Skip to content
All library documents

Interpolating SABR Volatility for Nonstandard Swaption Expiries

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

Summary

The document considers how to price swaptions at expiry and tenor points that were not directly calibrated, using a SABR volatility cube built from market observations. One proposed practice is to interpolate total implied variance across the cube for the desired point, then use that surface in pricing. This differs from linearly interpolating SABR parameters such as alpha, beta, and rho and applying the model formula afterward.

The answer warns that parameter interpolation can produce at-the-money volatility inconsistent with direct interpolation of the market matrix and may create arbitrage in the surface. It mentions interpolating at-the-money volatility as another possible choice, while stressing that surface construction is subjective. A second answer suggests alternatives such as smoothing or interpolating the implied volatility surface and recalibrating SABR, and notes that beta may be held fixed by convention. No empirical comparison establishes one method as universally best.

Key ideas

  • A calibrated SABR cube can be extended to nonstandard swaption points by interpolating total implied variance.
  • Interpolating SABR parameters can yield at-the-money volatilities that differ from direct market-matrix interpolation.
  • Parameter interpolation may introduce arbitrage into the constructed volatility surface.
  • Surface interpolation choices are subjective, and the document does not identify a universally correct method.
  • Holding beta fixed or interpolating a volatility surface before recalibration are alternatives mentioned.

Tags

OptionsDerivatives pricingVolatilityFixed incomeStatistics
Full text
# SABR for swaptions


# SABR for swaptions












We calibrate SABR on each expiry and tenor combination using market data. (e.g. 1mx10y, 3mx10y etc.) Then how about the non-standard expiry like 2.5mx10y? Do I linear interpolate the alpha, beta, rou parameters from 1mx10y and 3mx10y? Thanks

## Answer by user35980 (score 1)

https://quant.stackexchange.com/a/76848

This is a fair question. In my experience with most implementations, once you build the SABR parameter (expiry/swap-tenor) matrices by calibrating them to the ATMs/supplied skew data, what you end up with is a SABR interpolated vol cube object. This object is then fed into your pricer and when you price a non-standard expiry it's the total implied variance that's interpolated. This guarantees an arbitrage free vol surface.

Interpolating SABR parameters in time and applying Hagan's formula can lead to ATM implied vols that may be different to what u'd get if you interpolated the points on the ATM swaption matrix directly - especially for long-dated expiries. Moreover, this may also lead to arbitrage in the surface construction. The approach of interpolating the ATMvols instead of alpha mentioned by Antoine Conze would be one way to address this. But bear in mind this doesn't make one approach "more correct" than the other - vol surface interpolation is a highly subjective area.

## Answer by Amit Kumar Jha (score 0)

https://quant.stackexchange.com/a/76860

For non-standard expiries or tenors, interpolating the SABR parameters is one approach. However, there are some considerations to keep in mind:

Interpolation Method: Linear interpolation is straightforward and commonly used, but it might not always capture the nuances of the market. Other interpolation methods, such as spline interpolation, can provide smoother transitions between points.

Interpolation of Volatility Surface vs. Parameters: Instead of interpolating the SABR parameters directly, another approach is to interpolate the implied volatility surface generated by the SABR model for the given expiries and tenors. Then, for the non-standard expiry/tenor, you can calibrate the SABR model to the interpolated volatility surface to obtain the parameters. This method ensures that the interpolated volatilities are consistent with market observations.

Stability of Parameters: The β parameter is often kept fixed (e.g., β=0.8 for swaptions) based on market conventions, so you may not need to interpolate it. For the other parameters, ensure that the interpolated values lead to a stable and well-behaved volatility surface.

Market Dynamics: Always keep in mind the market dynamics and conditions when interpolating. During periods of high market stress or significant shifts in the yield curve, simple interpolation methods might not capture the true market sentiment

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.