Optimal Static-Dynamic Hedges for Exotic Options under Convex Risk Measures

Abstract

We study the problem of optimally hedging exotic derivatives positions using a combination of dynamic trading strategies in underlying stocks, and static positions in vanilla options when the performance is quantified by a convex risk measure. We establish conditions for the existence of an optimal static position for general convex risk measures, and then analyze in detail the case of expected shortfall with a power loss function. Here we find conditions for uniqueness of the static hedge. We illustrate the computational challenge of computing the market-adjusted risk measure in a simple diffusion model for an option on a non-traded asset.

Keywords: risk measures, hedging