Solving SABR in Exact Form and Unifying it with LIBOR Market Model

41 Pages Posted: 19 Oct 2009

See all articles by Othmane Islah

Othmane Islah

Lloyds Banking Group; Quantuply Ltd

Date Written: October 15, 2009


SABR stochastic volatility model is appealing for modeling smile and skew of option prices. Hagan, who first proposed this model, derived a closed form approximation for european options and showed that it provides consistent and stable hedges. Here I prove a new exact closed formula for the joint probability density of underlying and volatility processes, when correlation is zero. I argue that this formula remains a very good approximation when correlation is different from zero. I deduce from this expression different formulae for European options. After reviewing the Libor Market Model and its stochastic volatility extensions, I will show how to specify a unified SABR-LMM with a smile, where the term structure of skew is captured, and where closed formulae for caplets and robust approximations for swaptions are available.

Keywords: SABR model, stochastic volatility, CEV model, Bessel process, European option pricing,mixing approach, hypergeometric functions, modified Bessel functions, joint density of Geometric Brownian motion and its path integral, LIBOR Market Model, Stochastic Volatility LIBOR model, Term structure of skew

JEL Classification: C00, C60, G00, G12, G13

Suggested Citation

Islah, Othmane, Solving SABR in Exact Form and Unifying it with LIBOR Market Model (October 15, 2009). Available at SSRN: or

Othmane Islah (Contact Author)

Lloyds Banking Group ( email )

10 Gresham Street
London, EC2V 7AE
United Kingdom

Quantuply Ltd ( email )

United Kingdom

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