Approximating Swaption P&L with Rate Delta, Gamma, and Cross-Risks

Summary

The document explains a first-pass way to approximate a payer swaption’s profit and loss for a move in the underlying swap rate: use a Taylor expansion. Delta represents sensitivity to a parallel shift in the swap curve, while gamma captures convexity. This can help explain P&L when curve movements are small.

For a more detailed attribution, the answer suggests measuring rate sensitivities by tenor instead of assuming a parallel shift, since exposure is concentrated around the underlying swap’s maturity. Gamma can be represented as a risk-weighted measure focused on that maturity or as a matrix across tenor pairs. The answer also recommends including cross-gammas involving rates, implied volatility, and time. It offers no formulas or numerical example, and the approximation’s stated usefulness is limited to relatively small curve moves.

Key ideas

• A Taylor expansion using delta and gamma gives a first approximation of swaption P&L for small swap-rate moves.
• The delta described is sensitivity to a parallel shift in the swap curve, while gamma captures convexity.
• Tenor-specific rate sensitivities can improve attribution when curve changes are not parallel.
• Cross-gammas between rates, implied volatility, and time can add detail to a P&L explanation.

Tags

# Swaption PnL approximation/attribution


# Swaption PnL approximation/attribution












With a payer swaptions delta and gamma is there a method for approximating pnl for a given move in underlying swap rate? (An equivalent to the Taylor expansion for a vanilla call)

Thanks!

## Answer by Dimitri Vulis (score 1)

https://quant.stackexchange.com/a/57153

Yes, the Taylor expansion usually works well as the first approximation to explain the P&L when the curve doesn't move a lot. The "delta" (first derivative) is the sensitivity to the parallel shift of the swap curve. The "gamma" (second derivative) is the convexity.

You can get even better P&L explanation by

- including separate sensitivities to swap rates at different tenors (most of your sensitivity is to the rate for the maturity of your underlying), rather than assuming parallel shift.

- using more sophisticated IR gamma. Depending on your needs, you can use one number risk-weighted to ficus on yourt underlying's maturity. Or you can have a matrix with an entry for each tenor pair.

- include the cross-gammas between rates, implied volatility, and time.

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.