Inferring the Composition of a Trader Population in a Financial Market

There has been an explosion in the number of models proposed for understanding and interpreting the dynamics of financial markets. Broadly speaking, all such models can be classified into two categories: (a) models which characterize the macroscopic dynamics of financial prices using time-series methods, and (b) models which mimic the microscopic behavior of the trader population in order to capture the general macroscopic behavior of prices. Recently, many econophysicists have trended towards the latter by using multi-agent models of trader populations. One particularly popular example is the so-called Minority Game [1], a conceptually simple multi-player game which can show non-trivial behavior reminiscent of real markets. Subsequent work has shown that -at least in principle -it is possible to train such multi-agent games on real market data in order to make useful predictions [2,3,4,5]. However, anyone attempting to model a financial market using such multi-agent trader games, with the objective of then using the model to make predictions of real financial time-series, faces two problems: (a) How to choose an appropriate multi-agent model? (b) How to infer the level of heterogeneity within the associated multi-agent population?

This paper addresses the question of how to infer the multi-trader heterogeneity in a market (i.e. question (b)) assuming that the Minority Game, or one of its many generalizations [2,3], forms the underlying multi-trader model. We consider the specific case where each agent possesses a pair of strategies and chooses the best-performing one at each timestep. Our focus is on the uncertainty for our parameter estimates. Using real financial data for quantifying this uncertainty, represents a crucial step in developing associated risk measures, and for being able to identify pockets of predictability in the price-series.

As such, this paper represents an extension of our preliminary study in [6]. In particular, the present analysis represents an important advance in that it generalizes the use of probabilities for describing the agents’ heterogeneity. Rather than using a probability, we now use a finite measure, which is not necessarily normalized to unit total weight. This generalization yields a number of benefits such as a stronger preservation of positive definiteness in the covariance matrix for the estimates. In addition, the use of such a measure removes the necessity to scale the time-series, thereby reducing possible further errors. We also look into the problem of estimating the finite measure over a space of agents which is so large that the estimation technique becomes computationally infeasible. We propose a mechanism for making this problem more tractable, by employing many runs with small subsets chosen from the full space of agents. The final tool we present here is a method for removing bias from the estimates. As a result of choosing subsets of the full agent space, an individual run can exhibit a bias in its predictions. In order to estimate and remove this bias, we propose a technique that has been widely used with Kalman Filtering in other application domains.

Many multi-agent models -such as the Minority Game [1] and its generalizations [2,3] -are based on binary decisions. Agents compete with each other for a limited resource (e.g. a good price) by taking a binary action at each timestep, in response to global price information which is publicly available. At the end of each time-step, one of the actions is denoted as the winning action. This winning action then becomes part of the information set for the future. As an illustration of the tracking scheme, we will use the Minority Gamehowever we encourage the reader to choose their own preferred multi-agent game. The game need not be a binary-decision game, but for the purposes of demonstration we will assume that it is.

We provide one possible way of parameterizing the game in order to fit the proposed methodology. We select a time horizon window of length T over which we score strategies for each agent. This is a sliding window given by (w k-T , . . . , w k-1 ) at time step k. Here w k = -sgn(z k ) represents what would have been the winning decision at time k, and z k is the difference in the corresponding price-series, or exchange-rate, r k .

Each agent has a set of strategies which it scores on this sliding time-horizon window at each time-step. The agent chooses its highest scoring strategy as its winning strategy, and then plays it. Assume we have N such agents. At each time-step they each play their winning strategy, resulting in a global outcome for the game at that time-step. Their aggregate actions result in an outcome which we expect to be indicative of the next price-movement in the real priceseries. If one knew the population of strategies in use, then one could predict the future price-series with certainty -apart from a few occasions where ties in strategies might be broken randomly. The ne

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