The Smile of Certain Lévy-Type Models

31 Pages Posted: 8 Jul 2012 Last revised: 19 Apr 2013

See all articles by Antoine (Jack) Jacquier

Antoine (Jack) Jacquier

Imperial College London; The Alan Turing Institute

Matthew Lorig

University of Washington - Applied Mathematics

Date Written: April 18, 2013

Abstract

We consider a class of assets whose risk-neutral pricing dynamics are described by an exponential Lévy-type process subject to default. The class of processes we consider features locally-dependent drift, diffusion and default-intensity as well as a locally-dependent Lévy measure. Using techniques from regular perturbation theory and Fourier analysis, we derive a series expansion for the price of a European-style option. We also provide precise conditions under which this series expansion converges to the exact price. Additionally, for a certain subclass of assets in our modeling framework, we derive an expansion for the implied volatility induced by our option pricing formula. The implied volatility expansion is exact within its radius of convergence.

As an example of our framework, we propose a class of CEV-like Lévy-type models. Within this class, approximate option prices can be computed by a single Fourier integral and approximate implied volatilities are explicit (i.e., no integration is required). Furthermore, the class of CEV-like Lévy-type models is shown to provide a tight fit to the implied volatility surface of S{&}P500 index options.

Keywords: CEV, Lévy, Local volatility, implied volatility, default

Suggested Citation

Jacquier, Antoine and Lorig, Matthew, The Smile of Certain Lévy-Type Models (April 18, 2013). Available at SSRN: https://ssrn.com/abstract=2101719 or http://dx.doi.org/10.2139/ssrn.2101719

Antoine Jacquier

Imperial College London ( email )

South Kensington Campus
London SW7 2AZ, SW7 2AZ
United Kingdom

HOME PAGE: http://wwwf.imperial.ac.uk/~ajacquie/

The Alan Turing Institute ( email )

British Library, 96 Euston Road
London, NW12DB
United Kingdom

Matthew Lorig (Contact Author)

University of Washington - Applied Mathematics ( email )

Seattle, WA
United States

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