# a property of zero coupon bond in Brigo/Mercurio's “Interest Rate Models”

Let $P(t,T)$ be the the value of a contract at time $t$. This contract guarantees its holder the payment of $1$ at time $T$.

consider $t<T<S$, when the interest rate is non-deterministic, do we have $$\frac{P(t,S)}{P(T,S)}=P(t,T)$$ ?

I think the answer is no, but Brigo gives it a proof when he calculate the forward rates(Page 11, paragraph after equation (1.18), the following is an image of page 11) the main idea is:

consider $A:=1/P(T,S)$ as an amount of currency held at $S$, on the one hand, its value at $t$ is $P(t,S)/P(T,S)$; on the other hand, its value at $T$ is $1$, then discount it back to $t$, we get its value at $t$ is $P(t,T)$, hence $$\frac{P(t,S)}{P(T,S)}=P(t,T)$$

Would you mind telling me what's wrong with this proof?

• Why does the amount $A:=1/P(T,S)$ held at $S$ has the value $P(t,S)/P(T,S)$ at $t$? – Gordon Sep 26 '18 at 17:52
• @Gordon by the def of $P(t,S)$: $1$ at $S$ equals $P(t,S)$ at $t$, hence $A$ at $S$ equals $P(t,S)A$ at $t$. – Lookout Sep 26 '18 at 18:02
• If $A$ is known at $t$, your argument is fine. Bu$1/P(T, S)$ is unknown at time $t$, for $t < T$. – Gordon Sep 26 '18 at 18:17 • I don't understand "It is easy to see from the above:...". I think you simply replicate an FRA, right? so it should be $P(t;t,S)/P(t;t,T)=1+(S-T)P(t;T,S)$? – Lookout Sep 27 '18 at 3:32