Arbitrage-free Evaluation of American-Style Options on Assets with Stochastics Variance Characteristics

Abstract

This paper presents an arbitrage-free framework for contingent claim valuation under stochastic volatility that does not hinge on the market price of volatility risk. It is contrary to the traditional literature, in which some restrictive equilibrium assumptions about investors' preferences must be imposed if one allows arbitrary correlation between the asset price and volatility increments. Our approach to stochastic volatility modelling relies on specifying the forward variance as a function of the market option prices under the no-arbitrage condition. The model is implemented by the Ho-Stapleton-Subrahmanyam (1995) multivariate binomial approximation procedure, but with an extension to permit the multiplicative factor to be a function of stochastic volatility within a recombining context. In conjunction with the lattice-based algorithm, the generalised Geske-Johnson (1984) technique is employed to accelerate the computational efficiency when valuing American options. The key to the model is that it exactly matches the volatility structure inferred from a portfolio of actively traded options, yet is simple enough to be used for pricing a wide class of derivative securities within a reasonable time frame. We investigate how stochastic volatility influences the early exercise premium of an American option. The magnitude of this effect depends upon the moneyness of the option, the time to maturity, the volatilities of the state variables, as well as the correlation between them.