On the variance and skewness of the swap rate in a stochastic volatility interest rate model

Abstract

This paper provides new insight in the distribution of the (forward par) swap rate in a stochastic volatility model for the dynamics of the forward rate curve. First the swap rate dynamics are obtained in a multi-curve environment with deterministic spread. Then, the variance of the swap rate is derived making use of a result on the distribution of random variables generated by extended square-root diffusion processes. Also, the skewness is derived by Itô calculus. These results give rise to moment-matching swaption price formulas which are expected to permit a fast approximate calibration of the model.