Stochastic control and optimization of assets trading

Abstract

Stochastic optimization is an optimization method which involves probabilistic ingredients, such that there is random noise in the problem objectives or constrains, and/or there is randomness in the algorithm when making choice in the search direction. Due to the unpredictable nature of financial markets, stochastic optimization has been drawing gradually greater attention in the field. This dissertation presents stochastic optimization methods from different approaches on two common models in financial markets, namely mean reversion model and regime-switching model.

When the price of an asset is governed by mean reversion model, the objective of an investor is to find the threshold buy and sell prices such that the overall return (with slippage cost imposed) is maximized. This work provides those threshold prices that allows buying, selling and short selling of an asset. A dynamic programming approach is employed to ensure the optimality in the first part of the dissertation. It shows that the solution of the original optimal stopping problem can be achieved by solving four algebraic equations. In the last part of the dissertation, a stochastic approximation approach is implementated on the same problem for comparison. A recursive algorithm is designed to determine the threshold prices. In both aproaches, numerical examples such as Monte Carlo simulations and real market data are given for demonstration.

Considering trend-following trading strategies that are widely used in the investment world, the second part of this work provides a set of sufficient conditions that determine the optimality of the traditional trend-following strategies when the trends are completely observable. Again, a dynamic programming approach is used to verify the optimality under these conditions. The value functions are shown to be either linear functions or infinity depending on the parameter values. The results even reveal some counter-intuitive facts.