Exotic derivatives under stochastic volatility models with jumps

TL;DR

This paper develops explicit, analytically tractable formulas for pricing a variety of exotic derivatives under stochastic volatility models with jumps, enhancing computational efficiency and understanding of these complex financial instruments.

Contribution

It introduces new closed-form and integral formulas for exotic option prices within a specific subclass of stochastic volatility models with jumps, including variance swaps and double-no-touch options.

Findings

Closed-form Fourier transform formulas for vanilla and forward starting options.

Explicit approximation for variance swap prices.

Analytical formulas for double-no-touch and double knock-out options.

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.