Forward Rate And Its Derivatives Discussion

The HJM term structure modeling is regarded as a milestone in the arbitrage-free modeling of bond market. It is commonly known as the HJM methodology. One of its main features is that it covers a large variety of previously proposed models and provides a unified approach to the modelling of instantaneous interest rate and to the valuation of interest rate derivatives. The HJM methodology appears to be very successful both from the theoretical and practical viewpoints. Since the HJM approach is based on an arbitrage-free dynamics of the instantaneous continuously compounded forward rate, it requires a certain degree of smoothness with respect to the tenor of the bond prices and their volatilities. For this reason, working with such model is not always convenient. Therefore, we shall discuss a new term structure modeling of the lognormality of the forward rate in this paper.This paper is arranged in such a way. Firstly we quote some important concepts and results which are useful in the latter parts of the paper. In the second parts, we derive the lognormal model of forward libor rate under the corresponding forward martingale measure. The following important discussion is based on the upper model. In the third parts, we describe single-period forward swap and multi-period forward swaps. In the last, we discuss the pricing and hedging of some interest rate derivatives as Caps, Floors and Swaptions. On the basis of stochastic analysis and martingale theory ,we derive the pricing formula of Caps which is similar to the Black-Scholes futures formula.