Pricing Swaptions and Coupon Bond Options in Affine Term Structure Models

Abstract

We propose an approach to find an approximate price of a swaption in Affine Term Structure Models. Our approach is based on the derivation of approximate dynamics in which the volatility of the Forward Swap Rate is itself an affine function of the factors. Hence we remain in the affine framework and well known results on transforms and transform inversion can be used to obtain swaption prices in ways similar to bond options (i.e. caplets). We demonstrate that we can also obtain a closed form formula for the approximate price which is based on square-root dynamics for the swap rate. The latter approximation is extremely fast while remaining accurate. The method can be easily generalized to price options on coupon bonds. Computational time compares favorably with other approximation methods. Numerical results on the quality of the approximation are excellent. Our results show that in Affine models, analogously to the LIBOR Market Model, LIBOR and Swap rates are driven by approximately the same type of (in this case affine) dynamics.