Hedging Under the Heston Model with Jump-to-Default

12 Pages Posted: 24 Sep 2007  

Peter Carr

New York University (NYU) - Courant Institute of Mathematical Sciences

Wim Schoutens

KU Leuven - Department of Mathematics

Date Written: September 21, 2007

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.

Keywords: hHeston, hedging, variance swaps, CDS, jump to default

JEL Classification: C60

Suggested Citation

Carr, Peter and Schoutens, Wim, Hedging Under the Heston Model with Jump-to-Default (September 21, 2007). Available at SSRN: https://ssrn.com/abstract=1016109 or http://dx.doi.org/10.2139/ssrn.1016109

Peter P. Carr

New York University (NYU) - Courant Institute of Mathematical Sciences ( email )

251 Mercer Street
New York, NY 10012
United States

Wim Schoutens (Contact Author)

KU Leuven - Department of Mathematics ( email )

Celestijnenlaan 200 B
Leuven, B-3001
Belgium

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