A Stochastic Volatility LIBOR Market Model with a Closed Form Solution

Abstract

Since its initial publication the SABR model has gained widespread use across asset classes and it has now become the standard pricing framework used in the market to quote interest rate products sensitive to the non flat strike-structure of the market implied volatility. While very simple, the model’s use has always been based on the original study of its authors who derive a formula for pricing European options through a few approximating assumptions which are at times severely violated in the market. This thesis’ main theoretical goal is to set the path for a generalization of the SABR model which possesses a closed form solution free from assumptions about the magnitude of the model’s parameters. We propose such model and derive a closed form solution for the particular case in which the underlying forward rate and its volatility are uncorrelated. After using the solution for pricing caplets within a LIBOR Market Model framework we simplify an approximation for the swap rate developed by Piterbarg in order to use the same solution for the pricing of swaptions. We conduct the model’s calibration for short maturities using a computationally efficient approach which derives an approximation for the model’s implied volatility and uses it to fit the model to market quotes. Finally, we study the properties of the greeks of our model in comparison with those of the classical Black model.