A non-Gaussian option pricing model with skew

Abstract

Closed-form option pricing formulae explaining skew and smile are obtained within a parsimonious non-Gaussian framework. We extend the non-Gaussian option pricing model of Borland (2002 Quant. Finance 2 415–31) to include volatility–stock correlations consistent with the leverage effect. A generalized Black–Scholes partial differential equation for this model is obtained, together with closed-form approximate solutions for the fair price of a European call option. In certain limits, the standard Black–Scholes model is recovered, as is the Constant Elasticity of Variance (CEV) model of Cox and Ross. Alternative solution methods for that model are thereby also discussed. The model parameters are partially fit from empirical observations of the underlying distribution. The option pricing model then predicts European call prices which fit well to empirical market data over several maturities.

Acknowledgements

We wish to thank Jeremy Evnine, Roberto Osorio and Peter Carr for interesting and useful comments. Benoit Pochart is acknowledged for careful reading of the manuscript.

Notes

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Additional information

Notes on contributors

Jean-Philippe Bouchaud

E-mail: [email protected] and [email protected]