# Complete Analytical Solution of the Heston Model for Option Pricing and Value-at-Risk Problems: A Probability Density Function Approach

12 Pages Posted: 14 Jan 2015 Last revised: 6 Jun 2015

See all articles by Alexander Izmailov

## Brian Shay

Date Written: January 13, 2015

### Abstract

• The first ever explicit formulation of the concept of an option’s probability density functions has been introduced in our publications "Breakthrough in Understanding Derivatives and Option Based Hedging - Marginal and Joint Probability Density Functions of Vanilla Options -- True Value-at-Risk and Option Based Hedging Strategies" and "Complete Analytical Solution of the Asian Option Pricing and Asian Option Value-at-Risk Problems. A Probability Density Function Approach."

• In this paper we report similar unique results for pricing options in the presence of stochastic volatility (Heston model), enabling complete analytical resolution of all problems associated with options considered within the Heston Model. • Our discovery of the probability density function for options with stochastic volatility enables exact closed-form analytical results for their expected values (prices) for the first time without depending on approximate numerical methods and a Fourier transform that only abbreviates complex numerical integration procedure. • Expected value is the first moment. All higher moments are as easily represented in closed form based on our probability density function, but are not calculable by extensions of other numerical methods, such as a Fourier transform, now used to represent the first moment. • Our formulation of the density function for options with stochastic volatility within the Heston model is expressive enough to enable derivation for the first time ever of corollary closed-form analytical results for such Value-At-Risk characteristics as the probabilities that options with stochastic volatility will be below or above any set of thresholds at termination. Such assessments are absolutely out of reach of current published methods for treating options within the Heston model.

• All numerical evaluations based on our analytical results are practically instantaneous and absolutely accurate.

Keywords: Heston Model, Stochastic Volatility, Options, Put Options, Call Options, Synthetic Options, Greeks, Black Scholes, Trading, Hedging, Market Making, Risk Management, VaR, Options’ Portfolio, Probability Density Function, Probability of Default, , Insurance, Variable Annuity

JEL Classification: A10, A20, A22, A23, B40, C1, C10, C13, C15, C20, C30, C40, C50, C60, D40, D46, G1, G10, G20, G22

Suggested Citation

Izmailov, Alexander and Shay, Brian, Complete Analytical Solution of the Heston Model for Option Pricing and Value-at-Risk Problems: A Probability Density Function Approach (January 13, 2015). Available at SSRN: https://ssrn.com/abstract=2549033 or http://dx.doi.org/10.2139/ssrn.2549033