On the RND under Heston's stochastic volatility model

TL;DR

This paper characterizes the risk-neutral density functions compatible with Heston's stochastic volatility model, providing explicit distributions and validating their applicability through simulations and real market data analysis.

Contribution

It identifies a class of scale-family distributions that satisfy Heston's valuation, offering explicit RNDs and demonstrating their practical use with empirical data.

Findings

Explicit RNDs for Heston's model derived from scale-family distributions.

Validation of these RNDs with market data and simulations.

Demonstration of the applicability of these distributions in real-world option pricing.

Abstract

We consider Heston's (1993) stochastic volatility model for valuation of European options to which (semi) closed form solutions are available and are given in terms of characteristic functions. 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 the option price under Heston's SV model. In fact, we also show that any RND with mean being the forward spot price that satisfies Hestons' option valuation solution, must be a member of a scale-family of distributions in that mean. As particular examples, we show that one-parameter versions of the {\it Log-Normal, Inverse-Gaussian, Gamma, Weibull} and the {\it Inverse-Weibull} distributions are all members of this class and thus provide explicit risk-neutral densities (RND) for Heston's pricing model. We demonstrate, via exact calculations and Monte-Carlo simulations, the applicability and suitability of these explicit RNDs using already published Index data with a calibrated Heston model (S\&P500, Bakshi, Cao and Chen (1997), and ODAX, Mr\'azek and Posp\'i\v{s}il (2017)), as well as current option market data (AMD).