The Mean-Variance Hedging in a Bond Market with Jumps

We construct a market of bonds with jumps driven by a general marked point process as well as by a ? ⁿ -valued Wiener process based on Björk et al. 6 Björk , T. , Kabanov , Y. , and Runggaldier , W. 1997 . Bond market structure in the presence of marked point processes . Math. Finance 7 : 211 – 223 .Crossref], Web of Science ®] , Google Scholar]], 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(? _{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 ?*-martingales introduced by Choulli et al. 8 Choulli , T. , Krawczyk , L. , and Stricker , C. 1998 . ?-martingales and their applications in mathematical finance . The Annals of Probability 26 ( 2 ): 853 – 876 . Google Scholar]], 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.