Revisited multi-moment approximate option pricing models a general comparison (Part 1)

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), the log-normal Gram- Charlier series expansion (Jarrow and Rudd, 1982) and the Edgeworth series expansion (Rubinstein, 1998). The purpose of this paper is to compare these different multimoment approximate option pricing models. We first recall the link between the riskneutral density and moments in a general statistical series expansion framework under the martingale hypothesis. We then derive analytical formulae for several 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 consequently revisit the approximate option pricing models under study. We also get for these models the analytical expressions of implied probability densities, implied volatility smile functions and several hedging parameters of interest.