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
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:
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.
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