A Non-Gaussian Option Pricing Model with Skew

L. Borland and J. P. Bouchaud
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L. Borland: Evnine-Vaughan Associates
J. P. Bouchaud: Science & Finance/Capital Fund Management

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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 L. Borland (Quantitative Finance, {\bf 2}, 415-431, 2002) 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 methods of solution to that model are thereby also discussed. The model parameters are partially fit from empirical observations of the distribution of the underlying. The option pricing model then predicts European call prices which fit well to empirical market data over several maturities.

Date: 2004-02, Revised 2004-03
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