How to Beat a Stochastic Target

Abstract

We examine the situation when the investor wants to outperform a certain benchmark (typically an index) by active trading in the underlying asset. No arbitrage argument says that it is impossible to outperform the index in all possible outcomes, however it is possible to maximize the probability of reaching the goal, or alternatively, minimize the expected shortfall. Using arguments of stochastic optimal control, it is possible to find an explicit trading strategy which minimizes expected shortfall risk. Similar strategy maximizes probability of beating the index. The problem of beating the index is analogous to hedging an option with insufficient initial capital. Previously developed techniques of Follmer and Leukert for efficient hedging can be applied in this case as well.