Why Bond Price Ratios Do Not Equal Earlier Maturity Prices

Summary

The question examines whether the ratio of two zero-coupon bond prices, with one price evaluated at a later date, equals the price of a bond maturing on that date when interest rates are stochastic. It challenges an argument that treats a later-date amount as though it could be valued at the earlier date in two interchangeable ways.

The material points to a response that emphasizes specifying each bond's valuation date alongside its maturity. That distinction is essential: bond prices at different dates are random variables under stochastic rates, so a ratio involving a future bond price cannot be handled as if all terms were known at the initial date. The document provides no full derivation or detailed resolution, and its central value is identifying a subtle date-indexing issue in fixed-income valuation rather than presenting a complete pricing method.

Key ideas

• Bond prices depend on both the valuation date and the maturity date.
• Under stochastic interest rates, a future bond price is random from the perspective of an earlier date.
• Arguments comparing bond price ratios must keep valuation dates explicit.
• The document raises a valuation question but does not include the referenced explanation in full.

Tags

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


# 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?

## Answer by Magic is in the chain (score 1)

https://quant.stackexchange.com/a/41932

Just adding the valuation date/time to the bonds identifier will make it clearer. See below:

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