Bron Bag Seminar: Semi-static variance-optimal hedging with self-exciting jumps

The aim of this paper is to investigate a variance-optimal, semi-static 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 Hawkes type. More precisely, we aim at hedging a claim at time $T>0$, using a basket of available contingent claims, 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 semi-static type: one part has to be dynamic (i.e., continuously rebalanced) and another one will be static (i.e., buy-and-hold). We discuss in detail a specific example in which the approach proposed is applied, i.e. a variance swap hedged by means of European options, and we provide a numerical illustration of the results obtained.