Pricing Cancelable swap

Question

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?

Answer 1

On the second question, you have the choice to pay $(S_t - K)$ at $t$ or $(S_T - K)$ at $T$. The value at $t$ of deciding to pay now versus later is:

Value at $t$ of paying $S_t - K$ at $t$ - Value at $t$ of not paying $S_T - K$ at $T$.

$$= -(S_t - K) + e^{-r(T-t)} (F(t,T) - K)$$

where $F(t,T)$ is the forward price of the index.

Now $F(t,T) = S_t e^{(r-d)(T-t)}$ where $d$ is the div yield of the index

so we have $$-(S_t - K) + e^{-r(T-t)} (S_t e^{(r-d)(T-t)} - K)$$

which comes out to

$$S_t (e^{-d(T-t)} - 1) + K(1-e^{-r(T-t)})$$

the first term is negative and represents the fact that you have to pay divs between $t$ and $T$ if you exercise early. The second term is positive (assuming positive rates) and represents the value of getting $K$ earlier. Hence we have a tradeoff. If divs are zero and rates are positive, you always exercise early. If rates are negative, you never exercise early. The most value of exercising early comes when rates are high and positive, and dividends are zero.

Answer 2

Another take at question 2: the value of the contract to you would be given by $V_0$:

$$\begin{align} V_0&=\min_{0\leq\tau \leq T}\mathbb{E}^Q\left[e^{-\int_0^{\tau}r(s)ds}(S_\tau-K)\right] \\[6pt] &=\min_{0\leq\tau \leq T}\left\{e^{-\int_0^{\tau}d(s)ds}S_0-e^{-\int_0^{\tau}r(s)ds}K\right\} \end{align}$$

Assuming a zero dividend yield and a constant risk-free rate:

$$\begin{align} V_0&=\min_{0\leq\tau \leq T}\left\{S_0-e^{-r \tau}K\right\} \\[6pt] &=1_{\{r>0\}}\left(S_0-K\right)+1_{\{r<0\}}\left(S_0-e^{-rT}K\right) \end{align}$$

With stochastic rates $-$ letting $P_{0,t}$ be the price of a zero-coupon bond with maturity $t$:

$$\begin{align} V_0&=\min_{0\leq\tau \leq T}\left\{S_0-P_{0,\tau}K\right\} \\[6pt] & \Leftrightarrow \max_{0\leq\tau \leq T} P_{0,\tau} \end{align}$$

Conclusions are the same as @dm63: with positive interest rates you exercise at contract inception $0$ whereas with negative rates you exercise at maturity $T$.