Nonparametric bounds for European option prices

Abstract

There is much research whose efforts have been devoted to discovering the distributional defects in the Black-Scholes model, which are known to cause severe biases. However, with a free specification for the distribution, one can only find upper and lower bounds for option prices. In this paper, we derive a new nonparametric lower bound and provide an alternative interpretation of Ritchken's (1985) upper bound to the price of the European option. In a series of numerical examples, our new lower bound is substantially tighter than previous lower bounds. This is prevalent especially for out-of-the-money (OTM) options where the previous lower bounds perform badly. Moreover, we present that our bounds can be derived from histograms which are completely nonparametric in an empirical study. We first construct histograms from realizations of S & P 500 index returns following Chen, Lin, and Palmon (2006); calculate the dollar beta of the option and expected payoffs of the index and the option; and eventually obtain our bounds. We discover violations in our lower bound and show that those violations present arbitrage profits. In particular, our empirical results show that out-of-the-money calls are substantially overpriced (violate the lower bound).​