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

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

dlogP(t,s) = f(t,x) dt − ∫0xdy 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

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

where Cov [dtf(t,x),dtf(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) − ∫0x dyEt,dtf(t,y) + 1/2 ∫0x dy ∫0x dy′ Cov dtf(t,y) dtf(t,y′)] –

0x dy [dtf(t,y) − Et,dtf(t,y)] —– (3)

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

Et,dtf(t,x) ≡ Et [dtf(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)+ ∫0x dy Et,dtf(t,y)/dt − 1/2 ∫0x dy ∫0x dy′ c(t,y,y′) + o(1) —– (5)

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

c(t,y,y′)dt = Cov [dtf(t,y)dtf(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,

Et,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 Et,dtf(t,y)/dt. Taking the derivative with respect dt to x of the no-arbitrage condition (5), we obtain

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