The Edgeworth Expansion

In further work Jarrow and Rudd [44], Turnbull and Wakeman [72] applied the EE technique to derive the price of an Asian option and later on Collin-Dufresne and Goldstein [19] derived a series expansion for the pricing of swaptions assuming a 3-factor Gaussian- and CIR interest rate model. The main drawback of the above applications of the EE comes from the dependency of the series expansion on the underlying model dynamics, meaning that the Taylor series expansion has to be adapted for every interest model dynamics¹. This makes their approach intractable and quite cumbersome for a wider range of use. Going forward we extend the EE technique of Jarrow and Rudd [44], Turnbull and Wakeman [72] and Collin-Dufresne and Goldstein [19] by generalizing the series Expansion up to an arbitrary order M. Then, the model structure enters only by the computation of the moments of the underlying random variable. Hence, the option pricing technique has been widely separated from the underlying model structure. Finally, by plugging the moments in the EE scheme we are able to approximate the probability density function (pdf) of the underlying random variable.