American option pricing with imprecise risk-neutral probabilities

Abstract

The aim of this paper is to price an American option in a multiperiod binomial model, when there is uncertainty on the volatility of the underlying asset. American option valuation is usually performed, under the risk-neutral valuation paradigm, by using numerical procedures such as the binomial option pricing model of Cox et al. [J.C. Cox, S.A. Ross, S. Rubinstein, Option pricing, a simplified approach, Journal of Financial Economics 7 (1979) 229–263]. A key input of the multiperiod binomial model is the volatility of the underlying asset, that is an unobservable parameter. As it is hard to give a precise estimate for the volatility, in this paper we use a possibility distribution in order to model the uncertainty on the volatility. Possibility distributions are one of the most popular mathematical tools for modelling uncertainty. The standard risk-neutral valuation paradigm requires the derivation of the risk-neutral probabilities, that in a one-period binomial model boils down to the solution of a linear system of equations. As a consequence of the uncertainty in the volatility, we obtain a possibility distribution on the risk-neutral probabilities. Under these measures, we perform the risk-neutral valuation of the American option.