Empirical investigation of a quantum field theory of forward rates

A new test of a wide class of interest rate models is proposed and applied to a recently developed quantum field theoretic model and the industry standard Heath-Jarrow-Morton model. This test is independent of the volatility function unlike other tests previously proposed in the literature. It is found that the HJM model is inconsistent with the data while the quantum field theoretic model is in significant agreement with data. We also show that a portion of the spread between long and short term interest rates is explicable in terms of this model. Physicists have been working on several aspects of financial research over the last decade [1]. One of the most important and as yet unsolved problems in finance is the modelling of interest rates. The interest rates at any point in time form a usually continuous curve (current interest rates for different times in the future) called the forward rate curve (FRC). We denote these rates by f(t, x) where t is the current time and x is the time in the future for which the forward rate applies. For example, f(1, 2) is the interest rate one year from now for an instantaneous deposit to be made 2 years into the future. The earliest interest rate models (eg., Vasicek [2]) dealt only with the spot rate (current interest rate for the present time) and the forward rate curve was treated as a derived quantity. These were found to be inconsistent with the observed data. The Ho-Lee [3] and HJM[4] models were developed to deal with this problem by modelling the entire FRC rather than just the spot rate. The HJM model is, however, limited in the sense that the Brownian motions on which the HJM economy depends are independent of x. One way of removing this restriction is by formulating the forward rate curve as a quantum mechanical string as in Baaquie [5] (we will refer to this model as the quantum field theoretic model). Several empirical tests of the HJM model have been performed (eg., Bühler, Uhrig-Homburg, Walter and Weber [6], Flesker [7], Sim and Thurston [8]) with mixed results. All of the tests assume a certain form for the volatility function σ. In this paper, we propose a test which is independent of the volatility function. The test is applied to the HJM and quantum field theoretic model which was introduced in [5]. The HJM model can be formulated as a limit of the model in [5] which is briefly reviewed below. The one factor quantum field theoretic model models the forward rates as ∂f(t, x) ∂t = α(t, x) + σ(t, x)A(t, x) (1)