Mid-curve swaption

I would like to know how the mid-curve swaption could inform us about forward volatility.

In my understanding it is a swaption on a forward starting swap.

Let us say the midcurve swaption expires in 1y. The underlying swap starts 1y after expiry and matures 10y latter. As the forward starting swap could be expressed as a bascket of a long 1y-11y forward swap and short 1y-1y forward swap.

In my opinion, all such a product expresses is the implied vols 1y-1y and 1y-11y swaptions and the correlation between thier underlyings.

So I don't get where the forward volatility comes from.

• You are right about almost all things. Now additionally to the above consider the 2Y10Y benchmark swaption. You can use it, and the 1Y1Y10Y to derive the information about the 1Y10Y_1Yfwd vol, as per Helin's answer (without any further infor). You are correct about the correlation between instruments being important for pricing midcurves from benchmarks. A good reference for this and examples is Darbyshire: Pricing and Trading Interest Rate Derivatives.
– Attack68
Jul 22, 2018 at 16:55

You can only infer forward vol by pairing a mid-curve option with a spot option. It's easier to go through an example (I'll use 5y x 5y vol since I have the sketch below handy...) One decomposition of the 5y5y spot vol is as follows:

• 1y forward 4y x 5y vol: this is the implied vol of an option starting in 1 year, expiring 4 years thereafter, and eventually settling into a spot 5-year swap.
• 1y mid-curve vol on 4y5y rate: this is the volatility of a swaption expiring in 1 years, then settling into a 4y forward 5y swap.

So given the spot and mid-curve vols, it's straightforward to back out the corresponding forward vol.

• Very clever the idea of pairing. However still it isn't straightforward to me. I'm intuitively thinking about going long spot swaption vs short the mid-curve one. But it is similar to say "to be long forward volatility on stock market one can buy long dated option and short short-dated one". But we know we can have forward volatility information by only using path dependent pay-offs. In the mid-curve case, i think we don't have any path dependency of the pay-off.
– Jiem
Jul 19, 2018 at 23:32

You are correct. The midcurve swaption expresses the volatility of the forward swap rate , not the "forward volatility". The latter refers to the price of an option whose strike price will be determined at a future date.

• Interesting conversation - considering midcurve swaptions, say a 1y1y1y, what is defined the at the money forward for this i.e. where put call parity is obeyed and where a payer and receiver have the same value? Is it the 2y1y linear forward? Jul 24, 2020 at 10:13
• @user35980 that forward is today’s 2yr-1yr forward rate.
– dm63
Jul 24, 2020 at 11:04