Pricing of options on assets with level dependent stochastic volatility

Abstract

Many asset classes, such as interest rates, exchange rates, commodities, and equities, often exhibit a strong relationship between asset prices and asset volatilities. This paper examines an analytical model that takes into account this level dependence of volatility. We demonstrate how prices of European options under stochastic volatility can be calculated analytically via inverse Laplace transformations. We also examine a Hull-White stochastic volatility expansion. While a success of this expansion in approximate computation of option prices has already been established empirically, the question of convergence has been left unanswered. We demonstrate, in this paper, that this expansion diverges essentially for all possible stochastic volatility processes. In contrast to a majority of volatility expansion models reported in the literature, we construct expansions that explicitly show the contribution of all of the variance moments. Such complete expansions are very useful in analyzing properties of option prices, as we demonstrate by examining why empirical volatility surfaces plotted as a function of the rescaled strike can sometimes exhibit striking time invariance.