Pricing Swaptions After the Underlying Swap Has Started

# Can a swaption be priced when the underlying swap has already started before the option’s exercise date?


# Can a swaption be priced when the underlying swap has already started before the option’s exercise date?












I would like to confirm whether a swaption is still well-defined and how it should be priced when the underlying swap starts before the swaption’s exercise date.

Typically, a European swaption assumes that the entire swap begins at or after the exercise date $T_{\text{ex}}$. In that case the swap value at exercise can be written as

$$A(T_{\text{ex}})\,(S_{T_{\text{ex}}} - K),$$

and standard annuity-measure pricing applies.

However, consider the situation where the swap schedule is

$$T_0 < T_{\text{ex}} < T_1 < T_2 < \dots$$

so the swap has already started when the swaption can be exercised.

My questions:

- Does the standard transformation used for many non-standard swaptions (adjusting the annuity and strike to map into a standard swaption) still hold in this case, or does it break because part of the swap is already known?

- Are there references or papers that discuss pricing swaptions whose underlying swap has already partly accrued or fixed before the exercise date?

## Answer by dm63 (score 2)

https://quant.stackexchange.com/a/85218

I assert that this is an exotic that cannot be reliably priced using simple adjustments to strike and/or annuity. It depends on the co-evolution of short term rates versus long term rates between $T_0$ and $T_{ex}$. Specifically, the evolution of realized SOFR, versus the evolution of the $F(T_{ex},T_N)$ (the forward swap rate for the remaining swap beyond $T_{ex}$.

We can come up with some arbitrage bounds:

On the low side, we can say that the instrument in question (call it the 'OP') has greater value than a simple swaption $SN(T_0, T_0, T_N)$ expiring on $T_0$ into the same underlying swap. Proof: copying the exercise decision of the simple swaption will exactly match the cashflows. Plus you have some extra time value.

On the high side, I would claim that the OP is worth less then a portfolio consisting of $CAP(T_0,T_{ex})$ + $SN(T_{ex}, T_{ex}, T_N)$ where the cap covers the short realized swap and the swaption covers the rest. (I assumed a payer swaption; if the OP is a receiver swaption, substitute floor for cap). Proof: again, copying the exercise decision of the OP will replicate the cashflows. In this case, the portfolio will do better in some cases where the short dated swap and the long dated swap are on opposite sides of the strike, illustrating the dependence on long rate/short rate correlation. Intuitively, these cases should be relatively rare, which would mean that this upper bound should be quite tight.