This paper contains an investigation on a particular kind of non-linear rational expectations equilibrium in financial markets. By adopting an agent based computational finance (ACF) paradigm we will analyze whether using real options theory in financial markets is useful. Do agents who incorporate option value -when forming their trading prices - obtain higher profits? Real Options based valuation can be a valuable tool when the agents interacting in the market are homogenous in their cognitive abilities to understand and learn market dynamics, even if they are heterogenous in their ideas or ''market theories''. Speed of the learning process is another factor which crucially determines the relevance of an options based approach to valuation. If we introduce a small portion of traders acting on the basis of different strategies, we observe drastic deviations from the rational benchmark. Moreover the profitability of ROV based traders is inferior to random guessing, and to neural network based strategies.

Keywords:

Agent Based Models; Artificial Markets; Classifier Systems; Economic Simulation; Market Microstructure; SWA

Introduction

1.1

During the last ten years Real Options have frequently been considered as one of the most powerful and sophisticated tools of corporate finance theory.

1.2

Despite the almost general academic the adoption of ROV (Real Options valuation) in the corporate and financial world has been slower than expected. This phenomenon can be explained basically in three ways:

• By the status quo's tiranny
• By the scarcity of option based contexts
• By the coordinated nature of value

1.3

Business world's inertia could be considered to be responsible for the only moderate success of the Real Options technique in corporate management practice. It took about 20 years for NPV to become a standard in valuation. This explanation turns the issue essentially into a problem of vintage human capital.

1.4

This first explanation sounds reasonable but is not fully satisfactory. If Real Options are such a significant improvement in the valuation science, the competitive advantage generated by adopting a similar tool when all other players rely on NPV, should be so great as to stimulate a quick spread of knowledge and skills. From this point of view, it would just be a matter of time for ROV to become dominant.

1.5

The second concurrent explanation places a deeper and more serious problem regarding the intrinsic relevance of ROV. According to this idea, Real Options approach is justified only in particularly uncertain and highly non-linear environments, which supposedly are rare. In a simple context, where the option value linked to managerial flexibility is negligible, or where it could be approximated by relying on the NPV ''linearization'', the investment in skilled personnel and the longer period of time required to perform a more sophisticated analysis might not be acceptable. For those who share this view, the scarcity of ROV uses in actual business practice is not caused by the inability of agents to gain extra-profits from a powerful tool. They believe that ROV is valuable in specific contexts (oil investments, pharmaceutical R&D, etc.), while there would be no significant advantages by employing it in many other ones.

1.6

Finally an even more serious hypothesis regarding the opportunity of using ROV could be raised, based on the coordinated nature of the process which is implicit in any valuation exercise. A value which is beyond the internal limits of the organization that elaborates the valuation, should generally be accepted on a market. Market value is the result of a network of nested expectations based on the choices of other agents acting in the market. When the agents do not share a common ''view of the world'' (i.e. they are not aligned in using the same valuation technique), we enter the ''wilderness'' outside the rational expectation equilibria that have been the central focus of financial economics until now. Are we sure that in such a context using a more sophisticated tool such as ROV is profitable? In other words, if a fraction of the agents present in the market takes its decisions by relying on strategies which are different from the option based one, can we be sure that we will find ourselves adopting ROV, in the pleasant (for us!) situation of an evolutive path converging to an equilibrium where ROV dominates? If this is not true we will not get extra-profits from the use of ROV because of a coordination failure. According to this view Real Options analysis is rarely used because currently only in a few markets an ROV based equilibrium has emerged.

1.7

In this study we will try to explore these three hypothesis by looking for a dynamic story that can tell us if we can expect ROV to become a widely diffused standard in the next ten years in accordance with the first explanation, or whether we will observe a process of specialization in the use of Real Options which will restrict the practice to well defined contexts, or finally if their adoption depends on some ''critical mass'' phenomenon.

1.8

By adopting an agent based computational finance (ACF) paradigm we will observe a population of traders who are heterogenous in rationality and beliefs, and who trade shares in an artificial financial market that represent a firm endowed with a growth option. Is the capability to incorporate the growth option in the valuation useful? In other words, do agents who incorporate this option value when forming their trading prices obtain higher profits?

1.9

We study such a problem in an evolutionary perspective by looking at what kinds of equilibria emerge in the complex environment of our artificial financial market in different contexts.

1.10

We will find that Real Options based valuation can be a valuable tool when the agents interacting in the market are homogenous in their cognitive abilities to understand and learn market dynamics, even if they are heterogenous in their ideas or ''market theories''. The speed of the learning process is another factor which crucially determines the relevance of an options based approach to valuation. When learning is more intense, the frequency of the updates in agents' beliefs generate some difficulties in coordinating the valuation scheme. In such conditions the emergence of a common vision of value is more problematic.

1.11

We will explain these considerations more clearly and in details after examining the issues that we have briefly mentioned in this introduction. A brief overview of the literature which applies agent based methodology to financial issues, is presented in section 2. Section 3 introduces the main characteristics of our artificial financial market and its relationship with the models already present in the literature. The simulation results are described in section 4.

1.12

Final observations and comments about possible future extensions of this research conclude the paper.

Financial Economics in Agent Based Literature

2.1

The standard asset pricing models are generally based on the assumption that identical investors who share rational expectations models on assets future prices and who instantaneously and rationally discount all market information into prices behave on financial markets. Neoclassical theory suggests a sequence of conclusions which are hard to believe: i.e that in equilibrium, trading volume is very low, that prices are not serially correlated, that technical trading should be totally absent, that the return on risky assets would be lower when compared with observed ones. At the same time standard financial market models are affected by at least one great theoretical paradox concerning information gathering: the Grossman-Stiglitz's one.

2.2

The market in this theoretical view is rational and efficient. On the other hand if we read a newspaper or if we participate in a conversation on financial topics anywhere outside an academic institution we can frequently hear expressions such as ''the market is nervous'', or ''depressed'' or ''excited''. Is this organicistic use a purely linguistic phenomenon or do economic agents actually perceive ''the market'' as an entity with its own personality?

2.3

The ''noise trader approach'' has been introduced in the theory of rational expectational equilibria to overcome some of the difficulties listed above. Assuming that the noise traders act on the market according to a different kind of rationality from the perfectly rational agents, the research on financial topics has historically taken the first step towards the heterogeneity of agents' rationality. Noisy rational expectational equilibria theory has many advantages when compared with the homogenous, fully rational, vision of financial market, but it still fails to illustrate the endogenous nature of expectations under heterogeneity. Moreover the noise trader approach, although it criticizes the theory of rationality employed in mainstream literature, does not try to build an alternative theory of cognition in financial markets.

2.4

As noted by Le Baron (2000), financial markets are an important application for agent based modeling styles. In this field, there is great appeal in starting from the bottom-up with simple adaptive, learning agents, which stress interactions, heterogeneity and learning dynamics in groups of traders. The issue of heterogeneity even if it is not new in finance (differences in information and in rationality have been mainly explored as we mentioned above) could be exploited to the maximum degree with an agent-based approach.

2.5

To study the evolution of financial market systems where the underlying assumption is that agents cannot deductively compute any kind of equilibrium because of the complexity of the system, agent based computational models are probably the best candidate methodology. In other words we want to study a market environment in which it is impossible or at least ''hard'' for the agent to analytically derive an equilibrium strategy. Under these circumstances agents have to rely on evolutive rationality strategies, and in this case agent based models can be considered as "complements to mathematical theorizing" in the taxonomy introduced by Axtell (2000). An important literature has been developed, by the means of analytic tools, by Bray (1982) and Blume and Easley (1982), (1992) on the convergence of rational expectations from a mispecified model through a learning dynamic. Some of the models presented in the ACF literature can be considered as an alternative way of exploring the issue of how to coordinate out of equilibrium rationality. In this case simulations and analytic studies may be perfectly synergic.

2.6

The ACF literature presents greatly variegated themes and conceptual frameworks. It is difficult to find a unitarian perspective to survey the studies which have appeared in recent years. We can interpretate a large part of the ACF literature along the lines of the so called ''mind / no mind dilemma'' illustrating how important macro properties have been achieved by the interactions of micro components deprived of relevant intelligence (Zero intelligence traders)^[1] and how on the other hand artificial markets have been built by starting from traders endowed with different degrees of cognitive abilities (''Minded'' agents).^[2] Furthermore agent based approach has been used to study the issue of equilibrium selection in infinite rational equilibria environments (Equilibrium selection)^[3]. Finally, the properties of artificial time series generated in ACF models are discussed by a part of the literature (Fitting actual time series)^[4].

2.7

Software platforms and conceptual structures are not common and broad comparisons across models are difficult. However first results show that certain feature of financial data which remain puzzling to single representative agents models, may not be hard to replicate in a multi-agent world.

2.8

For the purposes of a comparison between different valuation techniques, as the one we want to build, agent based paradigm seems a perfect fit. A heterogeneity of evaluating schemes can be put in competition in a dynamic environment, the artificial financial market, to understand the main properties of this complex system by the means of simulations.

Testing in an Artificial Environment

3.1

To test whether a Real Options approach to valuation is in some way valuable in terms of realized profitability, with respect to other techniques, we created an artificial experiment by working on SUM (Seemingly Unrealistic Market), the general artificial financial market architecture introduced by Terna (2001). The simulation has been coded in Swarm.^[5]

3.2

SUM, is an example of Swarm's potential at work. It allows the researcher to investigate an artificial financial market populated by different kinds of operators. The great modularity of the platform makes easier to implement agent with extremely heterogeneous cognitive devices plugged in (in our case artificial neural networks, classifiers systems and rule of thumbs coexist). Moreover SUM gives us the opportunity to describe the trading's institutional details (for instance the book's formation process) without practically any restriction.

3.3

The first part of the section examinates, with standard analytic tools, the existence conditions for an ROV based rational expectations equilibrium in financial markets. Section 3.5 introduces an operative firm which has the right to exercise a growth option, and derives a simple closed-form solution to valuate this option. The next sub-section presents a text-book's version of assets allocation's problem with CARA utility functions. This framework is becoming the standard in the construction of such kinds of models. We then derive some rational expectations equilibria for agents behaving according to linear NPV valuation's rules and to non-linear ROV based ones.

3.4

The second part of this section is a description of the simulation. Sub-section 3.15 analyzes the relationships between this version of SUM and the ACF literature. In this context, the artificial traders introduced by Terna (2001) are examined, as well as the market clearing mechanism. Sub-section 3.29 is focused on classifiers system based agents. We use this class of agents to model agents' capability to ''learn'' an ROV based REE. These agents are endowed with an initial database of rules containing both ROV and NPV valuation techniques. We will study in the next section which rules will prevail when the classifiers system based agents are the only kind of traders in the market, and when they have to compete with all the others Terna's agents.

General settings

3.5

Let us then introduce the setting of our artificial experiment by stressing the main differences with the Santa Fe stock market (ASM - artificial stock market) and with the standard version of SUM.

3.6

We consider a market in which heterogeneous agents trade on a two assets space: a risk less bond, infinitely supplied, paying risk less interest rate r and a risky stock paying a stochastic monthly^[6] dividend^[7] which is a constant fraction of the stochastic cash-flow gained by the firm. Furthermore we assume that the firm associated with the risky stock is a mono-project one.^[8] The value of the firm can be expressed as:

where V(d_t) is the NPV value of the project already in place and F(d_t) is the value of the growth option that the firm can exercise when the cash flow is above its trigger value d*.

3.7

We will assume that the dividends follow a geometric Brownian motion process with a ''rational jump'' connected with the exercise of the expansion optino .

with is the cash flow's average rate of growth. is the scale multiplicative factor of the cash flow (under our assumptions equal to the dividend). It indicates how big the impact of new investments on cash flow will be when the expansion option is exercised. I is the exercise cost of the option, the value of the investments required to expand the scale. We assume and I to be common knowledge.

3.8

We assume that the management of the firm uses flexibility to generate value; in particular, it considers whether to exercise the expansion option or not. We are then innovating the existing ACF literature not only by adding a fundamental process in the SUM environment but also assuming that the dynamics of the firm's cash flow is affected by a strong non-linearity in presence of managerial flexibility.

3.9

Quarterly the management of the firm decides whether to exercise the expansion option by getting a cash flow equal to and making investments of amount I or to keep the expansion optino F(d_t) alive and so obtaining the dividend d_t from the project which is already implemented.

3.10

The stochastic process of the cash flow is then subject to ''jumps'' under the optimal control in terms of Real Options theory of the management ^[9]. A threshold value d* would be the trigger to exercise the option and make the stochastic process of the cash flow d and its related process V jump.

3.11

We are interested in understanding whether or not, by incorporating this notion of option based stochastic discontinuity in the cash flow process, traders acting in a financial market could obtain a superior performance when competing with an heterogenous population of traders. To recall the title of this study we would like to explore the opportunities of obtaining extra-profits trading on the basis of a Real Options strategy when the assets traded in the artificial financial market are actually managed according to the Real Options theory.

3.12

We need to dedicate some space to highlight the main results of financial market microstructure theory in order to illustrate how the strategies of our classifiers system based agents relate to fully rational models.

Rational financial markets

3.13

In a standard neo-classical setting each agent attempts, at each period, to optimize her allocation between a risk free asset and a risky stock. Assuming that the agent i has a CARA (constant absolute risk adversion) utility function from the standard maximization problem we get the investor's i demand function for the risky asset at time t:

where is the agent's degree of relative risk aversion. The risky asset's demand is a function of the expected excess return E[ p_t+1+d_t+1] which the trader can obtain by investing in this asset rather than the riskless one. The model is closed by determining the clearing price p by setting the demand equal to the supply (i.e.. by assuming one share supplied per agent):

3.14

Agents then use their information I_i,t which include historical dividend sequence and past price sequence to form their expectations of the next period's price and dividend E[p_t+1+d_t+1]. Then they compute their desired portfolios by determining demand curves and by passing the demand to some kind of market clearing mechanism which calculates p_t. By this time exchanges are realized. This sequence repeats itself over time.

Solving an NPV linear homogeneous rational expectations equilibrium

3.15

We will derive the homogenous linear rational expectations equilibrium (REE) which is central to many studies of ACF.^[10] The fundamental assumption to solve this class of models is that the risky asset market price is a linear function of dividends:

This assumption is perfectly in line with an NPV valuation process of firms.

3.16

To define the investor demand is crucial to determine the expectation E[p_t+1+d_t+1]. We will calculate this quantity in two different cases: when the dividend process is an AR(1) and when dividends are distributed according to a geometric Brownian motion.

3.17

When dividends are distributed according to an AR(1) process and the agents conjecture that there is a linear relationship between price and dividend, the equilibrium price will be:

3.18

If the dividends are distributed according to a geometric Brownian motion process we identify two possible rational expectations equilibria:

3.19

By looking at equation above we find that p_t is a real number only when . The conditions for the existence of a linear rational expectations equilibrium when the dividends process is a geometric Brownian motion appears extremely restrictive given the values of stock prices variances and the investors' risk aversion generally assumed in the ACF^[11]. Nevertheless it seems reasonable to investigate whether a non-linear specification of the relationship between price and dividend could be more useful in finding an REE for the case of dividends distributed according to a geometric Brownian motion. Real Options provide a fundamentalist explanation of this non-linear specification. We will illustrate our results in the next sub-sub-section.

An ROV non-linear rational expectations equilibrium

3.20

Here we study a case of rational expectation equilibrium consistent with the situation described in section 3.5. In this case the investors incorporate the notion of growth option in their decisions which is available to the firm. The consequence is that when the growth option is unexercised , the price of the risky asset is a non-linear function of the current dividend d_t.

3.21

This non-linear specification of the price function is somewhat similar to the intrinsic bubble on stock assets prices presented in Froot and Obstfeld (1991). The Authors consider bubbles that derive all their variability from the exogenous economic fundamentals and none from extraneous factors. The explanatory power of such a specification comes partly from the ability to generate persistent deviations that appear to be stable over long periods. Here we present a theory of asset pricing which is in some sense ''observationally equivalent'' to the intrinsic bubble one. Our growth option model provides an explanation for the behavior of asset prices in line with the empirical evidence of Froot and Obstfeld. By formalizing asset prices with non-linear equations like (8) we identify a bubble-like behavior which finds its origin in Real Options theory.

3.22

We will guess for the coefficients c₀, c₁, c₂, c₃, relying on the result obtained in the linear AR(1) case, presented in the previous sub-section, for the coefficient c₀ and c₁ and on the coefficients for the expansion's option for the coefficients c₂ and c₃. We are fully aware of the loss of generality induced in our results by this ''guessing'' approach, but we should not, on the other hand, forget our research goal, which is to test whether the ROV technique is a superior tool for valuation in financial markets. Our behavioral restriction is very important if someone wishes to establish if, in general, a vector c₀, c₁, c₂, c₃ that constitutes an REE exists, but it is not so important if we limit ourselves our study to NPV and ROV based equilibria.

3.23

We use the following non-linear rational equilibrium specification of the pricing function:

3.24

By putting the functional form above in the\ financial market equilibrium equation we get^[12]:

3.25

The condition above states that the ratio between the expectations on and should be equal to 1 plus the risk less rate r , the expansion option regime, in order to be a rational expectations equilibrium. This statement is, in some way, similar to the bubble growth condition of Froot and Obstfeld. This ''unstable'' condition may help us to explain phenomena like the recent New Economy boom and burst. An equilibrium with agents who incorporate the growth option value of firms is ''fragile'', as we noticed above. The collapse of high tech firms' valuation may be explained in terms of a ''jump out'' of the ROV based REE due to some disturbances in the expectations on .

3.26

We are now interested in investigating what happens outside this rational setting. Is it possible that artificial cognitive agents learn how to reach this REE? Will they form their expectations coherently with the theory above illustrated or will we observe the emergence of different behavioral strategies? The following two sub-sections complete the description of the artificial financial market.

Artificial market architecture

3.27

Our artificial setting is differentiated from the standard one in many respects that we can summarize in the two following points:

• Heterogeneity: the agents formulate expectations separately and are heterogenous in their cognitive abilities^[13]. In our artificial environment, as well as in Terna's SUM, agents are heterogenous also with regards to the degree of rationality, as will be shown. They neither communicate their expectations nor their buying or selling intentions to each other.
• Market Clearing: with regards to this aspect, a high degree of realism is implemented in the model. The book's formation mechanism of the Italian computerized stock exchange is coded in the model. The agents send their orders to the book with the related limit prices. The book immediately performs the orders if a counterpart is found in its log, otherwise it records the orders so as to match them later. The book is cleared at the beginning of each trading day and a price formation procedure is explicated to start the negotiations again. The issue of how market structure affects viability and stability has been extensively studied in financial microstructure theory. Various characteristics of the trading mechanism affect the transmission of information on prices. This process is important for several reasons: to evaluate how institutional features affect agents' abilities of learning; to develop an understanding of why some institutional arrangements dominate in some market settings and not in others; to study the relationship between stability and performance of the market. Usually two trading mechanisms are analyzed in theoretical studies: the sequential trade models employ a scheme in which each agent faces a bid-ask price and each single trade is realized at the quoted price. The batch models are characterized by orders batched together to transact a single market-clearing price without any bid-ask spread. In many of these models, agents are allowed to submit only simple buy or sell orders and cannot enter contingent limit orders. Clearly, in absence of limit orders, there are no books of unfilled orders and no difference in the order flow information available to traders. An analysis of the current financial markets reveals a rather more complicated structure of the trading mechanism compared to the schemes which are generally implemented in theoretical papers. Our effort towards realism is then justified by considering the importance of trading rules on market performance^[14].

---

Table 1: Agent Classes present in the original SUM

---

Agent	Rationality
Random	None
Market Imitating	Imitation
Stop Loss	Rule of thum
Forecasting	ANN based forecasting
Cross Targets	Cognitive abilities

---

3.28

We will use the various classes of agents discussed in detail by Terna (2001) and which are present in our model, focusing on classifiers system based agents, that are the most relevant to test the effectiveness of NPV vs. ROV. Table 1 illustrates the different degrees of rationality represented in the model.

NPV, ROV and classifiers agents

3.29

We introduce in the SUM environment a new class of classifiers system agents, based on Ferraris (1999) CW library.

3.30

We assume that each of the agents of this new class possesses a multiplicity of forecasting models, as in ASM, at any time. Every forecasting model can be considered as a ''theory of market'' and the agent has to decide which of the theories she has in her toolbox better fits the current state of the market and the fundamental nature of the assets. We will focus particularly on agents having different opportunities of recognizing and valuing Real Options, affecting the operations of the risky assets. Agents learn in our environment, as in the Santa Fe experiment, not by updating parameters, but by discovering which of their hypotheses obtains superior results. Is it correct to think that firm X has an expansion option? Can the firm exercise a contraction option in the case of unfavorable states of the nature? Would it be better not to waste time with these option based considerations and to remain near the traditional NPV valuation of the asset? These are just sample questions arising in the cognitive minds of the agents populating the artificial financial market.

3.31

Arthur et al. (1997) find at least 3 great advantages in a similar cognitive approach:

• It avoids the bias introduced by a fixed, shared expectational functional form.
• It allows the individuality of expectations to emerge over time, by the means of the genetic algorithm associated with prior beliefs.
• It provides an appealing description of cognitive reasoning in which different agents might ''cognize'' different patterns and arrive at different forecasts from the same set of information.

3.32

Agents' subjective expectational models are represented by classifiers systems. Each classifier has a conditioning part which contains a market condition that may be fulfilled by the current state of the nature, a forecasting formula for the next period's price and dividend and an action part which contains the rules to calculate the expectations on next cycle values. The state of the nature is encoded in a vector of J bits. Each agent possess M individual classifiers, representing the set of market theories she knows and uses the most accurate of those that are active (matched by the current state of the nature vector). The states of the nature are encoded according to the scheme container in table 2.

3.33

The first four binary descriptors reflect the current price in relation to the current dividend and thus indicate whether the stock is above or below fundamental value at current price. We will call the first and the second bit of the bit string ''NPV'' bits. Binary descriptions 5-7 are the ''technical trading'' ones that try to identify a trend in the data.

3.34

Binary descriptions 8-11 are the ''Real Options'' ones. The fifth and sixth bits encode the possibility that the firm might exercise an expansion option. The agent incorporates in her expectations the volatility of the underlying asset which is one of the key factors to evaluate such an option. Unlike the ROV bits, the bits at the seventh and eighth place condition agent expectations not on the correct standard deviation (the underlying asset, the project itself), but on the variance of the stock in the artificial financial market. These quantities may be systematically different in presence of asset bubbles. We will compare the relative performance of these two alternative approaches, observing the results of the ''quick and dirty'' approximation against the performance of a more ''scientifically grounded'' option evaluation. Both the variances and transmitted by the detectors to the classifiers systems and used to calculate the option value are computed by using an historical measure of these two volatilities^[15].

3.35

The conditioning part of each classifier corresponds to these binary market descriptors and thus consists of an 8 bit array, each position of which is filled with a 0,1 or #. A conditioning part matches the state of the nature when all its 0's, 1's and #'s match the 0's and 1's of the state of the nature vector^[16]. The conditioning part, when it is matched, activates the classifier and through an auction mechanism activates a corresponding forecasting expression.

3.36

In the experiment all forecasting rules use a non-linear combination of price and dividend:

3.37

Each classifier stores a specific value for a₁, a₂, a₃, a₄ and a current estimate of its forecast variance .

3.38

Classifiers with a high degree of generality have a small number of 0's and 1's in their conditioning vector. The genetic algorithm creates new classifiers by mutating the values in the conditioning vector or by recombining parts of different conditioning arrays.

3.39

Given the expectations formed by the classifiers system, the agent can calculate the desired risky asset holdings according to (5.7), compare it with its endowment and submit to the book an appropriate order:

3.40

We should point out that relation (11) holds only under the gaussianeity of stock prices. This will hold in the linear REE model but outside this regime it is not clear what the distribution of stock prices will be. In this case the demand function should be taken as given. We cannot rely on the maximizing assumption of a CARA utility function. In the next cycle, price and dividend are revealed and the accuracies of the active classifiers are updated.

3.41

Learning in this system is powered by the selection process of the classifiers and by the evolution of new ''ideas'' performed by the genetic algorithm. The learning space is constrained by the dimension of the bit string encoding the state of the nature. With 8 bits we can search over 256 different states of the nature. Classifiers that recognize states that rarely occur will be used unfrequently and their accuracy will be updated less often and, coeteris paribus, their precision will be lower.

3.42

This market information design allows us to study the ''emergence'' of valuation techniques. When ROV techniques are widely used by the agents, the bits coding the value of volatility should be massively included in the conditioning part of active classifiers and consequently the values of the parameters a₁, a₂, a₃, a₄ should incorporate the option value embedded in the risky asset.

Simulation Results

4.1

Before we start to study the properties of our artificial financial market we would like to point out the differences between an ROV based and a linear NPV based equilibrium. In the first case the price of the risky asset is always over the NPV equilibrium price. This is essentially due to the growth option which is incorporated by the agents in their pricing model.

4.2

In our experiments^[17] we would like to assess whether or not an ROV pricing regime may emerge in an artificial financial market, when traders are endowed with differentiated cognitive abilities. We will study the dynamics of the artificial financial market in two different settings: in the first one the simulation is populated by classifiers system based agents only, while in the second case the classifiers system agents interact in the artificial experiments with populations of traders who were present in the original SUM scheme introduced in 3.27

4.3

Each experiment will be run for 5.000 cycles, which are equivalent to approximately 20 years of real time, to allow asymptotic behavior to emerge if it is present^[18]. Table 3 synthesizes the main parameters employed in the simulations.

Homogenous cognitive abilities

4.4

In this section we restrict our analysis to simulations characterized by the presence of classifiers system based, agents which have been described in 3.29 (HOCA case henceforth). In this context all the agents are endowed with the same cognitive potential. Even if they don't share the same model of the economy, they have the capability to evolve a vision of the financial system which is in some ways quite similar to the rational benchmarks described in 3.15 and 3.16. As proved by Arthur et al. (1997) and in many of the subsequent studies, classifiers system based agents (CS agents henceforth) may be able to ''learn'' a linear rational expectations equilibrium. Now we will test the hypothesis that they are capable of reaching an ROV based expectations equilibrium when the dynamics of the fundamentals is coherent with the growth option scenario previously described.

4.5

The parameters governing the functioning of the classifiers systems are listed in table 4.

Table 3. Main simulation parameters

4.6

Forecasts are built, as we have seen in 3.29, as a non-linear function of price and dividend. For each matched bit string there is a corresponding vector composed of five real values which in turn corresponds to the forecast parameters and a conditional variance estimate. This vector is then mapped into the forecast equation to determine the agent's demand:

4.7

The parameters a₁, a₂, a₃, a₄ and and of the starting 32 rules of each classifiers system are set to random values distributed uniformly through the REE equilibrium forecast values. The first 32 rules constitute some sort of ''common knowledge.'' These 32 rules are set-up by using some kind of ''standard rationality'', which links the conditioning and the actioning part of each rule. For instance if the conditioning part is not imposing any requirement on ROV and Incorrect ROV bits then the value of the non-linear option based parameters are set as equal to zero. Such coherence is not present a priori in the rules generated by the classifiers system, because we would like to allow agents to have the incoherencies that are frequently observed among investors.

Table 4. Classifier system's parameters

4.8

We decided to reduce the number of initial rules in the mind of each agent, following Tay and Linn's (2001) objection to the original's ASM configuration. The Authors questioned whether a trader could manage a very large database of ideas^[19]. They propose as an alternative a fuzzy-logic based classifiers system which drastically reduces the number of initial rules. We choose instead to simply reduce the number of initial rules. Our results will show that many of the propositions derived by ASM are robust to large differences in the initial rules database's dimensions.

4.9

For each rule the five parameters remain fixed for the rest of the life of that forecasting rule. New values are set through the evolutionary engine of the classifiers systems.

4.10

To determine which classifier will determine the agent's actions and which rules will be favored by the selection process, we should describe the classifiers system's auction mechanism implemented in the simulation based on Goldberg (1989). When its conditioning part is matched, each classifier is required to bid an amount of its resources to a central auctioneer. Every bid is calculated according to:

where specificity is the relative specificity of the rule: the ratio between the number of # in both the action and the conditioning part and the length of the rule. The purpose of this cost is to make sure that each bit is actually serving a useful purpose in terms of a forecasting rule; s_j,t-1 is the strength of rules j at time t-1, W is a random real number between 0 and 1. Bid_j,t is the amount the auction's winner actually has to pay.

4.11

The reward for the activated classifier is based on the classifier's ability to predict the value of p_t+1+d_t+1.

4.12

The accuracy of the classifier j at time t is calculated according to an exponentially weighted average of the inverse of its squared forecast error:

4.13

The value of determines the horizon length that the agent considers relevant for forecasting purposes. If =1, then the rules only use the last period forecast error. If the agents use all past data then they would be making the implicit assumption that the world they live in is in a stationary state. The value of is critical and we are still waiting for a theory that will allow us to endogenize that parameter. We will use the value used by Arthur et al. (1997) and by Le Baron et al. (1999) of 75 in our simulation. The reward is then calculated by comparing the accuracy of the activated rule, with the mean accuracy of the previous activated rules. If the accuracy ^v2_j,t is below the average, the reward is equal to 1; if ^v2_j,t is equal to the average, the reward is 0; and finally if ^v2_j,t is greater than the average, the reward is equal to 1.

4.14

In each cycle the strength of each rule is then finally calculated according to:

4.15

A genetic algorithm is implemented in every cycle with probability equal to EvolutionProb. This evolutive engine allows the agents asynchronously to develop new rules, replacing some of the old ones. New rules are generated by applying the standard genetic GA's operators crossover and mutation. Rules are selected for reproduction on the basis of a Roulette Wheel random algorithm^[20] based on the strength of each rule.

4.16

We run two different sets of experiments. In the first set the genetic algorithm is activated, in each cycle, with probability equal to 0.1%. We call this regime the ''slow learning case''. In the second set of simulations the genetic algorithm will be activated with a higher probability (0.4%) and we will call this setting the ''fast learning case''. In ASM a well differentiated behavior emerges in correspondence to these two scenarios. We will explore the effect of learning speed in our environment.

Figure 1. Actual and REE prices in the HOCA Slow learning case

Discussion

4.17

First of all we compare the market price observed in the simulations with the REE price in figures 1 and 2. Given the initial disequilibrium in the expectations, agents need learning time to coordinate their forecasts and this leads to a higher price's volatility than the one simply connected with fundamentals dynamics. An inspection of the REE solution, under the assumption that the relation between the risky asset price and dividends is non-linear (ROV case), reveals that the market price observed in the model above the REE price. As an increased price variability reduces the market clearing price, coeteris paribus, because the expression for the intercept parameter in what is now the temporary market clearing equation, falls with increased variability. However, as has been discussed in 3.15 and 3.16, the size of this parameter is negligible compared with the others, when the dividends are distributed according to a geometric Brownian motion. This effect is therefore not very important in our case.

4.18

A visual inspection of the simulated paths reveals that actual prices in the artificial financial market tend to fluctuate around the REE ROV - based values in the slow learning case.

4.19

A certain degree of the variability which is present in the series can be attributed to the agent's stochastic learning process. It is difficult to estimate how big this effect is.

4.20

However as in ASM and in Tay and Linn (2001), the simulations reveal that the variability of actual prices is greater in the fast learning case. Table 5 reports some descriptive statistics on the price time series. The larger volatility in the fast learning case (EvolutionProbability 0.04) can be attributed to the more frequent evolution of the agents' ideas. This more intense switching in the agents' beliefs can give rise to higher volatility because they need time to adapt and coordinate their learning processes.

Figure 2. Actual and REE prices in the HOCA fast learning case

Table 5. Data on prices - HOCA

4.21

Table 6 contains some statistics on the excess return of the risky asset excR.

Table 6. Data on returns - HOCA

Table 7. Trading volume - HOCA

4.22

Regarding the standard deviation of excess returns, these results are fully in line with the order of magnitude observed in real financial markets, and in the other main artificial financial markets presented in the literature. It is interesting to note how our kurtosis' value are larger than those observed in reality. In all other models of artificial trading, researchers experienced exactly the opposite problem. Their kurtosis' values were too small compared with the real ones. Probably this totally different characteristic of our model is due to the non-linear relation, linking price and dividend expectations model. This non-linearity makes more difficult for the agents to coordinate their ''market views'', thus generating some form of mutual understanding which could greatly reduce the kurtosis of the risky asset excess return.

Trading volume

4.23

Figure 3 and 4 present observed trading volume over a simulation in the fast-learning and in the slow-learning case. Trading is active, consistent with real markets. Summary statistics for trading volume, Vol, are presented in table 7.

4.24

In the fast learning case, trading volume is on average higher than in the slow learning one. The variability of the trading volume is higher when the agents update their databases of cognitive rules more frequently. The values of volume autocorrelation are in line with the results obtained by Le Baron et al. (1999), Tay and Linn (2001), and Chen and Yeh (2001), and with the quantity observed in real financial markets.

4.25

Volume tends to decrease through time, because of the effect of the learning process. The more the agents reach a ''common vision'' of the risky asset price, the less they need to trade in the market. The no-trade condition, which affects fully rational models, is not reached even asymptotically given the nature of the evolutive learning process. Classifiers systems} attribute a considerable value to genetic heterogeneity, which preserve the model from converging towards an homogenous rational state.

Figure 3. Volume in the HOCA slow learning case

Figure 4. Volume in the HOCA fast learning case

Market efficiency

4.26

As we have already said, figure 1 and figure 2 show the ROV based REE prices and the market prices actually observed in the simulation. If the simulated prices after adjusting for volatility track the REE price, we should expect to see a constant difference between the two prices. The differences plotted are not constant across time, implying that the expectations held by the agents are not always consistent with an REE. This type of behavior is not uncommon in real financial markets. The historical record suggests that most of the time, prices of financial securities appear to be set in a ''semi-efficient'' fashion. However, there have been moments during which prices depart from their theoretical values and appear to exhibit behaviors unrelated to fundamentals}. This is specially evident during those periods that most observers would classify either as bubbles or as crashes; the New Economy is the most recent and conspicuous example probably in the actual markets. The last part of the fast learning case time series, for instance (figure 2), is an example of positive bubble where the market price formed in the artificial market fluctuates above the REE value for a period of a certain length^[21]. Loosely speaking we may conclude that an ROV based price is generally ''feasible'' in a population of agents which are heterogeneous in the initial set of ''market theories'', but are homogeneous in their cognitive abilities. As we will see, this result is not robust to a further degree of agent heterogeneity.

Classifier rules dynamics

4.27

If we compare our position in exploring the dynamics of our artificial financial markets with that of financial econometricians studying actual financial markets, we should remember that we have an important advantage over them. We can ''look inside the minds'' of the artificial traders. In other words, we can directly study the dynamics of classifier rules within each classifiers system rather than have to rely only on the statistical tools based on aggregate time series examined in the previous sub-section. This is one of the most important reasons for using a classifiers system cognitive device when modelling the ''mind'' of artificial agents. Classifiers systems, they have the advantage of explicitly showing which kind of rule, or ''market theory'', the agent is using to determine her behavior.

Figure 5. Percentages of the conditioning part's bit set - HOCA slow learning case

4.28

Figure 5 shows the average fraction of bits set for the slow learning case by dividing the simulation time in 5 intervals of 1000 cycles each. The percentages of bits set indicate the pair of bits in the conditioning part that are not ##, #1, 1#, 0# or #0, averaged over all the classifiers and the agents. In other words, this figure reports the fraction of rules that are conditioned by different pieces of information.

4.29

It is interesting to note that the only piece of information which has a steady state of setting is the pair of bits referring to the variance of the dividend process (ROV bits). In the HOCA slow learning case, agents understand through time that they are operating in an option based context and correctly form their expectations on the basis of the relevant pieces of information. Technical trading bits decline in importance as well as Fundamentals and Uncorrect ROV ones.

Heterogenous cognitive abilities

4.30

Having examined what happens when agents with classifiers system based rationality populate a financial market, we can now turn our attention to a much richer environment. In the heterogenous cognitive abilities case (HERA) we study populations of differentiated traders interacting in the SUM artificial financial market. Our goal now is to study what the impact of the heterogeneity of agents' cognitive abilities is on the results presented in the previous sections describing the HOCA case. More clearly: what happens when CS agents have to compete with different kinds of traders? Even more specifically, are we sure that the results previously achieved, in terms of capability of reaching an ROV based REE equilibrium, are robust to this extremely relevant perturbation of the underlying assumptions? Are agents endowed with some form of superior rationality and with a-priori knowledge capable of exploiting these advantages and of gaining extra-profits? If not, why should someone use a real options-oriented measure of value when estimating target prices in real financial markets? We would like therefore to explore, by means of our artificial environment, the opportunities available to a ''smart investor'' with ''brilliant ideas''. Our result will show that both these conditions are not sufficient to be sure of success.

Table 8

4.31

To answer such questions, we use differentiated populations of traders based on the behavioral types introduced by Terna (2001) and illustrated in 3.27. The core of the agents population (80% of the total) will still be composed by CS agents. We choose to keep the CS agents as by far the larger component of the population because we want to assess the effects of even small perturbations on the HOCA case. We ran some simulations with different population compositions and we can conclude that the results presented below are a fortiori true when CS agents are just a small subset of the total population. Our argument will be clearer after the exposition of our main findings.

4.32

The behavior of these agents depends on a set of behavioral parameters c taken from Terna's work without any significant changes.

4.33

We will now replicate many of the analysis performed in the HOCA case and we will compare the descriptive statistics obtained under the two different environments.

Table 9. Da on prices-HECA

Table 10. Da on returns-HECA

Discussion

4.34

Descriptive statistics on price time series are reported in table 9. The excess returns on the risky asset are examined in table 10.

4.35

Compared with the HOCA results presented in 4.30, the variability of prices is quite surprisingly slightly smaller in the HECA case. This reduction in the price's volatility is due to the classifiers systems capability to learn the fixed behavioral strategies of the other agents. Stop-loss agents, and market imitating ones, for instance, follow simple and fixed routines which can be easily predicted after some adaptation, by agents endowed with some form of cognitive abilities. While in the HOCA case the intense switching in the agent's beliefs give rise to higher volatility, in the HECA case the fixity of the rules of the thumb makes it easy to coordinate.

4.36

The skewness of prices remains substantially stable in all the four cases, while we should point out a great difference in the kurtosis' coefficients. In the HECA case the prices' distributions seems to have fatter tails than in the HOCA one.

4.37

In all the descriptive statistics on excess return we should signal a ''magnification effect'' in the HECA cases. The standard deviation as well as the skewness and the kurtosis and the autocorrelation are amplified by the presence of heterogenous agents. As in the HOCA case, the learning process's speed affects the time series, and so increase the dispersions.

Figure 6. Volume in then HECA slow learning case

Figure 7. Volume in then HECA fast learning case

Trading volume

4.38

Figure 6 and 7 present snapshots of the observed trading volume over simulations in the slow-learning case under heterogeneity of cognitive abilities. Trading is active also in the HECA simulations, like in real markets. Summary statistics for trading volume, are presented in table 11.

4.39

The variance and the autocorrelation of the trading volume are substantially stable in all the four simulations.

Table 11. Trading volume-HECA

Figure 8. Actual and REE prices in the HECA slow learning case

4.40

In these runs also we observe a rapid decrease of trading volume as the time goes by. After a comparatively short period (equivalent to two or three years of artificial time) trading volume is reduced by about 66% of its starting quantity and it then fluctuates almost steadily around its mean value.

Market efficiency

4.41

We plot in figures 8 and 9 the difference between the actual prices observed in the artificial financial market and the theoretical ones computed using the coefficients derived for the REE equilibrium.

4.42

A comparison of these figures with the HOCA graphs (figures 1 and 2) shows how by varying the composition of the agents populating the environment, we obtain a strong deviation from the ROV based path.

4.43

While in the HOCA setting the actual prices fluctuates around the REE values, the presence of heterogeneity among traders makes the price time series drastically depart from fundamentals. In the slow learning case periods of over and under valuation of the risky asset are alternated with periods characterized by valuations which are substantially in line with the rational expectations equilibrium values.

Figure 9. Actual and REE pices in the HECA fast learning case

4.44

When agents learn more frequently, the deviations from the ROV based prices are much more intense, particularly in the first part of the run. Someone could say that, during the first ten years, the actual and the REE path match purely by coincidence. Even if more than 80% of the population believes in the correct non-linear, option based relationship between price and dividends, the presence of traders choosing on the basis of ''not fully rational'', fixed strategies, is crucial. The results on the impact and the success of noise traders seem to fit well in this situation. Figure 10 helps us to answer one of the most relevant questions we asked ourselves at the beginning of this work. Is it valuable to use an ROV based approach for evaluating a risky stock in a financial market? In the figure we plot the average wealth of the CS agents compared with the average wealth of all the population.

4.45

CS agents are not able to exploit their superior a priori knowledge nor their more sophisticated ''mind''. They accumulate an average smaller wealth than the overall average. This means that they are systematically defeated (on average) by the others types of agents.

4.46

Figure 11 reveals which kinds of agents perform well in this financial market. Random agents are overall the best performing (the difference between the general mean wealth and their mean values is the most negative!). This result reminds us the old story of the little monkey and the Wall Street financial analyst. Chance is important even if just 2 agents out of 50 determine their choices randomly.

Figure 10. CS agents performance

Figure 11. Well performing agents in the HECA fast learning case

4.47

Furthermore, Cross-Target agents achieve a significative performance. By the results of the HECA fast learning case it seems that they are able to exploit the cognitive abilities implemented in their self-training artificial neural networks. The results of the third group of well performing agents, the stop-loss ones, is quite surprising and hard to interpretate by relying on financial theory. It would be necessary to study in a Monte-Carlo style if this good performance of stop-loss strategies is persistent on a larger sample of simulations before concluding anything about this class of agents. Clearly all our results should be read in the perspective of an agent based approach to financial modelling. The kind of generality achieved by simulations is totally different from the output of analytical investigations. Simulations can produce only sufficiency theorems. It is impossible to derive necessary conditions from them.

Figure 12. Percentages of the conditioning part's bit set - HECA fast learning case

4.48

We cannot say that a Real Options based approach to valuation will be necessarily defeated in a financial market, but we may conclude that under the hypotheses tested in our study it will. How general they are is probably one of the most important questions the reader should ask himself in order to judge the relevance of our findings.

Classifier rules dynamics

4.49

We will finally examine the evolution of the CS agents ideas in the HECA fast learning case by looking at figure 12. As we did with figure 5, for the HOCA slow learning case we analyze the percentage of pair of bits set in the conditioning part of the CS agents. Unlike what happened in the HOCA slow learning case, examined in 4.41, where agents correctly formed their expectations on p_t+d_t+1 by conditioning mostly on the variance of d_t, in this case the most important pair of bits is the one which encodes the variance of p_t (Uncorrect ROV).

4.50

In the heterogenous setting, agents perceive that the observed price is disconnected from fundamentalsand start to condition their expectations on different pieces of information. In the HOCA case the percentage of rules which had the Uncorrect ROV bits set was about 50% against the 70% of the HECA case. The fraction of rules that condition on NPV bits is now one half of the stable HOCA case. ROV bits itself are less relevant in this setting compared to the previous one. Moreover it is interesting to note that the Technical Trading bits maintain the same levels in both cases.

4.51

In a self-fulfilling way agents learn that market prices are not moving according to fundamentals and start to form their expectations more by conditioning on non-fundamental variables. These predictions make the prices more unstable and far from the REE equilibrium values in a self reinforcing way. This is the main mechanism that makes it possible that a small fraction of non-CS agents influences the market outcome so much. CS agents use their cognitive abilities to incorporate the behavior of all the other traders in their choices. By doing so they deviate from the rational path which they could reach and follow, when interacting in an homogenous setting. This alteration of the CS agents' strategies makes the market jump out of the REE path.

Conclusions