Adaptive Control Applied to Financial Market Data

This article describes a formal approach to decision making optimization in commodity futures markets. Our aim is to plan optimal decision at a given time, where we could decide to buy or sell a commodity contract or stay out of the market. The decision is made using dynamic programming with loss function equal to negative profit measured in money, where the probability density functions(PDF) are estimated using Bayesian learning. Predictive PDFs are computed using parametric models from exponential family, giving us easy to adapt systems. Trading costs (slippage and commission) are taken into account. The theory is supported by a series of experiments indicating the measure of success in predicting the market price movements. Addressed problem Using mathematical methods for prediction of financial markets has become popular at the end of last century with evolution of Quantitative Finance [Shreve(2004a); Shreve(2004b)]. Our main goal is to exploit dependence of price evolution on previous state of the world. To describe the state we use many different quantities previous price maxima and minima, variance, commitment of traders information or own engineered quantities taken out of trading experience. The apparatus used is general and normalized data of any kind can be used. Later on, in the experimental section, we present experiments using generally obtainable data only, mainly past prices. In this general setup we are allowed to buy a contract, sell a contract or hold our position and we try to maximize our profit. For the simplicity of the problem, we can have a maximal position of +1 contract and minimal position of -1 contract (for details about a futures contract see [J́ılek(2004)] or [Lee et al.(1990)Lee, Finnerty, and Wort]). The experiments should show if we are engaged in a game with positive or negative gain expectation and we can use money management rules to enhance our profit. Here we don’t enhance the model and use only the simplest setup. For the problem to be computationally solvable, we supply several restrictions and approximations specified carefully in third section of this article. Theoretical background For optimization of the position held in the market we use the mathematical apparatus of stochastic dynamic programming, we will give a brief summary of here. At first, we suppose a simple binomial model of stock price evolution. At a given time t ∈ 0, 1, 2, .. we choose an action at we buy (at = +1), sell (at = −1) or hold (at = 0). Then, between t and t+1 the price ∆t either rises or falls by 1 and again we choose our action. Therefore we get a sequence Q = (a0,∆0, a1,∆1, ..). If we are at a time t in the middle of such sequence, we can split it into two or more parts as (Pt, .., Ft), P standing for the past and F for the future. In place of the dots can be any number of listed members of Q. We mark sets with an asterisk so for example set of possible pasts at time t is P ∗ t . Theorem 1 (Stochastic dynamic programming) Let P ∗ t be a set of possible paths the price WDS'07 Proceedings of Contributed Papers, Part I, 187–192, 2007. ISBN 978-80-7378-023-4 © MATFYZPRESS