Abstracts

The purpose of the paper consists in developing a novel closed form approximation scheme for pricing European basket and spread options when the joint characteristic function of the corresponding log-prices is known in closed form. The idea consists in expanding the risk neutral density of the terminal underlying value in a Gauss-Hermite series around the normal density. Recently, Necula, Drimus and Farkas (2013, A General Closed Form Option Pricing Formula, SSRN id 2210359) have pointed out that the Gauss-Hermite expansion is more appropriate for heavy tailed log-return distributions than the Gram-Charlier expansion and, at the same time, it allows for a closed form European option pricing formula. In this paper, we obtain closed form pricing formulas for basket and spread options that depend on the Gauss-Hermite expansion coefficients of the risk neutral distribution of the terminal underlying value. We take advantage of the fact that the terminal underlying value in the case of these options is a linear combination of asset prices and provided that the joint characteristic function of the corresponding log-prices is known in closed form, we devise an efficient method for obtaining these Gauss-Hermite expansion coefficients. Our approach has several advantages over existing approximations. First, it is valid both for basket and spread options since no restriction is imposed that the terminal underlying value should be positive. Second, it can be employed beyond the simple case when log-prices are normally distributed since many heavy-tailed distributions have closed form characteristic functions. Finally, our approximation method is feasible for basket/spread options with more than two assets and hence it is appropriate for pricing energy spreads between three commodities such as clean spark spreads. We also conduct a study of the performance of the new approximation method.
Moreover, the option pricing formulas obtained in this paper can be interpreted as generalized Bachelier option pricing formulas, since the risk neutral density of the terminal underlying value (and not that of log-returns) is expanded around the normal density. As a curiosity, the method also allows to obtain a generalized Bachelier formula for pricing European options in the context of the Black-Scholes model, by expanding the log-normal density of the terminal price in a Gauss-Hermite series around the normal density.