Sequential Static-Dynamic Hedging for Long-Term Derivatives

Abstract

This paper presents a new methodology for hedging long-term financial derivatives written on an illiquid asset. The proposed hedging strategy combines dynamic trading of a correlated liquid asset (e.g. the market index) and static positions in market-traded options such as European puts and calls. Moreover, since most market-traded options are relatively short-term, it is necessary to conduct the static hedge sequentially over time till the long-term derivative expires. This sequential static-dynamic hedging strategy leads to the study of a stochastic control problem and the associated Hamilton-Jacobi-Bellman PDEs and variational inequalities. A series of transformations allow us to simplify the problem and compute the optimal hedging strategy.

Keywords: portfolio optimization, employee stock options,optimal stopping, Hamilton-Jacobi-Bellman PDE, variational inequality

JEL Classification: G12, G13, C68