The propagator is an important quantity that accounts for the correlation between forward rates in a parsimonious manner. The propagator D(x, x′; t, TF R) corresponding to the Lagrangian in definition is given by the following lemma where θ(·) denotes a Heavyside function.
Lemma:
Evaluation of Propagator
The propagator equals
D(x, x′; t, TFR) = μTFR/sinh(μTFR) [sinhμ(TFR −τ) sinh(μτ′)θ(τ −τ′) + sinhμ(TFR − τ′) sinh(μτ)θ(τ′ −τ) + 1/2cosh2(μTFR/2) {2coshμ (τ- TFR/2) coshμ (τ’- TFR/2) + sinh(μτ) sinh(μτ′) + sinhμ(TFR −τ) sinhμ(TFR −τ′)}] —– (1)
where τ = x − t and τ′ = x′ − t both represent time to maturities. Lemma is proved by evaluating the expectation E[A(t, x), A(t′, x′)]. The computations are tedious and contained in Baaquie but are well known in physics and described in common references such as Zinn-Justin. The propagator is seen as a special case of Lemma and is defined on the infinite domain −∞ < x, x′ < ∞ rather than the finite domain t ≤ x, x′ ≤ t + TFR. Hence, the propagator in Lemma converges to the propagator of as the time domain expands from a compact set to the real line. The effort in solving for the propagator on the finite domain is justified as it allows covariances near the spot rate f(t, t) to differ from those over longer maturities. Hence, a potentially important boundary condition defined by the spot rate is not ignored.
Observe that the propagator D(x, x′; t, TFR) in Lemma only depends on the variables τ and τ′ as well as the correlation parameter μ which implies that the propagator is time invariant. This important property facilitates empirical estimation when the propagator is calibrated to market data. To understand the significance of the propagator, note that the correlator of the field A(t, x) for t0 ≤ t, t′ ≤ t0 + TFR is given by
E[A(t, x)A(t′, x′)] = δ(t − t′)D(x, x′; t, TFR)
In other words, the propagator measures the effect the value of the field A(t, x) has on A(t′, x′); its value at another maturity x′ at another point in time. Although D(x, x′; t, TFR) is complicated in appearance, it collapses to one when μ equals zero as fluctuations in the x direction are constrained to be perfectly correlated. It is important to emphasize that μ does not measure the correlation between forward rates. Instead, the propagator solved for in terms of μ fulfills this role.
Remark: Propagator, Covariances, and Correlations
The propagator D(x, x′; t, TFR) serves as the covariance function for the field while σ(t, x) D(x, x′; t, TFR) σ(t, x′) serves as the covariance function for forward rates innovations. Hence, the above quantity is repeatedly found in hedging and pricing formulae presented in the next section. The correlation functions for the field and forward rate innovations are identical as the volatility functions σ(t,·) are eliminated after normalization.
As expected, the HJM drift restriction is generalized in the context of a field theory term struc- ture model. However, producing the drift restriction follows from the original HJM methodology as the discounted bond price evolves as a martingale under the risk neutral measure to ensure no arbitrage. Under the risk neutral measure, the bond price is written as
P(t0,T) = E[t0, t*] e−∫t0t∗ r(t)dt P(t* ,T) = ∫ DA e-∫t0t∗ dtf(t, t) e-∫t*Tdxf(t*, x) —– (2)
where DA represents an integral over all possible field paths in the domain t∗ dt T dx. The t0 t∗ notation E[t0,t∗][S] denotes the expected value under the risk neutral measure of the stochastic variable S over the time interval [t0,t∗].
Proposition: Drift Restriction
The field theory generalization of the HJM drift restriction equals
α (t, x) = σ (t, x) ∫tx dx’ D(x, x′; t, TFR) σ(t, x′) —– (3)
As expected, with μ equal to zero the result of Proposition reduces to
α (t, x) = σ (t, x) ∫tx dx’ σ(t, x′) —– (4)
and the one factor HJM drift restriction is recovered.
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