Numerical methods for interest rate derivatives

Abstract

It is well known that interest rate market is an important part of the financial market, and many models have been proposed to fit the market. In this research, we study numerical methods for interest rate derivatives under several models. We consider pricing American put options on zero-coupon bonds under a single factor model of short-term rate, and valuing caps under Lognormal Forward-LIBOR Model (LFM). Monte Carlo method and a novel PDE method are illustrated for pricing caps under one-factor and two-factor LFM. Also, the performance of a short rate model (CIR model) and the one-factor LFM for pricing interest rate derivatives is compared. Calibration experiments indicate that the LFM with zero correlations is closer to the real market than the CIR model, and the LFM with nonzero correlations is even better than the LFM with zero correlations in fitting the market data. More specifically, power penalty method is used for tackling the American put options on zero-coupon bonds for the first time. We choose the CKLS short rate model for the bond option pricing, then the option value satisfies a Linear Complementarity Problem (LCP), which is solved by the power penalty approach. Valuing caplets or caps under the one-factor LFM is usually carried out and resorted to the Black{174}s formula. However, we think that Black{174}s formula holds with the condition that the underlying forward rates are uncorrelated. When the underlying forward rates are correlated, Monte Carlo method is illustrated for pricing caps and European options on coupon-bearing bonds and swaptions under the one-factor LFM. Based on that, we extend the Monte Carlo method for pricing caps to the two-factor LFM. Calibration of interest rate models with market data indicates that the one-factor LFM is more practicable than the CIR model. On the other hand, we observe that caps have lower prices under the one-factor LFM than under the CIR model from numerical experiments. European options on coupon-bearing bonds under these two models are found possessing similar price behavior. Finally, we develop a PDE approach similar to that of Heston (1993) for pricing three-period caps under the one-factor LFM, and establish numerical schemes for solving the PDEs. This PDE approach is applicable when the underlying forward rates are correlated, and can be applied to evaluate caplets and caps under the one-factor LFM with stochastic volatility.