## 11.10 Conclusion

HJM develop a new methodology for modelling the term structure of interest rates. They make use of a process describing the evolution forward rates to derive a methodology for contingent claim valuation, which is free from arbitrage and independent of the market prices of risk. By modelling forward rates, the stochastic behaviour of the entire term structure, not just the short-term interest rate, is modelled at any point in time. This allows information from the term structure to be used to eliminate the dependence on market prices of risk.

For the single factor case, the HJM formulation does not add much to previously developed models such as the Hull-White extended Vasicek and BDT models. In fact, the complexity of the calibration and contingent claim valuation procedures may act as a deterrent. However, within a multi-factor context the elegance of the HJM framework is undeniable. The methodology provides a coherent framework allowing easy incorporation of additional factors. The resulting increase in computational time tends to be linear (as opposed to exponential increases exhibited by other models). This is because the non-Markovian nature of the model makes Monte Carlo simulation the valuation technique of choice. This allows easy valuation of path -dependent options, but does become problematic for American-style contingent claims.

# Chapter 12: Brace, Gatarek and Musiela Model

## 12.1 Introduction

### 12.1.1 A continuum of forward rates

All the models examined thus far have been based on instantaneous short- term or forward interest rates. This implies that the fundamental building blocks, that is default-free bonds , are assumed to be continuous (or smooth) with respect to the tenor. Even thediscretetimemodelssuchasHoandLee[ ** 27 ** ] (see Chapter 10) and Black, Derman and Toy [ ** 6 ** ] (see Chapter 8), which make use of a discrete set of discount bonds, assume these are extracted from an underlying continuum of default-free bonds. Such a continuum of default-free discount bonds is not actually traded, nor does the associated continuum of instantaneous short-term or forward interest rates exist.

This assumption need not be problematic , since calibration of the models often requires a discretisation of the continuous time processes. Additionally, traded instruments are only contingent on a discrete number of points on the yield curve. For example, the pricing and hedging of a forward contract on the discrete forward rate ^{ [1] } * f * ( * t, T, T * + ) requires the existence of two bonds * P * ( * t, T * ) and * P * ( * t, T * + ), maturing at the expiry and payoff times respectively. Similarly, a swap-based product depends on bonds maturing at the start of the swap and at the payment times of the fixed leg. Usually it is only a small set of discrete discount bonds that determines the price and associated hedge of such LIBOR-based ^{ [2] } instruments. Given a complete set of spanning forward rates, the required set of bonds may be recovered. Hence a complete set of spanning forward rates provides a sufficient description of the yield curve enabling the pricing of LIBOR-based instruments.

### 12.1.2 The lognormality assumption

Caps and floors are fundamental components within a swap and swap derivative market. A cap (floor) is a strip of caplets (floorlets) which are calls (puts) on an underlying forward rate. The market convention is to assume a lognormal structure for the forward rate process and hence to price each of these options using the Black futures formula. However, as shown in §11.5, allowing the instantaneous forward rate to assume a lognormal volatility structure causes it to explode in finite time. This implies that all forward rates cannot be lognormal under a single arbitrage-free measure. One could conclude that the market prices of caps and floors are flawed in some way and inconsistent with an arbitrage-free framework.

Brace, Gaterek and Musiela (BGM) [ ** 9 ** ] consider discretely compounded forward rates and show that a lognormal structure may be imposed while maintaining an arbitrage-free framework. In the HJM model of instantaneous forward rates, a single spot arbitrage-free measure is applied to all forward rates; while BGM assign, to each forward rate, a forward arbitrage-free measure defined by the settlement date of the associated forward rate. This model then justifies the use of the Black futures formula for pricing caps and floors ^{ [3] } .

^{ [1] } Here * f * ( * t, T, T * + ) represents the time * t * value of the forward rate applicable over the interval [ * T, T * + ].

^{ [2] } London Interbank Offer Rate. This is one of the most frequently used discrete forward rates.

^{ [3] } ere the observation needs to be made that the market does not appear to distinguish between forward measures, and hence forward probabilities, at different maturities