# Swaption on Forward-Starting Swap "Replication"?

Lately I was thinking about forward-starting swaptions vs. options on forward-starting swaps a bit, and I started wondering about the following:

Suppose we are at time $$T_0$$ (today) and we want to price a swaption that expires in $$T_1$$ and entitles us to enter into a swap which lives from $$T_2$$ to $$T_3$$. Clearly, I work in the setting $$T_0 < T_1 < T_2 < T_3$$.

I was asking myself whether it is reasonable (possible?) to approximate (replicate?) the price of above mentioned option by looking at a combination of the prices of:

• a spot ($$T_0$$) starting swaption with expiry $$T_2$$ that delivers the (then, i.e., at $$T_2$$) spot-starting swap and
• a forward-starting swaption that lives from $$T_1$$ to $$T_2$$ and delivers the (then, i.e., at $$T_2$$) spot-starting swap

I have drawn a little picture to illustrate what I mean ($$T_0=0$$ (today), $$T_1$$ is 1 year from today, $$T_2$$ is 3 years from today, and $$T_3$$ is 6 years from today):

I intuitively have the feeling that it's not working out, and my first line of thought is that it's because the swap underlying the three options is not 100% the same (although it's always the 3x6 swap, the forward starting swap seems more uncertain to me compared to the then-spot starting swap, as the optionality ends after 1y and not after 3y). Maybe someone can provide a little more information and/or some formulae that would confirm my conjecture?

## 1 Answer

The way to think about this is an option to enter a basket of two swaps. The basket contains these positions:

$$P_1$$: a long position in a swap that starts at $$T_1$$ and finishes at $$T_3$$

$$P_2$$: a short position in a swap that starts at $$T_1$$ and finishes at $$T_2$$.

This basket replicates the payoff of the forward starting swap. Denoting $$S(\tau_1, \tau_2)$$ as the swap rate for the swap starting at $$\tau_1$$ and ending at $$\tau_2$$, and $$A(\tau_1, \tau_2)$$ as the corresponding Annuity (PVBP), then the payoff (for a payer) can be written as:

$$$$\max \left \{ \underbrace{A(T_1, T_3) (S(T_1,T_3)-K)}_{P_1} - \underbrace{A(T_1,T_2) (S(T_1,T_2)-K)}_{P_2}, 0\right \}$$$$

This is effectively a spread option between two swap rates (obviously with some weights). The present value of the spread option therefore depends on the joint distribution between the two swap rates, $$S(T_1, T_2)$$ and $$S(T_1, T_3)$$. So you will not be able to perfectly replicate this payoff with vanilla swaptions, though some (upper / lower bound) approximations may be possible.