Laplace transforms of stochastic integrals and the pricing of Bermudan swaptions

Abstract

In this paper, we derive the Laplace transforms, i.e., expected values of the exponentials, of time integrals of a class of Extended Cox–Ingersoll–Ross (CIR) processes which are given as the product of a standard CIR process and a positive deterministic function. Assuming such a process for the short-rate gives rise to an affine term structure model for which we can now provide the bond price formulas that had so far not been accessible by attempting to solve the usual Riccati equation. By conditioning the exponentiated time integral on the potential states of the integrated process at its endpoint, we obtain the building blocks to evaluate certain interest rate derivatives, namely Bermudan swaptions. This is achieved by an algorithm that determines the exercise decision at each state by comparing the available payoff with an explicit lower bound built from such conditional Laplace transforms, without resorting to estimation techniques such as American Monte Carlo.