Multi-moment Approximate Option Pricing Models: A General Comparison (Part 1)

Abstract

After the seminal paper of Jarrow and Rudd (1982), several authors have proposed to use different statistical series expansion to price options when the risk-neutral density is asymmetric and leptokurtic. Amongst them, one can distinguish the Gram-Charlier Type A series expansion (Corrado and Su, 1996-b and 1997-b), log-normal Gram-Charlier series expansion (Jarrow and Rudd, 1982) and Edgeworth series expansion (Rubinstein, 1998). The purpose of this paper is to compare these different multi-moment approximate option pricing models. We first recall the link between risk-neutral densities and moments in a general statistical series expansion framework. We then derive analytical formulae for these different four-moment approximate option pricing models, namely, the Jarrow and Rudd (1982), Corrado and Su (1996-b and 1997-b) and Rubinstein (1998) models. We investigate in particular the conditions that ensure the respect of the martingale restriction (see Longstaff, 1995) and compare with option pricing models such as Black and Scholes (1973) and Hermite polynomial models (see Madan and Milne, 1994, Abken et al.,1996). We also get for these approximate option pricing models analytical expressions of implied probability distribution, implied volatility smile functions and several hedging parameters of interest, such as the Psi and the Chi that measure respectively the changes in the option price with respect to the changes in kurtosis and skewness.