Valuation of European-Style swaptions

Tese de mestrado em Matemática Financeira, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2010The present work focuses on the pricing of European-style interest rate swaptions, using the Edgeworth expansion [Collin-Dufresne and Goldstein (2002)] and the Hyperplane approxima-tions [Singleton and Umantsev (2002)], under multi-factor exponentially-affine models of the term structure. In a market without arbitrage opportunities, it is shown that an interest rate swaption can be priced as an option on a coupon-bearing bond. While the Edgeworth approx-imation suggests a cumulant expansion of the probability density function of the price of the underlying coupon-bearing bond, the Hyperplane approximation proposes a linearization of the exercise region, so that the same methods used when under one-factor models can be applied. Both methods are analyzed in detailed, and then implemented considering a three-factor Gaussian model, and different maturities for the underlying interest rate swaps, as well as a range of strike prices for each swaption. While there are almost no differences between the results yielded by both approximations, the Edgeworth approximation proves to be significantly slower as the time-to-maturity of the underlying swap increases. Moreover, the Edgeworth approximation is less .flexible, because it requires a closed-form solution for the moments of the distribution of the underlying asset (i.e. a coupon-bearing bond), which are not so readily available for non-affine term structure models