Quantum Field Theory and Evolution of Forward Rates in Quantitative Finance. Note Quote.


Applications of physics to finance are well known, and the application of quantum mechanics to the theory of option pricing is well known. Hence it is natural to utilize the formalism of quantum field theory to study the evolution of forward rates. Quantum field theory models of the term structure originated with Baaquie. The intuition behind quantum field theory models of the term structure stems from allowing each forward rate maturity to both evolve randomly and be imperfectly correlated with every other maturity. This may also be accomplished by increasing the number of random factors in the original HJM towards infinity. However, the infinite number of factors in a field theory model are linked via a single function that governs the correlation between forward rate maturities. Thus, instead of estimating additional volatility functions in a multifactor HJM framework, one additional parameter is sufficient for a field theory model to instill imperfect correlation between every forward rate maturity. As the correlation between forward rate maturities approaches unity, field theory models reduce to the standard one1 factor HJM model. Therefore, the fundamental difference between finite factor HJM and field theory models is the minimal structure the latter requires to instill imperfect correlation between forward rates. The Heath-Jarrow-Morton framework refers to a class of models that are derived by directly modeling the dynamics of instantaneous forward-rates. The central insight of this framework is to recognize that there is an explicit relationship between the drift and volatility parameters of the forward-rate dynamics in a no-arbitrage world. The familiar short-rate models can be derived in the HJM framework but in general, however, HJM models are non-Markovian. As a result, it is not possible to use the PDE-based computational approach for pricing derivatives. Instead, discrete-time HJM models and Monte-Carlo methods are often used in practice. Monte Carlo methods (or Monte Carlo experiments) are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. Their essential idea is using randomness to solve problems that might be deterministic in principle.

A Lagrangian is introduced to describe the field. The Lagrangian has the advantage over Brownian motion of being able to control fluctuations in the field, hence forward rates, with respect to maturity through the addition of a maturity dependent gradient as detailed in the definition below. The action of the field integrates the Lagrangian over time and when exponentiated and normalized serves as the probability distribution for forward rate curves. The propagator measures the correlation in the field and captures the effect the field at time t and maturity x has on maturity x′ at time t′. In the one factor HJM model, the propagator equals one which allows the quick recovery of one factor HJM results. Previous research has begun with the propagator or “correlation” function for the field instead of deriving this quantity from the Lagrangian. More importantly, the Lagrangian and its associated action generate a path integral that facilitates the solution of contingent claims and hedge parameters. However, previous term structure models have not defined the Lagrangian and are therefore unable to utilize the path integral in their applications. The Feynman path integral, path integral in short, is a fundamental quantity that provides a generating function for forward rate curves. Although crucial for pricing and hedging, the path integral has not appeared in previous term structure models with generalized continuous random processes.


Let t0 denote the current time and T the set of forward rate maturities with t0 ≤ T . The upper bound on the forward rate maturities is the constant TFR which constrains the forward rate maturities T to lie within the interval [t0, t0 + TFR].

To illustrate the field theory approach, the original finite factor HJM model is derived using field theory principles in appendix A. In the case of a one factor model, the derivation does not involve the propagator as the propagator is identically one when forward rates are perfectly correlated. However, the propagator is non trivial for field theory models as it governs the imperfect correlation between forward rate maturities. Let A(t,x) be a two dimensional field driving the evolution of forward rates f (t, x) through time. Following Baaquie, the Lagrangian of the field is defined as


The Lagrangian of the field equals

L[A] = -1/2TFR  {A2(t, x) + 1/μ2(∂A(t,x)∂x)2} —– (1)

Definition is not unique, other Lagrangians exist and would imply different propagators. However, the Lagrangian in the definition is sufficient to explain the contribution of field theory ∂A(t,x)∂x  that controls field fluctuations in the direction of the forward rate maturity. The constant μ measures the strength of the fluctuations in the maturity direction. The Lagrangian in the definition implies the field is continuous, Gaussian, and Markovian. Forward rates involving the field are expressed below where the drift and volatility functions satisfy the usual regularity conditions.

∂f(t,x)/∂t = α (t, x) + σ (t, x)A(t, x) —– (2)

The forward rate process in equation (2) incorporates existing term structure research on Brown- ian sheets, stochastic strings, etc that have been used in previous continuous term structure models. Note that equation (2) is easily generalized to the K factor case by introducing K independent and identical fields Ai(t, x). Forward rates could then be defined as

∂f(t, x)/∂t = α (t, x) + ∑i=1K σi(t, x)Ai(t, x) —– (3)

However, a multifactor HJM model can be reproduced without introducing multiple fields. In fact, under specific correlation functions, the field theory model reduces to a multifactor HJM model without any additional fields to proxy for additional Brownian motions.


Lagrangian of Multifactor HJM

The Lagrangian describing the random process of a K-factor HJM model is given by

L[A] = −1/2 A(t, x)G−1(t, x, x′)A(t, x′) —– (4)


∂f(t, x)/∂t = α(t, x) + A(t, x)

and G−1(t, x, x′)A(t, x′) denotes the inverse of the function.

G(t, x, x′) = ∑i=1K σi(t, x) σi(t, x’) —– (5)

The above proposition is an interesting academic exercise to illustrate the parallel between field theory and traditional multifactor HJM models. However, multifactor HJM models have the disadvantages associated with a finite dimensional basis. Therefore, this approach is not pursued in later empirical work. In addition, it is possible for forward rates to be perfectly correlated within a segment of the forward rate curve but imperfectly correlated with forward rates in other segments. For example, one could designate short, medium, and long maturities of the forward rate curve. This situation is not identical to the multifactor HJM model but justifies certain market practices that distinguish between short, medium, and long term durations when hedging. However, more complicated correlation functions would be required; compromising model parsimony and reintroducing the same conceptual problems of finite factor models. Furthermore, there is little economic intuition to justify why the correlation between forward rates should be discontinuous.


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