Exotic Derivatives under Stochastic Volatility Models with Jumps
47 Pages Posted: 14 Dec 2009 Last revised: 10 Oct 2010
Date Written: December 14, 2009
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: Suggested Citation