A quantum field theory of forward interest rates is developed as a natural generalization of the HJM model: the forward interest rates are allowed to have independent fluctuations for each future time. The forward interest rates are modeled as a two-dimensional Gaussian quantum field, leading to forward interest rates that have a finite probability of being negative. The model is consistent not for the interest rate sector but only for the bond sector and is consequently called the bond forward interest rates. The concept of a quantum field is briefly discussed in Appendix A. 7. The ‘stiff’ quasi-Gaussian model, together with the concept of market time, describes the forward interest rates. A differential formulation of forward interest rates' dynamics is obtained. Using a singular property of the forward interest rates' quantum field, a generalization of Ito calculus follows from the Wilson expansion. A derivation of a risk-neutral measure for zero coupon bonds is obtained based on the differential martingale condition.

Introduction

The complexity of the forward interest rates is far greater than that encountered in the study of stocks and their derivatives. A stock, at a given instant in time, is described by only one random variable (degree of freedom) S(t) and which is usually modeled using stochastic differential equations. In the case of interest rates, it is the entire interest rates yield curve f(t, x) that undergoes a random evolution. Clearly, at each instant, the most general random evolution is that the forward interest rates f(t, x), for each of future time x, should be an independent random variable.