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Hedge Portfolios in Markets with Price Discontinuities

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Abstract

We consider a market consisting of multiple assets under jump-diffusion dynamics with European style options written on these assets. It is well-known that such markets are incomplete in the Harrison and Pliska sense. We derive a pricing relation by adopting a Radon-Nikodym derivative based on the exponential martingale of a correlated Brownian motion process and a multivariate compound Poisson process. The parameters in the Radon-Nikodym derivative define a family of equivalent martingale measures in the model, and we derive the corresponding integro-partial differential equation for the option price. We also derive the pricing relation by setting up a hedge portfolio containing an appropriate number of options to "complete" the market. The market prices of jump-risks are priced in the hedge portfolio and we relate these to the choice of the parameters in the Radon-Nikodym derivative used in the alternative derivation of the integro-partial differential equation.

Suggested Citation

  • Gerald H.L. Cheang & Carl Chiarella, 2008. "Hedge Portfolios in Markets with Price Discontinuities," Research Paper Series 218, Quantitative Finance Research Centre, University of Technology, Sydney.
  • Handle: RePEc:uts:rpaper:218
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    File URL: https://www.uts.edu.au/sites/default/files/qfr-archive-02/QFR-rp218.pdf
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    References listed on IDEAS

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    1. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    2. Robert Jarrow & Dilip B. Madan, 1999. "Hedging contingent claims on semimartingales," Finance and Stochastics, Springer, vol. 3(1), pages 111-134.
    3. Robert Jarrow & Dilip Madan, 1995. "Option Pricing Using The Term Structure Of Interest Rates To Hedge Systematic Discontinuities In Asset Returns," Mathematical Finance, Wiley Blackwell, vol. 5(4), pages 311-336.
    4. Tomas Björk & Yuri Kabanov & Wolfgang Runggaldier, 1997. "Bond Market Structure in the Presence of Marked Point Processes," Mathematical Finance, Wiley Blackwell, vol. 7(2), pages 211-239.
    5. J. Michael Harrison & Stanley R. Pliska, 1981. "Martingales and Stochastic Integrals in the Theory of Continous Trading," Discussion Papers 454, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    6. David B. Colwell & Robert J. Elliott, 1993. "Discontinuous Asset Prices And Non-Attainable Contingent Claims," Mathematical Finance, Wiley Blackwell, vol. 3(3), pages 295-308.
    7. S. G. Kou & Hui Wang, 2004. "Option Pricing Under a Double Exponential Jump Diffusion Model," Management Science, INFORMS, vol. 50(9), pages 1178-1192, September.
    8. Schweizer, Martin, 1991. "Option hedging for semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 37(2), pages 339-363, April.
    9. Aase, Knut K., 1988. "Contingent claims valuation when the security price is a combination of an Ito process and a random point process," Stochastic Processes and their Applications, Elsevier, vol. 28(2), pages 185-220, June.
    10. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    11. Fabio Mercurio & Wolfgang J. Runggaldier, 1993. "Option Pricing For Jump Diffusions: Approximations and Their Interpretation," Mathematical Finance, Wiley Blackwell, vol. 3(2), pages 191-200.
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    Keywords

    incomplete markets; equivalent martingale measure; compound Poisson processes; Radon-Nikodym derivative; multi-asset options; integro-partial differential equation;

    JEL classification:

    • C00 - Mathematical and Quantitative Methods - - General - - - General
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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