On the RND under Heston's Stochastic Volatility Model

31 Pages Posted: 11 Feb 2021

See all articles by Ben Boukai

Ben Boukai

IUPUI Mathematical Sciences

Date Written: January 10, 2021


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, Mrázek and Pospíšil (2017)), as well as current option market data (AMD).

Keywords: Heston model, option pricing, risk-neutral valuation, calibration

JEL Classification: G10, G13

Suggested Citation

Boukai, Ben, On the RND under Heston's Stochastic Volatility Model (January 10, 2021). Available at SSRN: https://ssrn.com/abstract=3763494 or http://dx.doi.org/10.2139/ssrn.3763494

Ben Boukai (Contact Author)

IUPUI Mathematical Sciences ( email )

Indianapolis, IN 46202-3216
United States
3172746926 (Phone)

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