Exotic Derivatives under Stochastic Volatility Models with Jumps

47 Pages Posted: 14 Dec 2009 Last revised: 10 Oct 2010

Date Written: December 14, 2009

Abstract

In equity and foreign exchange markets the risk-neutral dynamics of the underlying asset are commonly represented by stochastic volatility models with jumps. In this paper we consider a dense subclass of such models and develop analytically tractable formulae for the prices of a range of first-generation exotic derivatives. We provide closed form formulae for the Fourier transforms of vanilla and forward starting option prices as well as a formula for the slope of the implied volatility smile for large strikes. A simple explicit approximation formula for the variance swap price is given. The prices of volatility swaps and other volatility derivatives are given as a one-dimensional integral of an explicit function. Analytically tractable formulae for the Laplace transform (in maturity) of the double-no-touch options and the Fourier-Laplace transform (in strike and maturity) of the double knock-out call and put options are obtained. The proof of the latter formulae is based on extended matrix Wiener-Hopf factorisation results. We also provide convergence results.

Keywords: Double-barrier options, volatility surface, volatility derivatives, forward starting options, stochastic volatility models with jumps, fluid embedding, complex matrix Wiener-Hopf factorisation

JEL Classification: G12, G13, C63

Suggested Citation

Mijatovic, Aleksandar and Pistorius, Martijn, Exotic Derivatives under Stochastic Volatility Models with Jumps (December 14, 2009). Available at SSRN: https://ssrn.com/abstract=1523091 or http://dx.doi.org/10.2139/ssrn.1523091

Aleksandar Mijatovic (Contact Author)

Imperial College London ( email )

Department of Mathematics
180 Queen's Gate
London, SW7 2AZ
United Kingdom

HOME PAGE: http://www3.imperial.ac.uk/people/a.mijatovic

Martijn Pistorius

Imperial College London ( email )

South Kensington Campus
Exhibition Road
London, Greater London SW7 2AZ
United Kingdom

HOME PAGE: http://www3.imperial.ac.uk/people/m.pistorius

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