Semistatic Variance-Optimal Hedging with Self-Exciting Jumps

The aim of this paper is to investigate a quadratic, that is, variance-optimal, semistatic hedging problem in an incomplete market model where the underlying log-asset price is driven by a diffusion process with stochastic volatility and a self-exciting jump process of the Hawkes type. More precisely, we aim at hedging a claim at time T > 0 by using a portfolio of available contingent claims so as to minimize the variance of the residual hedging error at time T. In order to improve the replication of the claim, we look for a hybrid hedging strategy of the semistatic type in which some assets are continuously rebalanced (the dynamic hedging component), and for some other assets, a buy-and-hold strategy (the static component) is performed. We discuss in detail a specific example in which the approach proposed is applied, that is, a variance swap hedged by means of European options, and we provide a numerical illustration of the results obtained.