A Framework for Dynamic Hedging under Convex Risk Measures

Abstract

We consider the problem of minimizing the risk of a financial position (hedging) in an incomplete market. It is well known that the industry standard for risk measure, the Value-at-Risk, does not take into account the natural idea that risk should be minimized through diversification. This observation led to the recent theory of coherent and convex risk measures. But, as a theory on bounded financial positions, it is not ideally suited for the problem of hedging because simple strategies such as buy-hold strategies may not be bounded. Therefore, we propose as an alternative to use convex risk measures defined as functionals on L ² (or by simple extension L ^p, p > 1). This framework is more suitable for optimal hedging with L ²-valued financial markets. A dual representation is given for this minimum risk or market adjusted risk when the risk measure is real valued. In the general case, we introduce constrained hedging and prove that the market adjusted risk is still a L ² convex risk measure and the existence of the optimal hedge. We illustrate the practical advantage in the shortfall risk measure by showing how minimizing risk in this framework can lead to a HJB equation and we give an example of computation in a stochastic volatility model with the shortfall risk measure

Mathematics Subject Classification (2000). 91G10, 91G20.