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The Compatible Bond-Stock Market With Jumps

  • DEWEN XIONG

    ()

    (Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, P. R. China)

  • MICHAEL KOHLMANN

    ()

    (Department of Mathematics and Statistics, University of Konstanz, D-78457, Konstanz, Germany)

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    We construct a bond-stock market composed of d stocks and many bonds with jumps driven by general marked point process as well as by an ℝn-valued Wiener process. By composing these tools we introduce the concept of a compatible bond-stock market and give a necessary and sufficient condition for this property. We study no-arbitrage properties of the composed market where a compatible bond-stock market is arbitrage-free both for the bonds market and for the stocks market.We then turn to an incomplete compatible bond-stock market and give a necessary and sufficient condition for a compatible bond-stock market to be incomplete. In this market we consider the mean-variance hedging in the special situation where both B(u, T) and eG(u, y, T)-1 are quadratic functions of T - u. So, we need to extend the notion of a variance-optimal martingale (VOM) as in Xiong and Kohlmann (2009) to the more general market. By introducing two virtual stocks $\widetilde{S}_1, \widetilde{S}_2$, we prove that the VOM for the bond-stock market is the same as the VOM for the new stock market $\bar{S}, \widetilde{S}_1, \widetilde{S}_2$. The mean-variance hedging problem in this incomplete bond-stock market for a contingent claim $H \in L^2(\mathscr{F}_{T^*})$ is solved by deriving an explicit solution of the optimal measure-valued strategy and the optimal cost induced by the optimal strategy of MHV for the stocks $\bar{S}, \widetilde{S}_1, \widetilde{S}_2$ is computed.

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    Article provided by World Scientific Publishing Co. Pte. Ltd. in its journal International Journal of Theoretical and Applied Finance.

    Volume (Year): 14 (2011)
    Issue (Month): 05 ()
    Pages: 723-755

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    Handle: RePEc:wsi:ijtafx:v:14:y:2011:i:05:p:723-755
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