A Nonparametric Approach to Derivative Asset Pricing

In this research we examine a new method for pricing European call options based on a nonparametric estimate of the probability density of the underlying asset’s returns. Such an approach allows the use of asymmetric and leptokurtic distributions. We estimate the density using a kernel estimation technique applied to random samples drawn from three particular underlying distributions: the Gaussian, asymmetric Laplace and mixed normal distributions. By assuming an exponential-affine form of the stochastic discount factor, three closed-form pricing formulas for each respective distribution allow for an exact calculation of the option price. Option prices calculated using the three analytic formulas are subsequently compared with option prices calculated using the nonparametric approach. It is thus possible to compare the nonparametric technique with both the exact and Black-Scholes formulas. In the case of the normal distribution we find excellent agreement between all three pricing formulas. Furthermore, in the case of the asymmetric Laplace distribution, we find clear evidence of over-pricing in the Black-Scholes scenario for atthe-money call options. Finally by using a weighted combination of two Gaussian distributions, the nonparametric estimation shows evidence of over-pricing by the Black-Scholes model for at-the-money options. In all cases nonparametric estimates of the option price are calculated without making any a priori assumption regarding the form of the underlying return distribution.