Exotic Derivatives under Stochastic Volatility Models with Jumps

Aleksandar Mijatović () and Martijn Pistorius ()
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Aleksandar Mijatović: University of Warwick, Department of Statistics
Martijn Pistorius: Imperial College London, Department of Mathematics

Chapter Chapter 17 in Advanced Mathematical Methods for Finance, 2011, pp 455-508 from Springer

Abstract: 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; 60K15; 91G20 (search for similar items in EconPapers)
Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-18412-3_17

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DOI: 10.1007/978-3-642-18412-3_17

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