Exotic derivatives under stochastic volatility models with jumps
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.
Cite
@article{arxiv.0912.2595,
title = {Exotic derivatives under stochastic volatility models with jumps},
author = {Aleksandar Mijatović and Martijn Pistorius},
journal= {arXiv preprint arXiv:0912.2595},
year = {2010}
}
Comments
Paper contains new convergence results. Section on the fluctuation theory has been extended. To appear in the forthcoming AMaMeF Springer volume. 47 pages, 1 figure