Approximating Optimal Trading Strategies Under Parameter Uncertainty: A Monte Carlo Approach

This paper considers the problem of a capital-limited investor with log utility who has the opportunity to invest in a security that follows a parametric price process. While the investor knows the form of the process, the exact parameter values are not known and must be inferred by observing the evolution of the security's price over time. The approach that will be described is applicable to any model and to single or synthetic securities. However, this paper will speci cally consider a synthetic security that follows an Ornstein-Uhlenbeck process, dSt = η(x̄ − St)dt + σdWt. The synthetic security will be formed by buying one asset and selling another. The Ornstein-Uhlenbeck process was chosen for two reasons. First, it has real-world applicability, for example as a model for pair trading. Pair trading has been practiced in industry since at least 1985 (Pole, 2007) and the profitability of a pair-trading strategy has also been examined in the literature.