Complete Analytical Solution of the SABR Model for Fixed-Income Option Pricing And Value-at-Risk Problems - A Probability Density Function Approach

Abstract

The first ever explicit formulation of the concept of an option’s probability density function 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). In this paper we report similar unique results for pricing options in the presence of stochastic volatility (SABR model), enabling complete analytical resolution of all problems associated with options considered within the SABR Model. Analogous results for the Heston Model for stochastic volatility have been reported earlier (see links: http://ssrn.com/abstract=2549033; http://ssrn.com/abstract=2609143 and http://ssrn.com/abstract=2605948).Our discovery of the probability density function for options with stochastic volatility within the SABR model enables exact analytical closed-form representations of their expected values (prices) for the first time without depending on approximate numerical methods. We demonstrate by means of these reference prices that approximate numerical methods introduce substantial errors, even as high as 200-300% in some ranges of parameters. Expected value is the first moment. All higher moments are as easily represented in analytical closed-form based on our probability density function, but are not calculable by extensions of any numerical methods now used to represent the first moment. Our formulation of the density function for options with stochastic volatility of the underlying securities within the SABR model is expressive enough to enable derivation for the first time ever of corollary analytical closed-form results Value-At-Risk (“VAR”) characteristics. These VAR characteristics are probabilities that options or portfolios of options together with the underlying securities will be below or above any set of thresholds at termination or at any time prior to termination. Such assessments are absolutely out of reach of current published methods for treating options within the SABR model.All numerical evaluations based on our analytical closed-form results are practically instantaneous and absolutely accurate.