“Stiff” Field Theory of Interest Rates and Psychological Future Time

The simplest field theory description of the multivariate statistics of forward rate variations over time and maturities, involves a quadratic action containing a gradient squared rigidity term. However, this choice leads to a spurious kink (infinite curvature) of the normalized correlation function for coinciding maturities. Motivated by empirical results, we consider an extended action that contains a squared Laplacian term, which describes the bending stiffness of the FRC. With the extra ingredient of a ‘psychological’ future time, describing how the perceived time between events depends on the time in the future, our theory accounts extremely well for the phenomenology of interest rate dynamics. Forward interest rate f(t, x) is the interest rate, agreed upon at time t, for an instantaneous loan to be taken at future time x > t, between x and x + dx. At any instant of time t, the forward rate curve (FRC) f(t, x) defines a kind of string which moves and deforms with time. Modelling the motion of this curve is of paramount importance for many financial applications [1]: pricing of interest rate derivatives, such as ‘caps’ that limit the rate of loans that individuals take on their housing, interest swaps, risk management (asset liability management), etc. The industry standard is the so-called HJM model [2, 3]. This model has recently been generalized in different directions [4, 5, 6], in particular by one of us [7, 8, 9], who has proposed a two-dimensional Euclidean quantum field theory for modelling the forward interest rate curve. The forward interest rate dynamics has a drift velocity α(t, x) and volatility σ(t, x); it is convenient to define a driftless noise field A(t, x) by ∂f ∂t (t, x) = α(t, x) + σ(t, x)A(t, x). (1) The noise field describes the external shocks of the economy on the different maturities of the forward rates; its statistics is governed by the exponential of an ‘action’ S[A], which gives the weight of a given path of A in the two dimensional space x, t, and is defined on the semi-infinite domain x ≥ t. The simplest action that factors in the one dimensional nature of the forward rate string was proposed in [7, 8, 9], and reads