Black-Scholes in a CEV Random Environment: A New Approach to Smile Modelling

24 Pages Posted: 29 Mar 2015 Last revised: 7 Jul 2015

See all articles by Antoine (Jack) Jacquier

Antoine (Jack) Jacquier

Imperial College London; The Alan Turing Institute

Patrick Roome

Imperial College London - Department of Mathematics

Date Written: March 27, 2015

Abstract

Classical (Ito diffusions) stochastic volatility models are not able to capture the steepness of small-maturity implied volatility smiles. Jumps, in particular exponential Levy and affine models, which exhibit small-maturity exploding smiles, have historically been proposed to remedy this (see Tankov for an overview). A recent breakthrough was made by Gatheral, Jaisson and Rosenbaum, who proposed to replace the Brownian driver of the instantaneous volatility by a short-memory fractional Brownian motion, which is able to capture the short-maturity steepness while preserving path continuity. We suggest here a different route, randomising the Black-Scholes variance by a CEV-generated distribution, which allows us to modulate the rate of explosion (through the CEV exponent) of the implied volatility for small maturities. The range of rates includes behaviours similar to exponential Levy models and fractional stochastic volatility models. As a by-product, we make a conjecture on the small-maturity forward smile asymptotics of stochastic volatility models, in exact agreement with the results in Jacquier and Roome for the Heston model.

Keywords: volatility asymptotics, random environment, forward smile, large deviations

Suggested Citation

Jacquier, Antoine and Roome, Patrick, Black-Scholes in a CEV Random Environment: A New Approach to Smile Modelling (March 27, 2015). Available at SSRN: https://ssrn.com/abstract=2586055 or http://dx.doi.org/10.2139/ssrn.2586055

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

Patrick Roome (Contact Author)

Imperial College London - Department of Mathematics ( email )

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

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