The Valuation of Caps, Floors and Swaptions in a Multi-Factor Spot-Rate Model

Abstract

We build a multi-factor, no-arbitrage model of the term structure of spot interest rates. The stochastic factors are the short-term interest rate and the premia of the futures rates over the short-term interest rates. In the three-factor version of the model, for example, the first factor is the three-month LIBOR, the second factor is the premium of the first futures LIBOR over spot LIBOR, and the third factor is the incremental premium of the second futures over the first. The model provides an extension of the lognormal interest rate model of Black and Karasinski (1991) to multiple factors, each of which can exhibit mean-reversion.

This method is computationally efficient for several reasons. First, we suggest calibrating the model to LIBOR futures prices, which enables us to satisfy the no-arbitrage condition without resorting to iterative methods. Second, we modify and implement the binomial approximation methodology of Nelson and Ramaswamy (1990) and Ho, Stapleton and Subrahmanyam (1995) to compute a multi-period tree of rates with the no-arbitrage property. The method uses a recombining two or three-dimensional binomial lattice of interest rates that minimizes the number of states and term structures over time. In addition to these computational advantages, a key feature of the model is that it is consistent with the observed term structure of futures rates as well as the term structure of volatilities implied by the prices of interest rate caps and floors. We use the model to price European-style and Bermuda-style swaptions and yield-spread options. To implement the methodology, we first calibrate the model to the caplet implied-volatility curve on a given day, and then use the model to price European-style swaptions. We find that the two-factor model, where the LIBOR mean reverts rapidly to a slowly mean-reverting second factor, overprices the swaptions relative to market quotations. However, introducing a third factor significantly reduces the overpricing. The calibrated model is used to price Bermudan-style swaptions and yield-spread options. Then, we re-calibrated the two-factor model simultaneously to caplet and swaption prices and use the model output to price Bermudan-style swaptions.