Agent-based analysis of asset pricing under ambiguous information

ions, pricing and other behavior is derived. Chapman and Polkovnichenko (2009) have recently shown that the lack of agent heterogeneity inherent in the representative agent approach may have important implications, particularly when the agent is not an expected utility maximizer. Specifically, these authors demonstrated that adding even one more agent to a market can qualitatively change the conclusions of an asset pricing study. Although a two-agent model suffices to demonstrate the weakness of a representative agent model, it may also fail to support the agent heterogeneity required to explain market behavior. Consider a two-agent model, where one agent opts not to trade in equilibrium. Our estimated equity premium would then correspond to half of the pool of traders not participating in that market. This level of granularity may inaccurately estimate equity premium since no trades are conducted and only half of the market is participating in price formation. Additionally, if our two-agent model corresponds to two different pricing strategies it may not be reasonable to assume that the representative agent opting out in equilibrium implies that no agents of that strategy would participate in the market we are trying to model. Because a single representative agent sets prices directly, such models need not consider how prices are actually determined by agent interaction through market mechanisms; however, this can lead to aggregate pricing that is inconsistent with real-world market operation. Consider the agent that has the lowest value among agents participating in a continuous double auction for a risky asset; if the market has a large number of orders outstanding then this agent’s bids are never seen by the market. His buy price is too low to reach the top of the order book and his sell orders match at the higher market bid. We can thus see that in a thick market, the prices of agents with low values for an asset should not factor into calculations of equity premium, since their prices are never seen by the market. With as few as three agents, market microstructure can affect the allocation of assets, and therefore should not be ignored out of hand, as this, in turn, can affect market pricing, efficiency, and other outcomes. 2.2. Agent-Based Modeling It might seem the way forward is to construct analytical models with greater numbers of agents; however, as the number of agents increases so does the complexity of analysis. Additionally, the incorporation of market microstructure can cause the complexity of the environment description to increase along with population granularity. Beyond a point, the only feasible computational approach is bottom-up simulation of agent behavior. Despite limited recognition by mainstream economic journals, ABM approaches have been employed extensively in the social sciences, including finance. LeBaron (2006) surveys the agent-based finance literature, and discusses the motivations and limitations of the approach. 2.3. Ambiguity Aversion and the Equity Premium Puzzle In standard models of asset pricing, investors demand a higher rate of return as the risk of an asset increases. This is a direct consequence of risk-averse utility: given a choice among two assets with the same expected value, the one with lower risk provides greater expected utility. This increase of rate of return on risky assets as opposed to risk-free alternatives is known as the equity premium. The equity premium puzzle refers to the apparent disparity between the observed equity premium and what would be predicted based on current models of investor risk preferences. If investors were not risk averse, classical economic theory would suggest that the price of stocks would rise until the point where expected return on stock was exactly equal to the expected return on risk-free assets. Since investing in stock has inherent risk, and traders are not in general risk neutral, stock prices should be lower than this point; however, even when accounting for risk aversion, the average return on stock is significantly higher than the return on treasury bills. Since this phenomenon was identified by Mehra and Prescott (1985), it has received a great deal of attention in finance research literature. DeLong and Magin (2009) provide an up-to-date survey characterizing the current state of knowledge and debate surrounding the U.S. equity premium. One path taken by economists to explain the equity premium puzzle is to posit forms of non-standard preferences or decision rules. One example is ambiguity aversion, a cognitive phenomenon famously identified by Ellsberg, wherein decision makers prefer actions where the chance elements are objectively clear, even at significant sacrifice of expected utility (Halevy, 2007). Epstein and Schneider (2008) (ES) argue that this aversion can be justified in a dynamic context, when information quality is taken into account. The quality of information revealed under an ambiguous prospect may be less useful, in proportion to the degree of ambiguity. To incorporate this ambiguity, ES extend the work of Gilboa and Schmeidler (1989) who demonstrate that preferences for known risks can be captured by worst-case reasoning over non-unique priors. ES show that worst-case reasoning amounts to an asymmetric response to information depending on its content, since the worst case when receiving positive news is that it is not very informative about future dividend movement while worst case when receiving negative news is that it is very informative about future dividend movement. With this rationalization they develop a model of asset pricing with an ambiguity-averse representative agent, and demonstrate that this can explain why even a market of well-diversified investors may still demand compensation for the idiosyncratic risk associated with each asset they hold, since diversification does not reduce ambiguity in the same way that it mitigates risk. This in turn could explain an equity premium, even among savvy investors, and other phenomena of interest. 3. EMPIRICAL GAME MODEL OF ASSET PRICING Taking the ES model as a starting point, we seek to address two questions. First, given the possibility of multiple strategies, is the ambiguity-averse strategy actually present in equilibrium? If traders gain no benefit from being averse to ambiguity, we would expect traders who are averse to ambiguity to be displaced by those who are not, calling into question the validity of modeling the whole market as a single ambiguity-averse trader. Second, in a model with agent heterogeneity and an active market mechanism, does pricing according to the ES model generate significant equity premium? To answer these questions, we constructed an ABM for asset pricing, and performed empirical game-theoretic analysis to evaluate strategy candidates. The full model includes elements that specify the market mechanism, asset definition, and agent strategies. 3.1. Market and Asset Models Many representative agent models, including ES, give little consideration to market microstructure. When analyzing price formation via ABM, however, it becomes necessary to specify a mechanism by which the market operates. In many stock markets, agent interaction is mediated through a continuous double auction (CDA) (Friedman and Rust, 1993), and we adopt this mechanism in our investigation. In a CDA, both buyers and sellers submit prices and are matched continuously. Following ES, the model we examine consists of two types of assets, one offering a fixed return and the other offering a variable return. Agents are able to exchange these two assets through the CDA by specifying how much of the risk-free asset they are willing to offer or accept for a share of the risky asset. 3.2. Agent Strategy Composition Given a market mechanism, we must also specify how agents will act within the mechanism to determine prices. The equilibrium bidding strategy for a CDA (or any dynamic market mechanism) in this context is unknown, thus we must evaluate a space of candidates to determine an appropriate composition of agent strategies in the model. We adopt a version of an approach called empirical game-theoretic analysis (EGTA). The EGTA framework (Wellman, 2006) performs agent-based simulation to generate sample payoffs for candidate strategy profiles, and from them induces a game form. The learned game model then serves as the basis for game-theoretic analysis, which identifies stable strategy profiles (e.g., Nash equilibria). In contrast to other popular equilibria discovery approaches, such as social learning (Ellison and Fudenberg, 1993), reinforcement learning (Hu and Wellman, 2003; Littman, 1994), or genetic programming (Chen, Duffy, and Yeh, 2002), finding Nash equilibria from an empirical game requires no assumptions about how agents arrive at equilibria, which by definition are outside the environment specification of the representative agent models that inform our study. Our set of strategy candidates starts with the representative agent employed in the ES model. This strategy calculates an asset price using the ES formulation of ambiguity aversion (AA), given its own private information. The strategy then bids in a straightforward manner based on that price. We then add a second strategy candidate, based on a standard Bayesian (B) pricing model, that likewise bids straightforwardly given its price calculation. These two strategies are then parameterized by whether or not they incorporate risk aversion into their pricing strategy. We offer the following justification for this strategy parameterization: even if an agent is averse to the risk presented by short term proposition of whether to hold shares of the risky asset, pricing according to the agents true value may not be expected-utility maximizing. If, as a trader, I observe that there is a significant equity premium enforced by the market, my expected earnings are increased by ignoring my risk aversion, since I am always able to acquire the asset at a discount relative to its expected return. There is a huge space of alternatives to these strategies, varying both the method of pricing, and mapping of prices to actual bids over time. Nevertheless, we focused on these strategies, given that our primary goal is to scrutinize the hypothesis that AA behavior is a plausible basis for modeling asset prices. Intuitively, for the AA strategy to be tenable, it should be minimally competitive with the natural alternative, B. Adding strategies to this mix only makes for a more stringent test. 3.3. Estimating the Empirical Game Our construction of an empirical game through simulation follows the basic steps outlined by Jordan, Kiekintveld, and Wellman (2007) for EGTA: 1. Approximate the full game by reducing the effective number of players. 2. Run simulations covering all distinct strategy profiles. 3. Apply variance reduction techniques to estimate the outcomes for each strategy profile in the reduced game. 4. Search for equilibria in the resulting empirical game. Our market simulation incorporates N = 120 agents. With the pool of four possible strategies, there are (123 3 ) = 302,621 distinct profiles, taking into account that agents’ roles are symmetric. Sampling from all of these profiles would be unmanageable; therefore, to contain the profile space further, we employ the hierarchical reduction proposed by Wellman et al. (2005b), where multiple agents are constrained to adopt the same strategy. Specifically, we analyze a four-player version of the game, where each player selects a strategy to be played by 30 trading agents. With this reduction, there are only (7 3 ) = 35 distinct strategy profiles to sample. Although we have restricted the strategic degrees of freedom (thus sacrificing some fidelity), we retain agent heterogeneity in terms of beliefs and preferences at the full granularity of 120 agents. Through averaging repeated simulations, we generate estimates of the expected payoff of playing each strategy profile in the underlying game. As in prior EGTA studies (Wellman et al., 2005a) as well as recent analyses of computer poker (White and Bowling, 2009), we apply variance reduction techniques in order to obtain statistically meaningful results with a feasible number of samples. For the experiments conducted in this paper, we employ the method of control variates (Lavenberg and Welch, 1981), though other traditional Monte Carlo tools may also provide significant variance reduction. Given an empirical game model, we can apply standard game-theoretic solution concepts, such as Nash equilibrium. For these experiments we used replicator dynamics (Schuster and Sigmund, 1983) to derive symmetric mixed-strategy εNash equilibria. Replicator dynamics considers a population playing strategies according to a distribution that is iteratively updated as a function of each strategy’s performance against the current distribution. Friedman (1991) demonstrated that fixed points of this iterative process correspond to Nash equilibria with respect to the fitness function used to evaluate performance. We examine markets where agents are risk averse to varying degrees. Agents in our simulations exhibit constant relative risk aversion (CRRA) according to the utility function used by Mehra and Prescott (1985), u(c) = c 1−α−1 1−α . For α = 0 this utility function collapses to c− 1, and for α = 1 utility is defined by limα→1 u(c) = log(c). Regardless of whether or not agents incorporate risk aversion into their pricing, when aggregating agent payoffs each agent’s payoff is calculated with respect to their CRRA utility function. This allows us to examine whether risk-averse pricing is found in equilibrium when agents can strategically ignore their aversion, and determine what effect risk aversion should have on the measured equity premium. 4. ASSET PRICING UNDER AMBIGUOUS INFORMATION Our case study addresses the scenario of asset pricing under ambiguous information investigated by Epstein and Schneider (2008). These authors model pricing of a risky asset (i.e., an equity security, or stock) by a representative agent that is averse to ambiguity. Though there are several available models of ambiguity aversion, we chose the ES formulation because it is prevalent in recent work in finance research. In our simulations, in each trading period, the agents— traders—are given a piece of news that is partially informative about future dividend payments. Traders use these signals to update their belief about the current value of stocks. With updated beliefs, the traders submit some constant number of single-unit limit orders to the order book, where they are processed according to CDA market rules. At the end of each quarter, q, traders receive dividend payments on the stock they own, and interest payments on their cash balance. Each quarter is defined to be some constant number of trading periods. We concern ourselves with equity premium estimates on the length of a year for consistency with prior work, where a year is defined to be four quarters. The remainder of this section provides further detail and describes the important assumptions of the simulation model. 4.1. Market Conditions Our market consists of two types of assets. A risk-free asset (cash) yields quarterly interest payments at a fixed rate, r. A risky asset (stock) yields quarterly dividend payments dq, which fluctuate according to the mean-reverting process used by ES: dq = κd̄ +(1−κ)dq−1 +δq, where δq is a shock to the dividend value at quarter q, δq ∼ Normal(0,σδ), and κ ∈ (0,1) is the degree of reversion toward the mean dividend value, d̄. If the dividend value drops below zero, traders are paid no dividend, though future dividend movement continues from the subzero value.