University of Washington - Department of Mathematics
We consider a mathematical model for a financial market and consider a trader who wants to optimize, by suitable trading, the value of his or her portfolio. The constraint in this optimization is given by a convex functional known as a convex risk measure. We propose a Monte-Carlo algorithm, who inputs are the joint law of the stock prices and the parameters of the convex risk measure, and whose outputs are the numerical values of the optimal trading strategy. We also prove the optimality of the output. Explicit theoretical evaluations of such trading strategies (known as hedging strategies) are extremely difficult, and we avoid the problem by resorting to computational methods. The main idea is to utilize the finiteness of the Vapnik-Cervonenkis dimension of a class of possible strategies.