A Complete Analytical Resolution of the Double Barrier Optionss Pricing within the Heston Model. A Probability Density Function Approach

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” (see links http://ssrn.com/abstract=2489601 and http://ssrn.com/abstract=2546430).• The first ever explicit formulation of the concept of an 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” and “Complete Analytical Solution of the American Style Option Pricing with Constant and Stochastic Volatilities: A Probability Density Function Approach” (see links http://ssrn.com/abstract=2549033 and http://ssrn.com/abstract=2554038).The probability density approach has allowed complete analytical resolution of not only all the pricing problems but for the first time complete analytical resolution of all the associated Value-At-Risk (VAR) problems by specifying probabilities of options, as stochastic quantities, to be below (above) of any threshold. • In this paper we report analogous results for pricing Double Barrier options in the presence of stochastic volatility (Heston model), even enabling complete analytical resolution of all problems associated with these options. • Our discovery of the analytical closed-form for the probability density function for the Double Barrier options with stochastic volatility enables exact results for pricing of these options for the first time without depending on approximate numerical methods. • Our formulation of the density function for the Double Barrier 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 the current published methods for treating options within or outside the Heston model.• All numerical evaluations based on our analytical results are practically instantaneous and absolutely accurate.