Forward Bond Prices and Forward-Measure Caplet Valuation

# Forward starting zero-coupon bonds


# Forward starting zero-coupon bonds












We trivially have that:

$$\frac{Z(t_0,t_1)}{Z(t_0,t_2)}=1+\tau L(t_0,t_1,t_2)$$

Where $L(t_0,t_1,t_2)$ is the forward Libor between $t_1$ and $t_2$, as of $t_0$.

Simply inverting this relationship then yields:

$$\frac{Z(t_0,t_2)}{Z(t_0,t_1)}=\frac{1}{1+\tau L(t_0,t_1,t_2)}$$

Could one interpret $\frac{1}{1+\tau L(t_0,t_1,t_2)}$ as a forward starting zero-coupon bond between $t_1$ and $t_2$, as of $t_0$?

I.e.:

$$\frac{Z(t_0,t_2)}{Z(t_0,t_1)}=Z(t_0,t_1,t_2)$$

If the above is true, then suppose we want to value a Caplet "set in arrears" (i.e. pay-off described in my last question).

This caplet pays $(L(t_1,t_1,t_2)-K)^{+}$ at time $t_1$. Valuing this caplet at $t_0$, choosing $Z(t_0,t_2)$ as Numeraire, we have:

$$C(t_0, T=t_1)=Z(t_0,t_2)\mathbb{E}^{t_2}_{t_0}\left[\frac{(L(t_1,t_1,t_2)-K)^{+}}{Z(t_1,t_2)}\right]$$

Using the identity:

$$Z(t_0,t_2)=Z(t_0,t_1)Z(t_0,t_1,t_2)$$

I get:

$$C(t_0, T=t_1)=Z(t_0,t_1)Z(t_0,t_1,t_2)\mathbb{E}^{t_2}_{t_0}\left[\frac{(L(t_1,t_1,t_2)-K)^{+}}{Z(t_1,t_2)}\right]=\\=Z(t_0,t_1)\mathbb{E}^{t_2}_{t_0}\left[(L(t_1,t_1,t_2)-K)^{+}\right]$$

And the problem at hand now seems trivial, since $L(t_1,t_1,t_2)$ is a martingale under $Z(t_0,t_2)$.

The above cannot be correct, since the answer is different to what @Gordon derived in my previous question linked above. So where have I gone wrong here?

## Answer by ir7 (score 4, accepted)

https://quant.stackexchange.com/a/61708

$Z(t_0,t_1,t_2)$ is the $t_1$-forward price of the ZC bond with maturity $t_2$, as of $t_0$. We have: $$ Z(t_0,t_1,t_2) = E_{t_0}^{t_1}[Z(t_1,t_2)]\not= Z(t_1,t_2).$$

With a not-trivially stochastic index, there is no way to take out $Z(t_1,t_2)^{-1}$ from under your conditional expectation operator until the running $t_0$ hits $t_1$. It is not $t_0$-measurable.

Note: To clarify @Novice555 comment below, we have ($L(t_1,t_2)=L(t_1,t_1,t_2)$):

$$ E^{t_1}_{0}[L(t_1,t_2)] \stackrel{(1)}{=} E^{t_2}_{0}\left[\frac{dQ^{t_1}}{dQ^{t_2}}\big\vert_{t_1} L(t_1,t_2) \right] $$

where $$ \frac{dQ^{t_1}}{dQ^{t_2}}\big\vert_{s} = \frac{Z(s,t_1)/Z(0,t_1)}{Z(s,t_2)/Z(0,t_2)},$$

hence $$ E^{t_1}_{0}[L(t_1,t_2)] = Z(0,t_1,t_2)E^{t_2}_{0}\left[Z(t_1,t_2)^{-1} L(t_1,t_2) \right]$$

$$ = Z(0,t_1,t_2) E^{t_2}_{0}\left[ L(t_1,t_2) \right] + Z(0,t_1,t_2) E^{t_2}_{0}\left[ L(t_1,t_2)^2\right] $$

$$ = Z(0,t_1,t_2) L(0,t_1,t_2) + Z(0,t_1,t_2) (t_2-t_1) E^{t_2}_{0}\left[ L(t_1,t_2)^2\right] $$

Note that all this can also be obtained from @Gordon's answer on caplets where the strike $K$ is set to $0$.

Also note that for ZC bonds, the same approach (replace $L$ by $Z$ in (1)) gives:

$$ E^{t_1}_{0}[Z(t_1,t_2)] = Z(0,t_1,t_2) E^{t_2}_{0}\left[Z(t_1,t_2)^{-1} Z(t_1,t_2) \right] = Z(0,t_1,t_2) $$

One way to sum up this subject is:

- the forward price of a ZC bond, $Z(\cdot, t_1,t_2)$, is a martingale under the $t_1$-forward measure, while

- the forward interest rate, $L(\cdot, t_1,t_2)$, is a martingale under the $t_2$-forward measure.