Why Lognormal Forward Rates Do Not Give a Simple Caplet on a Compounded Index

# Finding the distribution of $I(T_{1},T_{n})$ under an appropriate measure if the forwards are lognormal?


# Finding the distribution of $I(T_{1},T_{n})$ under an appropriate measure if the forwards are lognormal?












My question follows beneath the "lengthy" setting I describe:

Given a tenor discretization $0 = T_{0}< ... < T_{n} =T$,

and under the assumption that under $\mathbb P$, for all $i = 1,...,n-1$ the forward rates $L(T_{i},T_{i+1})$

(note : $L(T_{i},T_{i+1};t)$ denotes the forward rate over the period $[T_{i},T_{i+1}]$ evaluated at time $t$)

have the following dynamic

$dL(T_{i},T_{i+1})=L(T_{i},T_{i+1})(\mu_{i}^{\mathbb P}(t)dt+\sigma_{i}(t)dW^{\mathbb P}_{i}(t))$

where $W^{\mathbb P}_{i}$ is a $\mathbb P$ Brownian motion.

We know that under the martingale measure associated to the zero coupon bond maturing at $T_{i+1}$, $P(T_{i+1})$,

$dL(T_{i},T_{i+1})=L(T_{i},T_{i+1})\sigma_{i}(t)dW^{\mathbb Q^{P(T_{i+1})}}_{i}(t)$, and in particular:

$$d\log(L(T_{i},T_{i+1}))=-\frac{1}{2}\sigma_{i}^{2}(t)dt +\sigma_{i}(t)dW^{\mathbb Q^{P(T_{i+1})}}_{i}(t)$$

From this expression, assuming that $\sigma_{i}$ is some positive constant it is clear that

$\log(L(T_{i},T_{i+1};T_{i}))=\log(L(T_{i},T_{i+1};0))-\frac{1}{2}\sigma_{i}^{2}T_{i}+\sigma_{i}Z_{i}$, where $Z_{i} \sim \mathcal{N}(0,T_{i})$ such that

$$\log(L(T_{i},T_{i+1};T_{i}))\sim \mathcal{N}(\log(L(T_{i},T_{i+1};0))-\frac{1}{2}\sigma_{i}^{2}T_{i},\sigma_{i}^{2}T_{i})\; (*), $$

this means that the forward rates $L(T_{i},T_{i+1})$ above are lognormal.

From the above , we may obtain an analytical formula for the caplet with strike $K_{i}$ on the forward rate $L(T_{i},T_{i+1})$

$V_{\text{caplet on forward i}}(0)=P(T_{i+1};0)\mathbb E^{\mathbb Q^{P(T_{i+1})}}[\max(L(T_{i},T_{i+1};T_{i})-K,0)](T_{i+1}-T_{i})=P(T_{i+1};0)\left[L(T_{i},T_{i+1};0)\Phi\left(\frac{\log(\frac{L(T_{i},T_{i+1};0)}{K_{i}})+\frac{1}{2}\sigma_{i}^{2}T_{i}}{\sigma_{i}\sqrt{T_{i}}}\right)-K\Phi\left(\frac{\log(\frac{L(T_{i},T_{i+1};0)}{K_{i}})-\frac{1}{2}\sigma_{i}^{2}T_{i}}{\sigma_{i}\sqrt{T_{i}}}\right)\right](T_{i+1}-T_{i})$

Question:

Suppose I want to obtain an analytical formula (or even just simply the distribution under an appropriate measure) of a caplet type product on the following index:

$I(T_{1},T_{n}):=\frac{1}{T_{n}-T_{1}}\left(\prod\limits_{i=1}^{n-1}(1+L(T_{i},T_{i+1};T_{i})(T_{i+1}-T_{i}))-1\right)\; (*)$

Under the assumption of knowing that the forward rates are lognormal as above,

how do I find an analytic formula for a caplet on the index $I(T_{1},T_{n})$?

i.e. I want to find a measure $\mathbb Q$ and know the distribution of $I(T_{1},T_{n})$ under $\mathbb Q$ , in order to price the caplet:

$$ \max(I(T_{1},T_{n})-K,0)$$

Any ideas?