The perfect smile

Filling the gaps in the swaption volatility cube

How to deal with missing swaption volatility quotes.

Liquidity of swaptions versus caps/floors

Swaptions, caps and floors are popular OTC interest rate derivatives, used by banks and corporations to manage interest rate risks arising from their core business or from their financing arrangements.

The swaption market is approximately an order of magnitude larger than the equivalent cap/floor market. Nonetheless, the larger market volumes do not necessarily mean that the volatility quotes are liquid in all parts of the swaption volatility cube. Indeed, one often observes that the at-the-money swaption market is very liquid, however, for various tenors and expiries, the away-from-the-money quotes are missing or not at all reliable, especially when compared to corresponding cap/floor volatilities. The reason behind this can partially be explained due to different applications of swaptions versus caps/floors.

Volatility quotes depend on hedging applications

Swaptions are commonly traded to hedge against prepayment risks arising from fixed rate mortgages. Purchasing a swaption allows an issuer of a mortgage to “replace” the cash flows that would be lost in case of a prepayment. In general, prepayment is a risk when rates are declining. As a result, a prepayment hedge would often be an at-the-money (or slightly out-of-the money) swaption.

Caps/floors on the other hand are often used to hedge interest rate risk arising from contractual upper and lower rate bounds in floating-rate mortgages contracts. This would be the case, for example, in a mortgage contract which limits the variability of the floating rate from 0% to 3% (for instance). Given that the contractual bounds are typically away from the current level of interest rate, one would expect that caps/floors have liquid trades away from the money as well.

Completing the swaption volatility cube

The portfolio of a financial institution is typically very complex with instruments that require a range of volatility quotes. Therefore, although the data providers may not give quotes on the entire volatility cube, the trading floor and risk management of the financial institution are obliged to complete it.

How to fill the gaps in the swaption volatility cube?

There are several different ways by which one can “complete” the swaption volatility cube. The most common approach is based on the SABR model, which provides a way of interpolating volatilities between quoted strikes, as well as extrapolating beyond them. The SABR parameters (alpha, beta, rho and nu) can be easily calibrated when the market provides a number of reliable volatility quotes at different strikes, for a fixed expiry and tenor (see Skantzos et al. (2016)).

When, however, for a given expiry and tenor, one only has one quoted strike (typically the at the money point), the SABR approach cannot be applied directly. In this case, additional assumptions need to be made. Typical approaches would be to crudely assume a flat volatility smile, or to leverage some of the smile characteristics observed for different tenors or expiries (in case they would be available). This article, however, focuses on an alternative approach: using the information available from the cap/floor volatility surface to inform a swaption volatility smile.

Lifting from caps

There exists an intricate relationship between swaptions and caps/floors. Indeed, both instruments reference the same underlying interest rate curve. Whereas swaptions relate to forward swap rates, caplets/floorlets are driven by changes in forward rates. This relationship between the two instruments can be used to inform a swaption volatility smile from the cap/floor volatility surface, an approach referred to as: “lifting from caps”. In this article, we explore two different market practices:

  • Lifting from SABR (Hagan et al. (2004)): the SABR beta, rho and nu SABR parameters are taken from the caplet market. The alpha parameter is recalibrated to match the quoted at the money swaption volatility.
  • A structural approach: an explicit relationship is made between cap/floor volatilities and swaption volatilities by expressing the forward swap rate as a series of forward rates.

In the enclosed article, we present the key ideas and features of the two approaches and illustrate the two approaches on the EUR swaption market.

The perfect smile - Filling the gaps in the swaption volatility cube
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