), is the value, on a logarithmic scale, at which this expert expects the price to return in the long run. Discussion 3.17. We note that the fundamental value of our model F}(UJ) is different from expert to expert and it is not constant because of the dependence on ui. This is one of the differences with the model of Chiarella and lori (2002 and 2004) mentioned in the introduction. In practice, the fundamental value is usually a discounted sum of future earnings. As mentioned by Lux and Marchesi (1999) the most commonly used value represents a dis-counted sum of future dividends payments. Each expert t \u00a3 I determines their reference level for the next time interval using the following equation:

) = 0. Definition 3.22. A noise trader is an expert for which both o\/zt(u>) = 0 and (3\\(w) = 0. 37 Chapter 3. The Model Discussion 3.23. As mentioned by Bouchaud et al. (2003) and De Long et al. (1990) it is well known in the literature that these three types of traders need to be present in a model if we want to reproduce realistic financial market data. It is important to understand where the term noise trader comes from. For this purpose, we recall the discussion 3.5 where we pointed out the importance of the random liquidity demand r]f(u). If agent a \u00a3 A follows the advice from a noise trader expert i 6 I then using equation 3.18 she obtains the following reference level: V*t(w) = ^_(w) (3.24) Now using this reference level with the fact that Sf(ui) = <\/>t_(w) = \\og(pt_) and the equation 3.1 we obtain the agent excess demand which is simply given by: e?tpt-,w) = Ti?(u) (3.25) In other words, when an agent follows the advice from a noise trader her excess demand becomes simply her random liquidity demand. This complete the section on the description of the financial experts. In the next section we explain the process by which the agents decide from which expert they should follow advice. 3.4 The performance measure The performance measure defined in this section links the agents and the experts. Each agent a \u20ac A associates a performance measure to each expert i \u00a3 1. The probability to follow the reference level of expert i \u20ac I will depend on this performance measure. We first introduce the notion of conditional profit realized by expert i \u20ac I and than we define the performance measure. Definition 3.26. We define the immediate conditional profit associated with the expert i El, noted TX\\, as the profit that an agent will have realized between the time interval [t\u2014, t] if she had followed expert i recommendation. More precisely, 7 r ^ = ( Y > j - S t - ) ( e * - C * - ) . (3.27) We note that using equation 3.18 ir\\ can be rewritten as follow: TTJ = [a\\(Lj)(Ft(w) - & _ M ) - &__(W))](pt -pt_) (3.28) Now we can define the performance measure. 38 Chapter 3. The Model Definition 3.29. The performance measure an agent a \u00a3 A associates to the expert i G I at time t, denoted U^'1, is the discounted sum of the past profit the expert i \u00a3 I recommendation would have generate. Mathematically, it is given by: 1 i U?'* = y -. (3.30) ~ Q (1 + 7A)T _? Discussion 3.31. We note that if an agent has a smaller discount factor 7, and then a longer time horizon h (using equation 3.9), she will give more weight to past profit than an agent with a higher discount factor 7. This is totally consistent. The agent with a shorter time horizon is less interested in the profits realized long ago because they will not wait for very long to cancel their orders from the book and so for them these profits will never be realized. Discussion 3.32. At this time, we want to bring the reader's attention to the importance of the discount factor 7. Each agent a G A has a discount factor, denoted 7\u00b0. This discount factor enters in three different equations: \u2022 First, it enters in the determination of the agent time horizon ha through equation 3.9. \u2022 Second, it enters in the calculation of the reserve prices R^'huy and R^'sel1 through equations 3.12 and 3.13 respectively. \u2022 Finally, it is used to discount the past profit in the calculation of the performance measure of the experts through equation 3.28. Finally, we use the performance measure of each expert as the parameter of an expo-nential distribution. We then draw a number from these distributions for each expert and the agent will choose the expert with the highest number. This completes the section linking the agents and the financial experts. In the next section we present a short algorithm to help understanding the complete model. 3.5 Order Submission Mechanism In this section we present an easy algorithm to explain the complete model step by step. This section is particularly relevant for understanding how the computer simulation is created. The first thing that need to consider a discrete version of our model. We then take t = 0 as the first time and the subsequent times are simply t = 1, t = 2, etc. In particular, this will change the notation of t\u2014 for t \u2014 1, t for t \u2014 2 and t+ for t + I. 39 Chapter 3. The Model 1. For each agent a G A we draw a number from a Poisson distribution with constant parameter 11. If we let, 9a(u) be the result of such a drawing for each agent a \u20ac A, then we chose the agent a with the highest 6. a = a r g m a x { 0 n ( \u00bb } . (3.33) 2. Now that we have determined which agent will be trading for the next period we need to determine from which financial expert this agent will pick up the reference level for this period. This is done by calculating the performance measure C\/\"'1 of every expert i g I given by the equation 3.30: 4 i n a , i = >p Th Then, we draw a number for each expert from an exponential distribution with pa-rameter and choose the expert with the highest draw. 3. Now that we have determined our expert i we can calculate her recommendation ip\\ for this period using equation 3.18: \u00a5>!(\u00ab,) :=