On the Class of Risk Neutral Densities under Heston’s Stochastic Volatility Model for Option Valuation

Abstract

The celebrated Heston’s stochastic volatility (SV) model for the valuation of European options provides closed form solutions that are given in terms of characteristic functions. However, the numerical calibration of this five-parameter model, which is based on market option data, often remains a daunting task. In this paper, we provide a theoretical solution to the long-standing ‘open problem’ of characterizing the class of risk neutral distributions (RNDs), if any, that satisfy Heston’s SV for option valuation. We prove that the class of scale parameter distributions with mean being the forward spot price satisfies Heston’s solution. Thus, we show that any member of this class could be used for the direct risk neutral valuation of option prices under Heston’s stochastic volatility model. In fact, we also show that any RND with mean being the forward spot price that satisfies Heston’s option valuation solution must also be a member of the scale family of distributions in that mean. As particular examples, we show that under a certain re-parametrization, the one-parameter versions of the log-normal (i.e., Black–Scholes), gamma, and Weibull distributions, along with their respective inverses, are all members of this class and thus, provide explicit RNDs for direct option pricing under Heston’s SV model. We demonstrate the applicability and suitability of these explicit RNDs via exact calculations and Monte Carlo simulations, using already published index data and a calibrated Heston’s model (S&P500, ODAX), as well as an illustration based on recent option market data (AMD).