Mean-Variance Hedging When There Are Jumps

In this paper, we consider the problem of mean-variance hedging in an incomplete market where the underlying assets are jump diffusion processes which are driven by Brownian motion and doubly stochastic Poisson processes. This problem is formulated as a stochastic control problem and closed form expressions for the optimal hedging policy are obtained using methods from stochastic control and the theory of backward stochastic differential equations. The results we have obtained show how backward stochastic differential equations can be used to obtain solutions to optimal investment and hedging problems when discontinuities in the underlying price processes are modelled by the arrivals of Poisson processes with stochastic intensities. Applications to the problem of hedging default risk are also discussed. Key words– Jump diffusion, stochastic intensity, doubly stochastic Poisson process, mean-variance hedging, incomplete markets, backward stochastic differential equations, default risk. AMS subject classifications (2000): 91B28, 91B30, 60J75, 30A36