Skip to content
All library documents

Inferring Forward Swap Volatility from Mid-Curve Swaptions

Article Quant Q&A · Author: Jiem
Read the original

Summary

The document clarifies what a mid-curve swaption measures and how it can help infer forward volatility. A mid-curve swaption measures volatility of a forward-starting swap rate, but that is not itself the forward volatility of an option whose strike will be set at a future date.

To infer forward volatility, pair the mid-curve option volatility with the corresponding spot option volatility. The example describes combining a one-year mid-curve volatility on a forward swap rate with a one-year-forward spot swaption volatility to derive the associated forward volatility. The discussion is conceptual and provides no numerical derivation or model assumptions; actual inference depends on matching the relevant expiries, swap tenors, and underlying rates.

Key ideas

  • A mid-curve swaption is written on a forward-starting swap rate.
  • The volatility it expresses is distinct from forward volatility defined for an option with a future-determined strike.
  • Forward volatility can be inferred by combining a suitably matched mid-curve volatility with a spot option volatility.
  • The instruments must refer to compatible rate and tenor structures for the comparison to be meaningful.

Tags

Fixed incomeOptionsVolatility
Full text
# Mid-curve swaption


# 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.

## Answer by Helin (score 5)

https://quant.stackexchange.com/a/40829

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.

## Answer by dm63 (score 3)

https://quant.stackexchange.com/a/40828

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.