No-Arbitrage & Conditional Drift from the Covariance of Fluctuations. (Didactic 4)

From P(t,s) = exp {−∫₀^s−t f(t,x)dx}, we get

d_t logP(t,s) = f(t,x) dt − ∫₀^xdy dt f(t,y) —– (1)

where x ≡ s − t. We need the expression of dP(t,s)/P(t,s) which is obtained from (1) using Ito’s calculus. In order to get Ito’s term in the drift, recall that it results from the fact that, if f is stochastic, then

d_tF(f) = ∂F/df dtf + 1/2 ∫ dx ∫ dx′ ∂²F/∂f(t,x)∂f(t,x′) Cov [d_tf(t,x), d_tf(t,x′)] —– (2)

where Cov [d_tf(t,x),d_tf(t,x′)] is the covariance of the time increments of f(t,x). Using this Ito’s calculus, we obtain

dP(t,s)/P(t,s) = [dt f(t,x) − ∫₀^x dyE_t,dtf(t,y) + 1/2 ∫₀^x dy ∫₀^x dy′ Cov d_tf(t,y) d_tf(t,y′)] –

∫₀^x dy [d_tf(t,y) − E_t,dtf(t,y)] —– (3)

We have explicitly taken into account that d_tf(t,x) may have in general a non-zero drift, i.e. its expectation

E_t,dtf(t,x) ≡ E_t [d_tf(t,x)|f(t,x)] —– (4)

conditioned on f(t,x) is non-zero. The no-arbitrage condition for buying and holding bonds implies that PM is a martingale in time, for any bond price P. Technically this amounts to imposing that the drift of PM be zero:

f(t,x) = f(t,0)+ ∫₀^x dy E_t,dtf(t,y)/dt − 1/2 ∫₀^x dy ∫₀^x dy′ c(t,y,y′) + o(1) —– (5)

assuming that d_tf(t,x) is not correlated with the stochastic process driving the pricing kernel and using the definitions

c(t,y,y′)dt = Cov [d_tf(t,y)d_tf(t,y′)] —– (6)

and r(t) = f (t, 0). In (5), the notation o(1) designs terms of order dt taken to a positive power. Expression (5) is the fundamental constraint that a SPDE for f (t, x) must satisfy in order to obey the no-arbitrage requirement. As in other formulations, this condition relates the drift to the volatility.

It is useful to parametrize, without loss of generality,

E_t,dtf(t,x)/dt = ∂f(t,x)/∂x + h(t,x) —– (7)

where h(t, x) is a priori arbitrary. The usefulness of this parametrization (7) stems from the fact that it allows us to get rid of the terms f(t,x) and f (t,0) in (5). Indeed, they cancel out with the integral over y of E_t,dtf(t,y)/dt. Taking the derivative with respect dt to x of the no-arbitrage condition (5), we obtain

h(t, x) = −1/2 ∫₀^x [c(t, y, x) + c(t, x, y) —– (8)

In sum, we have the following constraint that the SPDE for f (t, x) must satisfy

E_t [d_tf(t,x)|f(t,x)] = dt ∂f(t,x)/∂x −1/2 ∫₀^x dy (Cov[d_tf(t,y) d_tf(t,x)] + Cov[d_tf(t,x) d_tf(t,y)]) —– (9)

which, for symmetric covariance, lead to

E_t [d_tf(t,x)|f(t,x)] = dt ∂f(t,x)/∂x − ∫₀^x dy Cov[d_tf(t,y) d_tf(t,x)] —– (10)

This expression (10) is reminiscent of the fluctuation-dissipation theorem in the Langevin formulation of the Brownian motion, linking the drag coefficient (analog to the l.h.s.) to the integral over time of the correlation function of the fluctuation forces (analog of the second term of the r.h.s.). In the usual fluctuation-dissipation theorem, the drag coefficient is determined self-consistently as a function of the amplitude and correlation of the fluctuating force in order to be compatible with the equilibrium distribution. Similarly, here the no-arbitrage condition determines self-consistently the conditional drift from the covariance of the fluctuations. In contrast, notice however that the same quantity f enters in both sides of (9) and (10). Thus, the no-arbitrage condition imposes a condition on the structure of the partial differential equations that govern the dynamical evolution of the forward rates f(t,x).

 

 

 

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