 # What is the LIBOR Market Model and its Principal Applications in Finance?

The LIBOR Market Model (LMM), also known as the BGM or BGM/J model, is a model for the stochastic evolution of forward interest rates. Its main strength over other interest rate models is that it describes the evolution of forward rates that exist, at market-traded maturities, as opposed to theoretical constructs such as the spot interest rate.

Example

In the LMM the variables are a set of forward rates for traded, simple fixed-income instruments. The parameters are volatilities of these and correlations between them. From no arbitrage we can find the risk-neutral drift rates for these variables. The model is then used to price other instruments.

The history of interest-rate modelling begins with deterministic rates, and the ideas of yield to maturity, duration etc. The assumption of determinism is not at all satisfactory for pricing derivatives however, because of Jensen's Inequality.

In 1976 Fischer Black introduced the idea of treating bonds as underlying assets so as to use the Black-Scholes equity option formula for fixed-income instruments. This is also not entirely satisfactory since there can be contradictions in this approach. On the one hand bond prices are random, yet on the other hand interest rates used for discounting from expiration to the present are deterministic. An internally consistent stochastic rates approach was needed.

The first step on the stochastic interest rate path used a very short-term interest rate, the spot rate, as the random factor driving the entire yield curve. The mathematics of these spot-rate models was identical to that for equity models, and the fixed-income derivatives satisfied similar equations as equity derivatives. Diffusion equations governed the prices of derivatives, and derivatives prices could be interpreted as the risk-neutral expected value of the present value of all cashflows as well. And so the solution methods of finite-difference methods for solving partial differential equations, trees and Monte Carlo simulation carried over. Models of this type are: Vasicek; Cox, Ingersoll & Ross; Hull & White. The advantage of these models is that they are easy to solve numerically by many different methods. But there are several aspects to the downside. First, the spot rate does not exist, it has to be approximated in some way. Second, with only one source of randomness the yield curve is very constrained in how it can evolve, essentially parallel shifts. Third, the yield curve that is output by the model will not match the market yield curve. To some extent the market thinks of each maturity as being semi-independent from the others, so a model should match all maturities otherwise there will be arbitrage opportunities.

Models were then designed to get around the second and third of these problems. A second random factor was introduced, sometimes representing the long-term interest rate (Brennan & Schwartz), and sometimes the volatility of the spot rate (Fong & Vasicek). This allowed for a richer structure for yield curves. And an arbitrary time-dependent parameter (or sometimes two or three such) was allowed in place of what had hitherto been constant(s). The time dependence allowed for the yield curve (and other desired quantities) to be instantaneously matched. Thus was born the idea of calibration, the first example being the Ho & Lee model.

The business of calibration in such models was rarely straightforward. The next step in the development of models was by Heath, Jarrow & Morton (HJM) who modelled the evolution of the entire yield curve directly so that calibration simply became a matter of specifying an initial curve. The model was designed to be easy to implement via simulation.

Because of the non-Markov nature of the general HJM model it is not possible to solve these via finite-difference solution of partial differential equations, the governing partial differential equation would generally be in an infinite number of variables, representing the infinite memory of the general HJM model. Since the model is usually solved by simulation it is straightforward having any number of random factors and so a very, very rich structure for the behaviour of the yield curve. The only downside with this model, as far as implementation is concerned, is that it assumes a continuous distribution of maturities and the existence of a spot rate.

The LIBOR Market Model (LMM) as proposed by Miltersen, Sandmann, Sondermann, Brace, Gatarek, Musiela and Jamshidian in various combinations and at various times, models traded forward rates of different maturities as correlated random walks. The key advantage over HJM is that only prices which exist in the market are modelled, the LIBOR rates. Each traded forward rate is represented by a stochastic differential equation model with a drift rate and a volatility, as well as a correlation with each of the other forward rate models. For the purposes of pricing derivatives we work as usual in a risk-neutral world. In this world the drifts cannot be specified independently of the volatilities and correlations. If there are N forward rates being modelled then there will be N volatility functions to specify and N(N — 1)/2 correlation functions, the risk-neutral drifts are then a function of these parameters.

Again, the LMM is solved by simulation with the yield curve 'today' being the initial data. Calibration to the yield curve is therefore automatic. The LMM can also be made to be consistent with the standard approach for pricing caps, floors and swaptions using Black 1976. Thus calibration to volatility-and correlation-dependent liquid instruments can also be achieved.

Such a wide variety of interest-rate models have been suggested because there has not been a universally accepted model. This is in contrast to the equity world in which the lognormal random walk is a starting point for almost all models. Whether the LMM is a good model in terms of scientific accuracy is another matter, but its ease of use and calibration and its relationship with standard models make it very appealing to practitioners.

Brace, A, Gatarek, D & Musiela, M 1997 The market model of interest rate dynamics. Mathematical Finance 7 127-154

Brennan, M & Schwartz, E 1982 An equilibrium model of bond pricing and a test of market efficiency. Journal of Financial and Quantitative Analysis 17 301-329

Cox, J, Ingersoll, J & Ross, S 1985 A theory of the term structure of interest rates. Econometrica 53 385-467

Fong, G & Vasicek, O 1991, Interest rate volatility as a stochastic factor. Working Paper

Heath, D, Jarrow, R & Morton, A 1992 Bond pricing and the term structure of interest rates: a new methodology. Econometrica 60 77-105

Ho, T & Lee, S 1986 Term structure movements and pricing interest rate contingent claims. Journal of Finance 42 1129-1142

Hull, JC & White, A 1990 Pricing interest rate derivative securities. Review of Financial Studies 3 573-592

Rebonato, R 1996 Interest-rate Option Models. John Wiley & Sons Ltd

Vasicek, OA 1977 An equilibrium characterization of the term structure. Journal of Financial Economics 5 177-188