e s from S^ the excess r e t u r n i s repre-sented by the v e r t i c a l d i s t a n c e between the curve W| and the l i n e AM. I t can be c l e a r l y seen that i f the change i n stock p r i c e i s 'small' (stock p r i c e f a l l i n g w i t h i n S* and S** on the next day) the excess r e t u r n w i l l be negative. For a ' l a r g e ' change i n stock p r i c e (stock p r i c e f a l l i n g beyond S* or S** on the next day) the excess r e t u r n i s p o s i t i v e . The e f f e c t . o f u s i n g yesterdays N(d^) causes a p o s i t i v e b i a s which increases w i t h the s i z e of the stock p r i c e changes but decreases to zero i f the stock p r i c e remains unchanged. On the other hand the downward s h i f t i n the curve from W to causes a s m a l l negative b i a s independent of the stock p r i c e movement. The net r e s u l t w i l l depend on the s i z e of each b i a s . Hence, f o r ' l a r g e ' (small) changes i n stock p r i c e the p o s i t i v e ^ This was pointed out to me by Phelim Boyle. (negative) b i a s outweighs the negative ( p o s i t i v e ) thus i n c u r r i n g a net p o s i t i v e (negative) excess r e t u r n . Note, however, that the net negative excess r e t u r n cannot exceed PQ whereas the amount of the p o s i t i v e excess r e t u r n increases r a p i d l y w i t h the s i z e of stock p r i c e changes beyond S* or S**. Therefore, we w i l l expect that the d i s t r i b u t i o n of the excess r e t u r n to be skewed ra t h e r than normal. Having shown that excess r e t u r n i s indeed a f u n c t i o n of the magni-tude of changes i n the stock p r i c e , the i n t e r e s t r a t e adjustment f a c t o r w i l l be introduced. I n t e r e s t r a t e adjustment i s c a l c u l a t e d as (C1 - N(.d 1)S 1)rAt (4-3) w i t h r as the r i s k f r e e r a t e . As N(d^)S^ i s always g r e a t e r than the a d d i t i o n of t h i s p o s i t i v e constant to the excess returns w i l l tend to i n -f l a t e the amount of p o s i t i v e and reduce the amount of negative excess r e t u r n . T h e o r e t i c a l l y , t h i s i n t e r e s t r a t e adjustment f a c t o r i s supposed to o f f s e t the r e d u c t i o n i n the opt i o n values as we move from T to T - l . In f i g u r e 1, the r e d u c t i o n i n the o p t i o n v a l u e i s the v e r t i c a l d i s t a n c e between the curves W and W^ , the e f f e c t of the i n t e r e s t r a t e adjustment f a c t o r can be seen as moving the W^  curve to the same p o s i t i o n as the W curve thus e l i m i n a t i n g the negative b i a s discussed above. When t h i s occurs, the e f f e c t of the discontinuous hedging on excess r e t u r n i s a p o s i t i v e b i a s which gets l a r g e r the f u r t h e r the stock p r i c e on the next day i s away from S^. I f the i n t e r e s t r a t e adjustment f a c t o r i s i n s u f f i c i e n t , then the curve W., w i l l not move back to the same p o s i t i o n as W but w i l l be some-- 26 -where between the present W and W^ . I f t h i s occurs, the phenomenon of o b t a i n i n g negative excess r e t u r n s f o r small changes i n stock p r i c e s and p o s i t i v e excess r e t u r n s f o r l a r g e changes i n stock p r i c e s w i l l occur. As a v a l i d a t i o n and a l s o to determine i f the a n a l y s i s i s a p p l i c a b l e i n cases when the r a t e s of r e t u r n of stock p r i c e s do not f o l l o w a l o g -normal d i s t r i b u t i o n a simple s i m u l a t i o n study was done. 100 stock p r i c e s were obtained by s i m u l a t i n g the r a t e s of r e t u r n f i r s t l y from a log-normal and subsequently from a normal and t d i s t r i b u t i o n w i t h two degrees of freedom. The hedged p o s i t i o n s were formed and the excess r e t u r n c a l c u -l a t e d as before. A s o r t of the excess r e t u r n i n ascending order of stock p r i c e changes c l e a r l y shows th a t i n every case, a ' l a r g e ' jump i n stock p r i c e r e s u l t s i n a p o s i t i v e r e t u r n s w h i l e a ' s m a l l ' jump r e s u l t s i n a negative r e t u r n s . The phenomenon i s present whether the stock p r i c e r e -turns f o l l o w a log-normal, normal or t d i s t r i b u t i o n . This r e s u l t tends to suggest that the i n t e r e s t r a t e f a c t o r described i n the previous paragraph i s i n s u f f i c i e n t . ^ ^ Histograms of the excess r e t u r n s were p l o t t e d f o r the three d i s t r i b u t i o n s . As expected they show a skewness to the l e f t . As an i l l u s t r a t i o n , the histogram of excess r e t u r n using a log-normal d i s t r i -b u t i o n i s presented i n f i g u r e I I . In summary, the i n t e r e s t r a t e adjustment f a c t o r tends to reduce the range w i t h i n which stock p r i c e movements w i l l cause a negative excess r e -t u r n . But i t i s , however, not b i g enough to e l i m i n a t e the negative r e -t u r n a l t o g e t h e r . Therefore the e f f e c t of using d i s c r e t e i n s t e a d of con-tinuous hedging w i l l cause the excess r e t u r n f o r a p a r t i c u l a r day to be a f u n c t i o n of the stock p r i c e changes. For the p e r i o d over which the option e x i s t s , the d i s t r i b u t i o n of excess r e t u r n s w i l l be a f u n c t i o n of the d i s t r i -b u t i o n of r a t e s of r e t u r n of stock p r i c e s of that p e r i o d . I f the d i s t r i b u t i o n of ^ In Appendix I I I , i t i s shown that the i n t e r e s t r a t e adjustment f a c t o r i s i n s u f f i c i e n t by X e _ r T [ Z ( d 9 ) _ 2 ^ _ ]At where Z i s the normal de n s i t y f u n c t i o n . 1 2Vf-* * *• * * ; ; * ' * t -* * *_ _._ ._ .. ._ * * * * * * * * *. * * * * * * * 30 ! * * * * * * * * * * * * * A * * 1 s . * * * * ***** + * . .... —A—-c f -•• • - • • * * * * * * * * * * * * * * * * + * * ***** * ***** * * * * ********* * * * * * * * * * * * * .*. * ***.*.*..* *..*. -, 042 ,032 -.022 -,012 ,002 0,008 0,016 0, 028 O.OiH FIGURE I I : Histogram of Excess Returns f o r a Log-Normal D i s t r i b u t i o n ISO Excess Return ($) - 28 -r a t e s of r e t u r n of stock p r i c e s during the period i s almost uniform, the d i s t r i b u t i o n of the excess r e t u r n w i l l probably be skexred to the l e f t . On the other hand, i f the d i s t r i b u t i o n i s concentrated around the mean then we w i l l expect the reverse to be t r u e . 5. THE CANADIAN OPTION MARKET v ' In t r o d u c t i o n : C a l l options were f i r s t traded on the Montreal Stock Exchange on September 15, 1975 through' the Montreal Option C l e a r i n g Corporation. On September 29, 1975, the Canadian Option C l e a r i n g Corporation (COCC) was set up as a successor to the Montreal Option C l e a r i n g Corporation. The f u n c t i o n of COCC was described as " i s s u i n g and a c t i n g as the primary o b l i g o r on the Exchange Traded Options and f o r c l e a r i n g t r a n s a c t i o n s i n Options". The shares of the COCC were d i v i d e d e q u a l l y between the Montreal Stock Exchange and the Toronto Stock Exchange. I t s name was subsequently amended to Trans Canada Options Inc. As of J u l y 1977, options on 23 stocks are being traded on both exchanges. (A l i s t i n g of these stocks i s given i n Appendix I I . ) D e s c r i p t i o n s of Options Options on underlying s e c u r i t i e s are permitted only i f the underly-i n g s e c u r i t i e s meet the f o l l o w i n g c o n d i t i o n s : i ) The underlying s e c u r i t y i s l i s t e d on one of the Exchanges. i i ) The i s s u e r of the underlying s e c u r i t y i s i n c o r p o r -ated, organized or continued under the laws of Canada or a province or t e r r i t o r y thereof, i i i ) The i s s u e r and i t s c o n s o l i d a t e d s u b s i d i a r i e s have had a net income, a f t e r taxes but before e x t r a -ordinary items net of tax e f f e c t , of at l e a s t $500,000 f o r each of the l a s t 3 f i s c a l years. This s e c t i o n draws h e a v i l y from the prospectus of Trans Canadian Option Inc., May 6, 1977. - 30 -i v ) The i s s u e r and i t s c o n s o l i d a t e d s u b s i d a r i e s have not d e f a u l t e d during the past 3 f i s c a l years i n the payment of any dividends, or s i n k i n g fund i n s t a l -ment, i n t e r e s t and p r i n c i p a l of any borrowed money or i n payments of r e n t a l under long-term l e a s e s , v) The i s s u e r earned i n three of the l a s t f i v e f i s c a l y ears, any d i v i d e n d s , i n c l u d i n g the f a i r market value of any stock d i v i d e n d s , paid i n each such year on a l l c l a s s e s of s e c u r i t i e s . There may be exceptions to requirements ( i i i ) to (v) of the above. Besides the above requirements, the Corporations a l s o e s t a b l i s h e d c e r t a i n g u i d e l i n e s f o r the s e l e c t i o n of the u n d e r l y i n g s e c u r i t i e s . They are: i ) The i s s u e r of the u n d e r l y i n g s e c u r i t i e s s h a l l have outstanding a minimum of 5,000,000 shares, i i ) The i s s u e r s h a l l have a minimum of 5,000 r e g i s t e r e d shareholders. i i i ) T o t a l t r a d i n g v a l u e i n the u n d e r l y i n g s e c u r i t y on a l l stock exchanges on which i t i s l i s t e d s h a l l have been at l e a s t 800,000 shares per year i n each of the past two years. i v ) The market p r i c e of the u n d e r l y i n g s e c u r i t y s h a l l be at l e a s t $5 per share. M a t u r i t y dates of options on u n d e r l y i n g s e c u r i t i e s traded on both exchanges or only on the Montreal Stock Exchange are r e s t r i c t e d to Febru-ary, May, August and November. At any p o i n t i n time only options w i t h the next three m a t u r i t y dates w i l l be opened f o r t r a d i n g , e.g. i n June 1977, - 31 -options w i t h m a t u r i t y at August, November and February w i l l be opened f o r t r a d i n g . Trading on options once opened, subjected to c e r t a i n r e s t r i c -t i o n s , w i l l remain opened u n t i l the t h i r d F r i d a y of the e x p i r i n g month.'-As p r i c e s of the underlying stock f l u c t u a t e , new options w i t h higher or lower e x e r c i s e p r i c e w i l l be introduced by the exchange. The new e x e r c i s e p r i c e w i l l d i f f e r from the o l d p r i c e i n m u l t i p l e of $2.50 f o r shares t r a d i n g below $25.00, $5.00 f o r shares t r a d i n g between $25.00 and $50.00, $10.00 f o r shares t r a d i n g between $50.00 and $100.00, $20,00 f o r shares t r a d i n g between $100.00 and $200.00 and l a s t l y $25.00 f o r shares t r a d i n g above $200.00 Departure from t h i s general p r a c t i c e i s permitted i f the r e s u l t would be to provide b e t t e r l i q u i d i t y f o r options covering a p a r t i c u l a r u n d e r l y i n g s e c u r i t y . Options are traded i n m u l t i p l e s of 100 and are e x e r c i s a b l e any time a f t e r issuance u n t i l e x p i r a t i o n except i n the f o l l o w i n g circumstances: i ) The number of options that can be e x e r c i s e d covering the same un d e r l y i n g s e c u r i t i e s i s r e s t r i c t e d to 1,000 co n t r a c t s or 100,000 shares, i i ) The Exchange can r e s t r i c t the e x e r c i s e of any options i f i t f e e l s that such a c t i o n i s a d v i s a b l e i n the i n t e r e s t of m a i n t a i n i n g a f a i r and o r d e r l y market. Adjustment of Options Adjustment to options are necessary as a r e s u l t of d i v i d e n d s , stock s p l i t s or r e o r g a n i s a t i o n . In the case of d i v i d e n d s , no adjustment i s made to any of the terms of the exchange traded options. The r u l e i s that i f the holder of a c a l l o ption f i l e s an e f f e c t i v e e x e r c i s e n o t i c e p r i o r to the ex-dividend date, then he or she i s e n t i t l e d to that dividend even though the w r i t e r to whom the e x e r c i s e i s assigned may not r e c e i v e the - 32 -assignment n o t i c e u n t i l a f t e r the ex-dividend date. Stock s p l i t s , stock dividends or other stock d i s t r i b u t i o n which i n -crease the number of outstanding shares of the i s s u e r have the e f f e c t of p r o p o r t i o n a t e l y i n c r e a s i n g the number of shares of the u n d e r l y i n g stock covered by the options and p r o p o r t i o n a t e l y decreasing the e x e r c i s e p r i c e as of the ex-date. In cases where stock s p l i t or stock d i s t r i b u t i o n r e s u l t s i n the d i s t r i b u t i o n of one or more whole shares f o r each share outstanding then an adjustment w i l l be made "to the number of options c o n t r a c t . Adjustment as a r e s u l t of r e o r g a n i z a t i o n w i l l be made w i t h respect to e x e r c i s e p r i c e or u n i t of t r a d i n g , i f the c o r p o r a t i o n considered such adjustments to be f a i r to the holders and w r i t e r s of such options. L i m i t a t i o n s on Trading The number of options of the same un d e r l y i n g s e c u r i t y r e g a r d l e s s of mat u r i t y dates which may be hel d by a s i n g l e i n v e s t o r or group of i n v e s t o r s i s r e s t r i c t e d to 500 co n t r a c t s or 50,000 shares. The l i m i t on the number of options of the same und e r l y i n g s e c u r i t y on any ma t u r i t y date i s 1,000 con t r a c t s or 100,000 shares. These l i m i t s are a p p l i c a b l e to both buying and s e l l i n g options. The exchange i s empowered to order l i q u i d a t i o n of any p o s i t i o n exceeding these l i m i t s or to order any other s a n c t i o n s . Besides these l i m i t s , margin requirements are imposed on the op t i o n w r i t e r . These requirements d i f f e r between Exchange members. However, as an i n d i c a t i o n of these requirements, we take a look at the margin r e q u i r e -ment the Exchange imposed on i t s v a r i o u s members. Each Exchange member i s r e q u i r e d e i t h e r i ) to deposit w i t h the Corporation the un d e r l y i n g s e c u r i t y , represented by the opt i o n . In t h i s i n -stance, an escrow r e c e i p t issued by approved - 33 -i n s t i t u t i o n or i i ) to maintain w i t h the Corporation, each s e c u r i t y of the government of Canada or a province, or bank of c r e d i t , equalled to 30% of the market p r i c e of the u n d e r l y i n g s e c u r i t i e s , increased (reduced) by the d i f f e r e n c e between the market p r i c e of the s e c u r i t i e s and the e x e r c i s e p r i c e . The amount of deposits r e q u i r e d are adjusted d a i l y . - 34 -6. THE DATA D a i l y o p t i o n p r i c e s traded on the Montreal Stock Exchange were c o l l e c t e d from the Montreal Gazette f o r the p e r i o d 15th September 1975 to 31st December 1976. They were sorted i n t o e i g h t m a t u r i t y dates s t a r t -ing w i t h those maturing on November 1975 and ending w i t h those maturing on August 1977. This gives a t o t a l of options on 18 u n d e r l y i n g stocks. (A l i s t i n g of the stocks are given i n Appendix I I . ) The c l o s i n g stock p r i c e s traded on the Montreal Stock Exchange f o r the p e r i o d 15th September 1974 to 31st December 1976 were c o l l e c t e d from the Toronto Globe and M a i l . Dividends i n f o r m a t i o n was obtained from the o f f i c i a l monthly p u b l i c a t i o n s of the Montreal Stock Exchange f o r the same p e r i o d . The p r i c e s were checked f o r accuracy by p i c k i n g v a r i o u s sample and t r a c i n g these back to the o r i g i n a l source document. Furthermore, p r i c e s w i t h a percentage change of 10% or more were v e r i f i e d . Besides the above, estimates of the r i s k l e s s r a t e of i n t e r e s t and the v a r i a n c e of the stock's r a t e of r e t u r n d i s t r i b u t i o n were r e q u i r e d . For the r i s k l e s s i n t e r e s t r a t e , the average weekly i n t e r e s t r a t e of the 90-day tre a s u r y b i l l s over the l i f e of the option was used. These r a t e s were obtained from the Bank of Canada Review f o r the p e r i o d concerned. In the case of the v a r i a n c e of the stock's r a t e of r e t u r n d i s t r i b u -t i o n , two estimates were used, namely a d a i l y adjusted past v a r i a n c e and a constant v a r i a n c e . On any day, the past v a r i a n c e i s c a l c u l a t e d using the r a t e s of r e -t u r n of the stock f o r the past 12 months. For subsequent days, t h i s past v a r i a n c e i s updated by dropping the e a r l i e s t observation and t a k i n g i n t o c o n s i d e r a t i o n the l a t e s t r a t e of r e t u r n . For purposes of c a l c u l a t i n g the variances the r a t e s of r e t u r n were adjusted f o r d i v i d e n d s . The constant - 35 -v a r i a n c e i s a r r i v e d at by t a k i n g the average of the d a i l y adjusted past v a r i a n c e over the l i f e of the o p t i o n . - 36 -7. TEST AND RESULTS 7.1 Tests of the Black-Scholes Model The a n a l y s i s i n Section 4 suggests the p o s s i b i l i t y of a b i a s i n using d i s c r e t e i n s t e a d of continuous hedging. Such a b i a s i s dependent on the d i s t r i b u t i o n of the r a t e s of r e t u r n of the stock p r i c e f o r the period.concerned. Hence, p r i o r to using the Black-Scholes model f o r t e s t -i n g market e f f i c i e n c y , i t w i l l be necessary to determine i f such a b i a s e x i s t s f o r the t e s t p e r i o d , and i f i t e x i s t s , the extent of the b i a s . Therefore, a t e s t s i m i l a r to the 'buy a l l options at model p r i c e s ' (Buy a l l ) s t r a t e g y performed by Black and Scholes [1] was c a r r i e d out whereby ( i ) a constant v a r i a n c e r a t e and a c t u a l c l o s i n g stock p r i c e s from the Montreal Stock Exchange were used, and ( i i ) the hedge p o s i t i o n was adjusted on a d a i l y b a s i s . For each day, the t h e o r e t i c a l model p r i c e and N(d^) were c a l c u l a t e d using equation (2.9). The a c t u a l c l o s i n g stock p r i c e f o r that day was used. A hedged p o s i t i o n was then set up by buying one opt i o n at the model p r i c e and s e l l i n g N(d^) stock at the market p r i c e . On the next day t h i s p o s i t i o n was l i q u i d a t e d and a new one e s t a b l i s h e d . The excess r e t u r n f o r h o l d i n g the hedged p o s i t i o n f o r one day i s c a l c u l a t e d as f o l l o w s : (AC - N(d )AS) - (C - N(d ) S ) r (7.1) This procedure was repeated f o r each op t i o n traded on the Montreal Stock Exchange from the day t r a d i n g on the opt i o n commenced to the e x p i r a -t i o n date. These excess r e t u r n s were examined and the f o l l o w i n g p o i n t s noted: ( i ) options w i t h the same und e r l y i n g stocks tend to have p o s i t i v e r e turns on the same day - 37 -( i i ) changes i n stock p r i c e s of a l a r g e (small) amount give r i s e to p o s i t i v e (negative) excess returns f o r the hedged p o s i t i o n s ( i i i ) almost 95% of the excess returns on the hedged p o s i t i o n f o r the f i r s t s i x m a t u r i t y dates are negative. Furthermore, the number of negative returns reduces s i g n i f i c a n t l y f o r the l a s t two m a t u r i t y dates, i . e . May and August, 1977 P o i n t s ( i ) and ( i i ) above i n d i c a t e that the excess returns are a f u n c t i o n of the stock p r i c e changes. P o i n t ( i i i ) suggests that the d i s -t r i b u t i o n s of the r a t e s of r e t u r n of stock p r i c e s f o r the f i r s t s i x matur-i t y dates tend to be 'peaked' ( i . e . concentrated around the mean) w h i l e the d i s t r i b u t i o n of the r a t e s of r e t u r n of stock p r i c e s f o r the l a s t two m a t u r i t y dates tend to be ' f a t - t a i l e d ' ( i . e . greater p r o b a b i l i t y of having a b i g r a t e of returns of the stock p r i c e s ) . To t e s t the above hypothesis, histograms of the r a t e s of r e t u r n of stock p r i c e s f o r each m a t u r i t y date of the 18 stocks were p l o t t e d . For every stock, a comparison of the histogram of the e i g h t maturity dates shows the above hypothesis to be t r u e . Four histograms ( f o r m a t u r i t y date; February, August, 1977 and 1978) of a stock are presented i n Appendix I f o r purposes of i l l u s t r a t i o n . The above observations i n d i c a t e the existence of a b i a s along the l i n e s i n d i c a t e d i n Section 4. To examine the d i r e c t i o n and extent of the b i a s these excess returns f o r each op t i o n were i n d i v i d u a l l y regressed against the excess r e t u r n on the market. The r e g r e s s i o n model was used: \ t - a + 3 V + e t ( 7 - 2 ) - 38 -M a t u r i t y Date No. of Options No. of a s i g n i f i c a n t -l y p o s i t i v e No. of a s i g n i f i c a n t -l y negative No. of a i n s i g n i f i -cant No. of g s i g n i f i -cant Average a Nov. 75 31 - 23 8 - -0.009715 Feb. 76 38 - 35 3 2 -0.006341 May 76 51 - 49 2 4 -0.004757 Aug. 76 55 - 55 - 2 -0.004271 Nov. 76 55 - 50 5 15 -0.003995 Feb. 77 61 - 42 19 6 -0.003367 May 77 60 - 12 48 3 -0.002492 Aug. 77 27 - 4 23 — -0.007971 Table I : Returns on the Hedged P o s i t i o n s Using a B u y a l l Strategy (Constant Variance) M a t u r i t y Date a t-a B t-B Nov. 75 -0.2526 -4.7879 1.1795 0.1516'-Feb. 76 -0.2296 -6.5931 1.8871 0.3638 May 76 -0.2369 -7.8078 5.3385 1.1374 Aug. 76 -0.2347 -8.1321 7.7844 1.5848 Nov. 76 -0.2046 -6.8352 10.2870 2.2230 Feb. 77 -0.1680 -4.0229 3.0571 0.5090 May 77 -0.0928 -1.7387 1.6400 0.2538 Aug. 77 -0.0367 -0.9175 3.5821 0.7388 ft Table I I I : Returns on the P o r t f o l i o s Using a B u y a l l Strategy (Constant Variance) S e r i a l c o r r e l a t i o n s of the r e s i d u a l s are not s i g n i f i c a n t . - 39 -where *Ht = excess r e t u r n on the hedged p o s i t i o n on day t excess r e t u r n on the Montreal Stock Exchange Composite Index on day t e t e r r o r term The slope c o e f f i c i e n t g i s i n t e r p r e t e d as the measure of r i s k w h i l e a, the i n t e r c e p t i s the amount of excess returns which can be earned a f t e r t a k i n g i n t o c o n s i d e r a t i o n any r i s k which the hedged p o s i t i o n might have. The a i s therefore an i n d i c a t i o n of the amount and d i r e c t i o n of the b i a s . The r e s u l t s of the r e g r e s s i o n were sorted according to the maturity date and the summarized r e s u l t s are presented i n Table I . These r e s u l t s show that ( i ) although most of the g's are i n s i g n i f i c a n t l y d i f f e r -ent from zero, the a's i n most cases are s i g n i f i c a n t l y negative ( i i ) the percentage of a's which are s i g n i f i c a n t l y nega-t i v e i s r e l a t i v e l y low f o r the l a s t two m a t u r i t y . dates The above r e s u l t s should be taken as i n d i c a t i v e r a t h e r than conclu-s i v e proof of the d i r e c t i o n and extent of the b i a s . As explained i n Section 4, we expect t h e ' d i s t r i b u t i o n of excess returns to be-skewed rather than normal depending on the d i s t r i b u t i o n of the r a t e s of r e -(9) t u r n of the stock p r i c e . With the excess returns not being normally d i s t r i b u t e d , the a's and 3's obtained above are unbiased but not the most 79) To be c e r t a i n , histograms of the excess returns of a few options were p l o t t e d and they were found to be skewed. - 40 -e f f i c i e n t estimators of the t r u e a and 3. Therefore i n t e r p r e t a t i o n of the r e s u l t s should be made wi t h c a u t i o n . In an attempt to overcome the above problems as w e l l as to d i v e r s i f y away e r r o r s i n v a r i a b l e s that can a f f e c t each p o s i t i o n i n d i v i d u a l l y , p o r t -f o l i o s are formed by summing d a i l y a l l the i n d i v i d u a l excess r e t u r n s on the hedged p o s i t i o n of the same mat u r i t y date. These p o r t f o l i o excess returns were test e d f o r normality using a Kolmogorov-Smirnoff goodness of f i t t e s t and the r e s u l t s are presented i n Table I I . Of the e i g h t p o r t f o l i o s we can conclude that only three of them ( i n d i c a t e d by '*') have excess returns that are normal. K-S D 0.05 0.01 K-S D 0.05 0.01 M a t u r i t y S t a t i s - C r i t i c a l C r i t i c a l M a t u r i t y S t a t i s - C r i t i c a l C r i t i c a l Date t i c s L e v e l L e v e l Date t i c s L e v e l L e v e l Nov. 75 0.1757 0. 1984 0.2378* Nov. 76 0. 2654 0.1000 0. 1198 Feb. 76 0.2637 0. 1303 0.1561 Feb. 77 0. 2374 0.1107 0. 1327 May 76 0.2296 0. 1037 0.1243 May 77 0. 1149 0.1442 0. ft 1728 Aug. 76 0.1891 0. 1003 0.1202 Aug. 77 0. 2011 0.2570 0. * 3130 Table I I Kolmogorov-Smirnoff goodness of f i t t e s t on excess r e t u r n The d a i l y p o r t f o l i o excess r e t u r n s were then regressed against the (11) market returns using equation (7.2). The r e g r e s s i o n r e s u l t s are pre-sented i n Table I I I . The a's were s i g n i f i c a n t l y negative f o r the f i r s t s i x m a t u r i t y dates and i n s i g n i f i c a n t f o r the l a s t two m a t u r i t y dates. I t i s i n t e r e s t i n g to note that the p o r t f o l i o of options maturing on November 1976 has a s i g n i f i c a n t p o s t i v e 3. For ten hedged p o s i t i o n s , the r e s i d u a l s obtained from the r e g r e s s i o n were p l o t t e d against the p r e d i c t e d Rjj t> the independent and t e s t e d f o r n o r m a l i t y . There appears to be no f u r t h e r v i o l a t i o n of the other assumptions of the r e g r e s s i o n model used. As the number of outstanding c o n t r a c t s each day i s constant f o r each mat u r i t y date, no problem of h e t e r o s k e d a s t i c i t y i s encountered. - 41 -No. of a No. of a No. of a No. of g Ma t u r i t y Date No. of Options s i g n i f i c a n t -l y p o s i t i v e s i g n i f i c a n t -l y negative i n s i g n i f i -cant s i g n i f i -cant Average a Nov. 75 31 - 25 6 5 -0.011330 Feb. 76 38 - 35 3 5 -0.008664 May 76 51 - 48 3 16 -0.006837 Aug. 76 55 - 51 4 17 -0.005955 Nov. 76 55 - 47 8 - -0.004899 Feb. 77 61 - 33 28 - -0.004105 May 77 60 - 3 57 2 -0.003994 Aug. 77 27 - - 27 - -Table IV: Returns on the Hedged P o s i t i o n s Using a B u y a l l Strategy ( D a i l y Adjusted Variances) M a t u r i t y Date a t-a 6 t-e Nov. 75 -0.3064 -5.7953 12.3820 1.5891 Feb. 76 -0.3135 -7.6482 9.6116 1.5745 May 76 -0.3280 -8.9023 15.3030 2.6855 Aug. 76 -0.3126 -9.1490 15.2220 5.8153 Nov. 76 -0.2486 -7.8127 3.8960 0.7924 Feb. 77 -0.1749 -3.3982 -4.0397 -0.5474 May 77 -0.0126 -0.1606 -0.6514,' -0.0689 Aug. 77 0.0639 1.0535 4.0845 0.5560 Table V: Returns on the P o r t f o l i o s * Using a B u y a l l Strategy ( D a i l y Adjusted Variance) S e r i a l c o r r e l a t i o n s of the r e s i d u a l s are not s i g n i f i c a n t . - 42 -To determine the s e n s i t i v i t y of the r e s u l t obtained, the t e s t was repeated using the d a i l y adjusted past v a r i a n c e . The r e s u l t s are pre-sented i n Table IV f o r i n d i v i d u a l hedges and Table V f o r p o r t f o l i o s . The use of d a i l y adjusted variance d i d not change the r e s u l t s s i g n i -f i c a n t l y . The observations made w i t h regard to the r e s u l t of the t e s t using constant v a r i a n c e are s t i l l v a l i d . However, there are two p o r t f o l i o s which have s i g n i f i c a n t l y p o s i t i v e 3's. The f a c t that s i g n i f i c a n t l y p o s i t i v e B's are c o n s i s t e n t l y obtained supports the point made i n Sectio n 4 that formation of p o r t f o l i o w i l l not n e c e s s a r i l y e l i m i n a t e a l l the r i s k due to d i s c r e t e hedging. 7.2 E f f i c i e n c y of the Montreal Option Market We w i l l attempt to t e s t the e f f i c i e n c y of the Montreal Option Market by using a simple s t r a t e g y . E f f i c i e n c y r e f e r s to the p o s s i b i l i t y of earn-ing higher than normal returns a f t e r t a k i n g i n t o c o n s i d e r a t i o n the r i s k taken. The s t r a t e g y used i s s i m i l a r to the 'buy ( s e l l ) a l l undervalued (12) (overvalued) options at market va l u e ' s t r a t e g y performed by Black and Scholes [ 1 ] . An option whose model p r i c e i s greater ( l e s s ) than i t s market p r i c e i s considered undervalued (overvalued). - 43 -On the f i r s t day the o p t i o n i s traded, equation (2.10) i s used to c a l c u l a t e the t h e o r e t i c a l o p t i o n p r i c e as w e l l as N(d^). This p r i c e i s compared to the option's c l o s i n g market p r i c e f o r the day. I f the o p t i o n i s undervalued (overvalued), a hedge p o s i t i o n i s e s t a b l i s h e d by buying ( s e l l i n g ) one o p t i o n at the market p r i c e and s e l l i n g (buying) N(d^) amount of shares. The hedge p o s i t i o n i s l i q u i d a t e d the next day and the excess r e t u r n i s c a l c u l a t e d as ^ t = (< " ^ P ^ ^ A - i ^ t - ^ - i ^ ^ t - ^ ^ A - i ^ - i ^ ' ( 7- 3 ) where = Return on the hedge at day t M C = Model option p r i c e at day t = A c t u a l ' (market) option p r i c e at day t S = Closed stock p r i c e at day t On second and subsequent days u n t i l m a t u r i t y , the model p r i c e i s assumed to be the market p r i c e and the excess r e t u r n i s c a l c u l a t e d using (7.4) - 44 -The assumption of the model p r i c e being the market p r i c e i s necessary as the secondary a c t i v i t y on the Montreal Option Market does not provide a d a i l y o p t i o n p r i c e f o r the m a j o r i t y of the options traded. However, we w i l l attempt to simulate t r a d i n g at the market p r i c e by adding to the d a i l y excess r e t u r n a p o s i t i v e amount. This amount i s c a l c u l a t e d as an annuity payment over the l i f e of the option equivalent to the d i f f e r e n c e between the i n i t i a l market p r i c e and the model p r i c e . The r a t i o n a l e behind t h i s a d j u s t -ment i s that as the o p t i o n approaches m a t u r i t y , the market p r i c e w i l l tend towards the o p t i o n . At the m a t u r i t y date, the market p r i c e w i l l be equal to the option p r i c e . Therefore the t o t a l of a l l the d i f f e r e n c e s between excess returns of two hedged p o s i t i o n s (one u s i n g model p r i c e s and the other using market p r i c e s ) over the l i f e of the o p t i o n i s approximately equal to the d i f f e r e n c e between i n i t i a l market and model p r i c e s . The above procedure i s used to a r r i v e at the adjusted d a i l y excess r e t u r n s f o r every option traded on the Montreal Option Market f o r the per-i o d 15th September 1975 to 31st December 1976. The i n d i v i d u a l excess returns were regressed against the market using equation (7.2). The t e s t was c a r r i e d out using the d a i l y adjusted past variances and the summarized r e -s u l t s are presented i n Table VI. For comparison purposes, p o r t f o l i o s are formed according to the eight m a t u r i t y dates. Each p o r t f o l i o c o n s i s t s of a l l the c o n t r a c t s outstanding on a p a r t i c u l a r day. A s e r i e s of d a i l y excess r e t u r n s on the p o r t f o l i o i s obtained by aggregating the excess returns each day over a l l con-t r a c t s outstanding. Since the i n d i v i d u a l options s t a r t t r a d i n g on d i f f e r e n t days, the number of contracts outstanding vary between days. Hence, the p o r t f o l i o r e t u r n on day t (R t ) 1 S d i v i d e d by the number of hedged p o s i t i o n s o u t -standing that day. The average p o r t f o l i o excess returns are then regressed - 45 -M a t u r i t y Date No. of Options No. of a s i g n i f i c a n t -l y p o s i t i v e No. of a s i g n i f i c a n t -l y negative No. of a i n s i g n i f i -cant No. of B s i g n i f i -cant Average a Nov. 75 31 13 5 13 5 0.007924 Feb. 76 38 12 10 16 5 0.003779 May 76 51 23 13 15 11 0.007453 Aug. 76 55 21 18 16 4 0.002296 Nov. 76 . 55 39 5 11 6 0.009672 Feb. 77 61 17 6 38 1 0.008062 May 77 60 12 - 48 2 0.011929 Aug. 77 27 1 - 26 1 0.004688 Table VI: Returns on the Hedged P o s i t i o n s Using a Buy and S e l l Strategy ( D a i l y Adusted Variance) M a t u r i t y Date a t-•a 3 t-B Nov. 75 0 .009515 34. 9260 -0 .0045 -0.1141 Feb. 76 0 .001566 8. 2921 0 .06204 2.2099 May 76 0 .001715 9. 5967 -0 .02135 -0.7419 Aug. 76 0 .002144 16. 7830 -0 .05281 -2.1135 Nov. 76 0 .009377 30. 1830 -0 .00382 -0.9349 Feb. 77 0 .002570 20. 661 0 .05067 3.4217 May 77 0 .004841 36. 05 7 0 .05333 3.7263 Aug. 77 0 .006291 35. 341 -0 .01647 -0.8518 Table V I I : Returns on the P o r t f o l i o s Using a Buy and S e l l Strategy ( D a i l y Adjusted Variance) S e r i a l c o r r e l a t i o n s of the r e s i d u a l s are not s i g n i f i c a n t . - 46 -against the market and the r e g r e s s i o n model used i s R tfiT = av^tT + e C R ^ ^ T ) + y t ( 1 3 ) (7.4) N t where R = p o r t f o l i o excess returns on day t Pt R„, = excess r e t u r n on the market on day t Mt N = number of hedged p o s i t i o n s outstanding on day t u = e r r o r term The r e g r e s s i o n l i n e was forced to go through the o r i g i n w i t h R /l~ p t N as the dependent v a r i a b l e and and R j ^ - 1 ^ ^ a s independent v a r i -a b l e s . The a's and (3's can be i n t e r p r e t e d as before. The r e s u l t s are presented i n Table V I I . The f o l l o w i n g p o i n t s can be noted from the r e s u l t s presented i n Tables VI and V I I . ( i ) Although the average of the s i g n i f i c a n t a's f o r the i n d i v i d -u a l hedged p o s i t i o n s are p o s i t i v e there are a l a r g e number of a's which are s i g n i f i c a n t l y negative. However, i t i s not known whether these negative a's are caused by market i n e f f i c i e n c y or the b i a s as a r e s u l t of us i n g d i s c r e t e hedging. ( i i ) The very s i g n i f i c a n t a's i n the case of p o r t f o l i o s w i l l tend to suggest that i t w i l l be p o s s i b l e to make excess returns on the market by f o l l o w i n g a simple buy and s e l l s t r a t e g y , e.g. I f a p o r t f o l i o c o n s i s t i n g of a l l the hedged p o s i t i o n s maturing on November ^ 1 3^ To overcome the problem of h e t e r o s k e d a s t i c i t y that a r i s e s , we m u l t i p l y the standard r e g r e s s i o n equation throughout by t/n^. A problem of m u l t i c o l l i n e a r i t y i s encountered here. This w i l l tend to i n f l a t e the standard e r r o r of the c o e f f i c i e n t s a'a and g's thus reduc-ing t h e i r r e l i a b i l i t y and un d e r s t a t i n g t h e i r s i g n i f i c a n c e . - 47 -1975 were he l d and adjusted d a i l y then (from Table VII) a d a i l y average r e t u r n of $0.95 w i l l be made w i t h zero (12) xnvestment. ( i i i ) Despite the negative b i a s caused by d i s c r e t e hedging, i t i s s t i l l p o s s i b l e to obta i n a s i g n i f i c a n t l y p o s i t i v e excess r e t u r n on a l l the p o r t f o l i o s . This w i l l i n d i c a t e that p r o f i t o p p o r t u n i t i e s do e x i s t i n the market. However i t i s do u b t f u l that when t r a n s a c t i o n c o s t s , taxes, d i f f -erent lending and borrowing r a t e s are taken i n t o consider-a t i o n , such p r o f i t o p p o r t u n i t i e s w i l l s t i l l e x i s t i n the market. ( i v ) Four of the eight p o r t f o l i o s have s i g n i f i c a n t 3's. The s i g n i f i c a n t 3's show that the r i s k on the p o r t f o l i o s i s not zero implying the existence of systematic r i s k . To t e s t i f the model can perform b e t t e r when dividends are taken i n t o c o n s i d e r a t i o n we d i v i d e a l l the options of each ma t u r i t y date i n t o four p o r t f o l i o s . The f i r s t p o r t f o l i o c o n s i s t s of options on stocks w i t h the lowest dividend y i e l d and the l a s t p o r t f o l i o c o n s i s t i n g of a l l options on stock w i t h the highest d i v i d e n d y i e l d . The excess r e t u r n s on the p o r t -f o l i o were c a l c u l a t e d i n the same manner and regressed against the market using equation (7.4). The a on Table V I I shows the d o l l a r amount to be made on a p o r t f o l i o c o n s i s t i n g of hedged p o s i t i o n s on one op t i o n . However, options are only traded i n m u l t i p l e s of a hundred, t h e r e f o r e the r e t u r n w i l l be $0.0095 x 100 = $0.95. - 48 -Although we would expect the model to perform b e t t e r thus r e s u l t i n g i n higher a's i n those cases when the assumptions of the model are b e t t e r met, the r e s u l t s show that the a's obtained are independent of the amount of dividend payment. In summary, t h i s s e c t i o n shows that the use of the model f o r t h i s p e riod w i l l r e s u l t i n a s i g n i f i c a n t negative b i a s f o r excess returns i n s i x of the eight maturity dates. Despite t h i s b i a s , we are able to make s i g n i f i c a n t p o s i t i v e p r o f i t s by f o l l o w i n g a simple buy and s e l l s t r a t e g y implying that the market i s i n e f f i c i e n t . However such p r o f i t opportuni-t i e s are u n l i k e l y to e x i s t i n the market when t r a n s a c t i o n c o s t s , d i f f e r e n t lending and borrowing r a t e s are taken i n t o c o n s i d e r a t i o n . - 49 -8. Summary and Conclusion Past e m p i r i c a l s t u d i e s using the Black-Scholes o p t i o n p r i c i n g model have e i t h e r assumed that the d i s c r e t e hedging w i l l not s i g n i f i c a n t l y a f f e c t the r e s u l t s i n any way or that i t causes u n c e r t a i n r e t u r n s , but such un-c e r t a i n t y could be d i v e r s i f i e d away by the formation of p o r t f o l i o . How-ever the analyses and t e s t s i n Sections 4 and 7.1 have shown that the use of d i s c r e t e hedging has r e s u l t e d i n a s i g n i f i c a n t b i a s which i s a f u n c t i o n of the d i s t r i b u t i o n of the r a t e s of r e t u r n on the stock p r i c e s . Consequent-l y , there i s a p o s s i b i l i t y that the hedge contains systematic r i s k s which cannot be d i v e r s i f i e d away by .the formation of p o r t f o l i o s . Despite t h i s b i a s , we are able to o b t a i n s i g n i f i c a n t l y p o s i t i v e excess returns when a simple buy and s e l l s t r a t e g y i s f o l l o w e d . This i n d i -cates that the Montreal Option Market i s i n e f f i c i e n t . Note that i f the model i s adjusted to take i n t o c o n s i d e r a t i o n d i s c r e t e hedging, we w i l l expect to obtain even higher p o s i t i v e excess returns as the model w i l l be able to b e t t e r d i f f e r e n t i a t e p r o f i t o p p o r t u n i t i e s . However, i t i s u n l i k e l y that such p r o f i t o p p o r t u n i t i e s w i l l e x i s t i f t r a n s a c t i o n c o s t s , e t c . are con-s i d e r e d . This paper does not provide a l l the answers as to the e f f e c t s of d i s c r e t e hedging on a model which assumes continuous time. However, i t does show that v i o l a t i o n of t h i s assumption i s of s u f f i c i e n t importance to warrant f u r t h e r i n v e s t i g a t i o n . Further research can be done i n the f o l l o w i n g areas ( i ) i n d e r i v i n g the Option P r i c i n g formula, higher order terms i n the expansion of dC i n equation (2.7) were ignored on the b a s i s that dt i s very s m a l l . This i s true only i n continuous time. However, as hedging - 50 -can only be done i n d i s c r e t e time, adjustment can be made to the Black-Scholes model to take i n t o con-s i d e r a t i o n such higher order terms. ( i i ) i n using the r e g r e s s i o n model we encounter v a r i o u s problems as the u n d e r l y i n g assumptions are v i o l a t e d . New techniques should t h e r e f o r e be derived f o r the purpose of t e s t i n g market e f f i c i e n c y . ( i i i ) d e r i v i n g an o p t i o n p r i c i n g model i n d i s c r e t e time. « 5 • ********** 1 * ' * * • * » * * * * * * * * ***** » J A * • 1 * * I * * * f * * * 2 0 ! * * * * * * * * * ********** * * i I * ' * * t * * * * t * * * * * * * * * * * * 1 t * * * ********** * * * * " A l b , * * * * * t * * * * A * * * * A * * * • * A ' • * * * * A * * * * A t * * * * A * * * * * A • * * * * A i * * * * * * * * * ' ! A i * * * * A * * * it A t * * * * A • ********** * * * A * * * * * A • * * * * * A i * * * * * A * * * ' * * A • * * * * * A ********** * * * * ********** * * * * * * A A 1 * * * * * * A * * * * * * * A * 1 * • * * * * A A • * * * * * * A A • * * * * * ' * A A A A A A A A A A * * * A A A A * * A * * * * * * A A ' " " * A * * * * * A * * A _ * * * * * A * A A A A A 0 , 9 4 4 0 , 9 5 4 0 . 9 6 U 0 , 9 7 4 0 . 9 6 4 0 , 9 9 4 1 . 0 0 4 1 , 0 1 4 1 , 0 2 4 1 , 0 3 4 1 , 0 4 4 1 , 0 5 4 I I * 0 5 4 . Rates of Return Appendix IA Rates of Return f o r S h e l l Canada From 16th Sepember to 20th February.1976 3t> 32 20 20 16 12 ******** * ' * ******** ******** *' * * * * * _* * * * ******** * " " * * * * * ******** ******** * * * •* ******** * * *******> ******** * * ************** * ********************** I-u.936 o,94b 0.9.S6 o.9t>6 0.9/6 0,966 o ,99 b 1 ,006 1 ,016 1 ,026 1 ,036 i , 046 1 ,056 i , 0b 6 1,076 [Rates of Re-tur n Appendix IB Rates of Returns f o r S h e l l Canada From 25th November to 20th August 1976 100 ************ ft ' ft ************ * * * * * « * * » * «« * * ************ * ' * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ************ * *********************** * * " * * o *..,, 0,960 0,970 0 ,9b0 0,990 1.000 1,010 Appendix IC 1.020 1,030 1,040 i ,05o Rates of - : Return Rates of Returns of S h e l l Canada From 26th May to 31st December 1976 1" ********** * * * *_ * * * * * i _ ********** * ** * * ******* ********** * ' * _* *... ********** ********** k * * * * * * * * * ********** * * ********** * * 0 « . . . 0 , 9 3 6 0 , 9 U 6 0 . 9 5 6 0 , 9 6 b 0 . 9 7 6 0 , 9 6 6 0 , 9 9 6 1 , 0 0 6 1,016 1 , 0 2 6 1 . 0 3 6 1 . 0 4 6 Rates of Return Appendix ID Rates of Returns f o r S h e l l Canada From 23rd November to 31st December - 55 -Appendix I I L i s t i n g of Underlying Stocks Whose Options are Traded on the Montreal Options Market as at 31st December 1976 1. A b i t i b i Paper Company 2. Alcan Aluminium 3. Bank of Montreal 4. B e l l Canada 5. Brascan 6. Canadian P a c i f i c 7. Gulf O i l 8. Imperial O i l 9. I n t e r - P r o v i n c i a l P i p e l i n e s 10. I n t e r n a t i o n a l N i c k e l Company 11. MacMillan B l o e d a l 12. Massey Ferguson 13. Moore Corporation 14. Noranda Mines 15. P a c i f i c Petroleum 16. S h e l l Canada 17. S t e e l Company of Canada 18. Trans Canada P i p e l i n e s APPENDIX I I I Inadequacy of the I n t e r e s t Rate Adjustment Fact o r The i n t e r e s t r a t e adjustment f a c t o r i s - [ C 1 - N(.d 1)S 1] rAt ( i ) S u b s t i t u t i n g equation (2.10) i n t o ( i ) we have - [ - X e " r T N ( d 2 ) ] rAt ( i i ) As the opti o n p r i c e i s a f u n c t i o n of both S and T, a change i n the optio n p r i c e dC can be expressed as dC = C TAt + C.gAS ( i i i ) where s u b s c r i p t s denote p a r t i a l d e r i v a t i v e s . Since we are only i n t e r e s t e d i n changes i n the option p r i c e as a r e s u l t of a change i n T, AS i s assumed to be zero. S u b s t i t u t i n g C T = { X e " r T [ Z ( d 0 ) - ^ r - + r N ( d 9 ) ] } A t ( i v ) wi t h Z as the normal density f u n c t i o n , i n t o equation ( i i i ) we have dC •= { X e " r T [ Z ( d 0 ) — + rN(d_)]}At (v) The d i f f e r e n c e between dC and the i n t e r e s t r a t e adjustment f a c t o r given i n equation ( i i ) i s there f o r e X e " r T [ Z ( d „ ) — + rN(d„)]At-[Xe"rT N(d„)]rAt 2 2 ^ 2 2 = X e " r T [ Z ( d 9 ) — ]'At Cvi) Equation ( v i ) w i l l be approximately zero as the hedged p o s i t i o n i s adjusted continuously. However i f d i s c r e t e hedging i s used, the i n t e r e s t r a t e adjustment f a c t o r w i l l not o f f s e t the re d u c t i o n i n the op t i o n p r i c e by t h i s amount. - 57 -B i b l i o g r a p h y 1. F i s c h e r Black and Myron Scholes, "The V a l u a t i o n of Option Contract and a Test of Market E f f i c i e n c y " , J o u r n a l of Finance, Papers and Proceedings of the T h i r t i e t h Annual Meeting of the American Finance A s s o c i a t i o n . December 1971: 399-417. 2. F i s c h e r Black and Myron Scholes (1973), "The P r i c i n g of Options and Corporate L i a b i l i t y " , J o u r n a l of P o l i t i c a l Economy, 81: 631-659. 3. F i s c h e r Black (1975), "Fact and Fantasy i n the Use of Options", F i n a n c i a l A nalyst J o u r n a l , 31: 36-72. 4. Phelim Boyle (1977), "Options: A Monte Ca r l o Approach", J o u r n a l of F i n a n c i a l Economics, 4: 323-338. 5. Cox, J.C. and Ross, S.A. (1975), "The P r i c i n g of Options f o r Jump Processes", Rodney White Centre f o r F i n a n c i a l Research. Working Paper 2-75, U n i v e r s i t y of Pennsylvannia. 6. Dan G a l a i (1975), "Tests of Market E f f i c i e n c y of the Chicago Board Options Exchange", J o u r n a l of Business: 167-195. 7. J . I n g e r s o l l (1976), "A T h e o r e t i c a l and E m p i r i c a l I n v e s t i g a t i o n of the Dual Purpose Funds: An A p p l i c a t i o n of Contingent Claim A n a l y s i s " , J o u r n a l of F i n a n c i a l Economics, Jan./March 1976: 83-124. 8. R.C. Merton (1973), "Theory of R a t i o n a l Option P r i c i n g " , B e l l J o u r n a l of Economics and Management Sciences, Spring 1973: 141-183. 9. Montreal Stock Exchange, "A Guide to the Option Market", 1975. 10. E.S. Schwartz (1975), "Generalized Option P r i c i n g Models: Numerical S o l u t i o n s and the P r i c i n g of a New L i f e Insurance Contract", Ph.D. T h e s i s , U.B.C. 11. C.W. Smith (1976), "Option P r i c i n g - A Review", J o u r n a l of F i n a n c i a l Economics, Jan./March 1976: 3-51. 12. E. Thorpe (1973), "Extensions of the Black-Scholes Option Model", 39th Session of the I n t e r n a t i o n a l S t a t i s t i c a l I n s t i t u t e (Vienna, A u s t r i a ) : 322-330. 13. E. Thorpe (1966), "Common Stock V o l a t i l i t i e s i n Option Formula", Proceedings, Seminar on A n a l y s i s of the S e c u r i t y P r i c e s . May 1976: 235-276. 14. Trans Canada Option Inc., Prospectus dated May 6, 1977. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0094470/manifest

Comment

Related Items