# Zero coupon bonds dynamics

The dynamics of a zero coupon bond under the risk neutral probability are: $$\frac{dP(t,T)}{P(t,T)}=rdt+\sigma(t,T) dW_t$$ What happens if I take the limit for the maturity $T$ going to small $t$? Do I obtain $$\frac{dP(t,t)}{P(t,t)}=rdt$$ even if I know that $P(t,t)$ is always equal to 1?

This probably has a very simple answer that I can't see at the moment.

• It sounds weird to me to take such limit in an SDE. You may want to solve the SDE and find the forward rate and then take such limit. – Gordon Mar 15 '17 at 14:05
• T is fixed, P(T,T) =1 is the "final condition" and the PDE determines P(t,T) for $t < T$. (If the problem is discretized, you would start by finding P(T-1,T) then P(T-2,T) etc.) – noob2 Mar 15 '17 at 14:43