Pricing American Interest Rate Options under the Jump-Extended Vasicek Model

Abstract

By introducing Brownian motion to finance, Black, Scholes, and Merton caused a quantum jump in sophistication and realism in the returns processes assumed in asset pricing models. Diffusion processes have taken us a long way, but for many markets there is considerable evidence that diffusions aren't enough to represent real world returns accurately. We need the possibility of occasional discrete jumps, as well. Short-term interest rates are especially prone to jumps because they are strongly influenced by administered rates, such as the Fed funds rate, that are changed in discrete increments. In this article, the authors develop a model of the short rate in the spirit of the Vasicek model, but with exponential up-jump and down-jump processes appended. The pricing equations for European options on zero coupon bonds and interest rate swaps can be obtain by cumulant or Fourier transform techniques, but for American exercise the authors also present a lattice model suitable for computing theoretical values and hedging parameters. The key to capturing the jump effects within a lattice framework is to allow multiple branching from each node, such that large jump moves can occur. Convergence properties of both techniques seem quite good.