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Hedging Under The Heston Model With Jump-To-Default

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  • PETER CARR

    ()
    (Bloomberg LP and Courant Institute of Mathematical Sciences, 731 Lexington Avenue, New York, NY 10022, USA)

  • WIM SCHOUTENS

    ()
    (K. U. Leuven, Department of Mathematics, Celestijnenlaan 200 B, B-3001 Leuven, Belgium)

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    Abstract

    In this paper, we will explain how to perfectly hedge under Heston's stochastic volatility model with jump-to-default, which is in itself a generalization of the Merton jump-to-default model and a special case of the Heston model with jumps. The hedging instruments we use to build the hedge will be as usual the stock and the bond, but also the Variance Swap (VS) and a Credit Default Swap (CDS). These instruments are very natural choices in this setting as the VS hedges against changes in the instantaneous variance rate, while the CDS protects against the occurrence of the default event.First, we explain how to perfectly hedge a power payoff under the Heston model with jump-to-default. These theoretical payoffs play an important role later on in the hedging of payoffs which are more liquid in practice such as vanilla options. After showing how to hedge the power payoffs, we show how to hedge newly introduced Gamma payoffs and Dirac payoffs, before turning to the hedge for the vanillas. The approach is inspired by the Post–Widder formula for real inversion of Laplace transforms. Finally, we will also show how power payoffs can readily be used to approximate any payoff only depending on the value of the underlier at maturity. Here, the theory of orthogonal polynomials comes into play and the technique is illustrated by replicating the payoff of a vanilla call option.

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    Bibliographic Info

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

    Volume (Year): 11 (2008)
    Issue (Month): 04 ()
    Pages: 403-414

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    Handle: RePEc:wsi:ijtafx:v:11:y:2008:i:04:p:403-414

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    Related research

    Keywords: Hedging; Heston model; stochastic volatility; default; credit default swap; variance swap; orthogonal polynomials;

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    Cited by:
    1. Claudio Fontana & Juan Miguel A. Montes, 2012. "A unified approach to pricing and risk management of equity and credit risk," Science & Finance (CFM) working paper archive 1212.5395, Science & Finance, Capital Fund Management, revised May 2013.

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