The competitive analysis of risk taking with applications to online trading

Competitive analysis is concerned with minimizing a relative measure of performance. When applied to financial trading strategies, competitive analysis leads to the development of strategies with minimum relative performance risk. This approach is too inflexible. Many investors are interested in managing their risk: they may be willing to increase their risk for some form of reward. They may also have some forecast of the future. We extend competitive analysis to provide a framework in which investors can develop optimal trading strategies based on their risk tolerance and forecast. We first define notions of risk and reward that are smooth extensions of classical competitive analysis. We then illustrate our ideas using the ski-rental problem. Finally, we analyze a financial game, the unidirectional conversion problem. In particular, we present an optimal risk-tolerant algorithm for the forecast that prices will reach a certain level at some point during the game, and give numerical results of the investor's reward for making such a forecast.