Consider a first hypothetical, a swap. Party 1 is paying 6 month Libor, semi-annually. Party 2. pays $1+3*(\frac{Index_\color{red}{T}}{Index_0}-1) $ only at maturity. Say the notional is 1. $Index_t$ is the closing value of some publicly available index.
Party 2 has the option, at any time $t < T$, to pay $1+3*(\frac{Index_\color{red}{t}}{Index_0}-1) $ and end the contract.
How would I price this contract? Is it appropriate to say that the libor leg will price at par on a reset date and the index related leg, at time $s$ will be valued at $1+3*(\frac{Index_\color{red}{s}}{Index_0}-1) $ ?
To me, the fact that this note can ended at any time, makes it a swaption and we cannot separate the two legs. The optimal choice of when to end the contract will depend both on the value of the index and the value of Libor. Suppose we did not have any market data on the Index, such as swaption vols etc, then how would we go about pricing the swap?
Now consider a Second hypothetical question. Consider only the second leg of the swap. You have the option to pick a time $s$ where $ 0 \leq s \leq T$, where you pay $1+3*(\frac{Index_\color{red}{s}}{Index_0}-1) $ . How do I pick the optimal time and what is the value of this contract at some time $t_1$?
We can simplify the payoff to $\frac{3}{Index_0}(Index_t - \frac{2*Index_0}{3})$,
which is like $scaling factor*(S_t-K) $
Suppose we fix the exercise time, the time where you can choose to pay K and receive $S_t$ to only be $t=T$ then this becomes a forward contract and the price of a forward contract, at $ t=t_1$ is certainly not $scalingfactor*(S_{t_1}-K)$.
In our contract, we have the option to "exercise" or to make this payment, at any $t \in (0,T)$. This adds more optionality to the value of this contract should be higher than that of a forward. How should I go about pricing this?