A Complete Analytical Solution of the Asian Option Pricing within the Heston Model for Stochastic Volatility: A Probability Density Function Approach

Abstract

The first ever explicit formulation of the concept of the 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.” See links: http://ssrn.com/abstract=2489601 and http://ssrn.com/abstract=2546430.

The first ever explicit formulation of the concept of the options’ probability density functions within the framework of stochastic volatility (Heston model) has been introduced in our publications “Complete Analytical Solution of the Heston Model for Option Pricing and Value-at-Risk Problems: A Probability Density Function Approach”, “Complete Analytical Solution of the American Style Option Pricing with Constant and Stochastic Volatilities: A Probability Density Function Approach” and “A Complete Analytical Resolution of the Double Barrier Option’s Pricing Within the Heston Model. A Probability Density Approach.” See links:http://ssrn.com/abstract=2549033 and http://ssrn.com/abstract=2554038 and http://ssrn.com/abstract=2605948.

In this paper we report complete analytical closed-form results for the European style Asian Options considered within the Heston model for Stochastic Volatility (SV).

Our discovery of the probability density function of the European style Asian Options with SV enables exact closed-form representation of its expected value (price) for the first time ever. Our formulation of the probability density function for the European style Asian Options with SV is expressive enough to enable derivation for the first time ever of corollary analytical closed-form results for such Value-At-Risk characteristics as the probabilities that an Asian Option with SV will be below or above any threshold at any future time before or at termination. Such assessments are absolutely out of reach of the current published methods for treating Asian Options even in the framework of constant volatility.

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

Keywords: Asian Options, Stochastic Volatility, Heston model, Put Options, Call Options, Synthetic Options, Greeks, Trading, Hedging, 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