Valuation of Caps and Swaptions under a Stochastic String Model

We develop a Gaussian stochastic string model that provides closed-form expressions for the prices of caps and swaptions that, under certain conditions, reduce to Black (1976) formulas. We also propose a stochastic string LIBOR market model that generalizes the models of Brace et al. (1997) and Longstaff et al. (2001a) and allows us to obtain cap prices using the Black (1976) formula. This model can be approximated by one that is compatible with the previous Gaussian models. Our main results are as follows: we show that one of the assumptions of Longstaff et al. (2001a) is incompatible with our model and, then, we obtain a possible explanation for some problems related to the relative valuation of caps and swaptions. We attain the observational equivalence of Kerkhof and Pelsser (2002) and we prove that, under the stochastic string approach, the models proposed by Brace et al. (1997) and Longstaff et al. (2001a) are more parsimonious than previously stated in the original papers. Finally, we derive a a close-form expression for the price of a portfolio option that is a multi-factor extension of the result obtained by Jamshidian (1989) and an analytical formula for swaption prices