All stochastic models of spot and forward interest rates are based on a finite number of degrees of freedom, and are precursors of the more general modelling of interest rates based on quantum field theory, which forms the subject of all the subsequent chapters.

The formalism of quantum field theory requires one to make a fairly large transition in the level of mathematical complexity. The path-integral and Hamiltonian analysis of stochastic interest rate models is undertaken to smoothen this transition, as well as for its intrinsic importance. The key ideas that will be later given a field theory generalization are introduced in stochastic models that have, at each instant, only a finite number of independent random variables.

Spot interest rate Hamiltonian and Lagrangian

The spot interest rate r(t) is the interest rate for an overnight loan at time t. Spot rate models are useful for modelling the short time behaviour of the interest rates' yield curve, as well as in the study of the stock market. Furthermore, since central bank policies intervene in determining the spot rate, jumps and discontinuities in the spot rate are particularly important, and need to be considered separately from the remaining yield curve.

We consider only the arbitrage-free, and not the empirical, martingale time evolution of the spot interest rate, as is required for pricing its derivatives. The interest spot rate models can hence be directly modelled using the Langevin equation.