The Mean-Variance Hedging in a Bond Market with Jumps

Abstract

We construct a market of bonds with jumps driven by a general marked point process as well as by a ℝ ⁿ -valued Wiener process based on Björk et al. [6], in which there exists at least one equivalent martingale measure Q ⁰. Then we consider the mean-variance hedging of a contingent claim H ∈ L ²(ℱ _{T 0} ) based on the self-financing portfolio based on the given maturities T ₁,…, T _n with T ₀ < T ₁ < … <T _n  ≤ T*. We introduce the concept of variance-optimal martingale (VOM) and describe the VOM by a backward semimartingale equation (BSE). By making use of the concept of ℰ*-martingales introduced by Choulli et al. [8], we obtain another BSE which has a unique solution. We derive an explicit solution of the optimal strategy and the optimal cost of the mean-variance hedging by the solutions of these two BSEs.

Keywords:

• Backward semimartingale equation (BSE)
• Bond market with jumps
• Mean-variance hedging (MVH)
• Variance optimal martingale (VOM)
• ℰ*-martingale

Mathematics Subject Classification:

• 90A09
• 60H30
• 60G44

Supported by National Natural Science Foundation of China under Grant No. 70671069 and Grant No. 10801097 and National Basic Research Program of China (973 Program) under Grant No. 2007CB814903.

D. X. thanks the University of Konstanz for its support and hospitality.