On Gaussian HJM framework for Eurodollar futures

The arbitrage-free term structure model of Heath, Jarrow and Morton is one of the standard tools for the theoretical analysis of fixed income securities and their associated derivatives. A specific HJM model is fully determined by a choice of volatility structure. This is attributed to the forward rate drift restriction of HJM models. Therefore, once the volatility structure is specified one can price–at least theoretically–any derivative in an interest rate market. The question which we have attempted to answer is what specific HJM model is consistent with the observed price of an Eurodollar futures contract ? Eurodollar futures, apart from being the worlds heavily traded futures are connected to LIBOR (London Inter Bank Offered Rate) and to domestic monetary conditions. The answer to the above question will help in pricing any new derivative on Eurodollar futures or price the one which is not heavily traded. ^ We restrict ourselves to HJM models with bounded and deterministic integrated volatility, known as Gaussian HJM. The reason why we restrict ourselves to Gaussian framework is for practical purpose. As a first step, we examine the question of normality of Eurodollar futures prices. Then we suggest a simple tool to measure the adequacy of different HJM structures that may be used to model Eurodollar futures price process. A plot of monthly realized volatility will be our machinery. ^ Estimation of volatility is the next major part of this thesis. Although it sounds like a standard Statistical procedure one must be careful in applying methods which are not suitable under arbitrage for example – Maximum Likelihood. However one of the procedures which we employ is ML! This is because of an interesting observation which we make from the estimates obtained through ML and by the method of realized volatility. The latter method is model free and within the confines of arbitrage theory. We then suggest the use of a two-stage procedure to estimate volatility whenever the method of ML or realized volatility poses optimization problems. Finally we conclude by discussing some properties of a one-dimensional Gaussian model which may be considered as an alternate to Ho-Lee and Vasicek.