The Risk-Neutral Measure and Option Pricing under Log-Stable Uncertainty

The fact that expected payo¤s on assets and call options are in…nite under most log-stable distributions led both Paul Samuelson (as quoted by Smith 1976) and Robert Merton (1976) to conjecture that assets and derivatives could not be reasonably priced under these distributions, despite their attractive feature as limiting distributions under the Generalized Central Limit Theorem. Carr and Wu (2003) are able to price options under log-stable uncertainty, but only by making the extreme assumption of maximally negative skewness. This paper demonstrates that when the observed distribution of prices is log-stable, the Risk Neutral Measure (RNM) under which asset and derivative prices may be computed as expectations is not itself log-stable in the problematic cases. Instead, the RNM is determined by the convolution of two densities, one negatively skewed stable, and the other an exponentially tilted positively skewed stable. The resulting RNM gives …nite expected payo¤s for all parameter values, so that the concerns of Samuelson and Merton were in fact unfounded, while the Carr and Wu restriction is unnecessary. Since the log-stable RNM developed here is expressed in terms of its characteristic function, it enables options on log-stable assets to be computed easily by means of the Fast Fourier Transform (FFT) methodology of Carr and Madan (1999), provided a simple extension of the FFT, introduced here, is employed