Choosing SABR Strikes for Mid-Curve Swaption Spread Volatility

Summary

The document frames a pricing question about mid-curve swaptions: options that expire before the underlying forward-starting swap begins. A common approximation constructs the forward swap's volatility from the volatilities of two spot-starting swaps and their correlation. The example uses two swaps with a common option expiry and different swap tenors, illustrating which market volatilities and correlation enter the calculation.

The unresolved issue is how to select the strikes for those component swaption volatilities when the volatility surface is skewed and modeled with SABR. The document asks whether the inputs should be evaluated at particular strikes, but supplies no answer or pricing derivation. It therefore identifies a practical modeling question rather than establishing a strike-selection rule. Any implementation would need an explicit convention for mapping the forward swap's strike into the component swap rates, and the text does not provide one.

Key ideas

• A mid-curve swaption expires before its underlying forward-starting swap begins.
• A common approach estimates forward swap volatility using two spot-starting swap volatilities and their correlation.
• The strike used to read each component volatility matters when the volatility surface is skewed.
• The document poses the SABR strike-selection problem but does not resolve it.

Tags

# Mid-curve swaption pricing - how to get the spread vol?


# Mid-curve swaption pricing - how to get the spread vol?












I believe I understand the following (from the accepted answer to the Quantitative Finance question called "volatility of a mid curve option"):

> A swaption in which the underlying swap starts at a date materially after the expiration date is called a midcurve swaption. The implied volatilities of these can not be obtained from the regular swaption surface. Market makers calculate implied volatilities for midcurves in a number of ways. One popular method is to compute the volatility of the forward swap using the volatilities of two spot starting swaps, and the correlation between them. For example , consider a midcurve option expiring in 1 year into a swap which starts 5 years later and ends 10 years later. The correct volatility can be computed from the 1yrx5yr volatility, the 1yrx10yr volatility , and the correlation between 5yr and 10 yr swaps for the next year.

But, my question is, what strikes should be used to get 1Y5Y volatility and 1Y10Y volatility? Specifically, if we use SABR model, what is the strike to be input to handle the vol skew?

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