Using Gamma Ladders to Estimate Swaption P&L for Large Curve Moves

Summary

The question asks how to use a swaption’s delta and gamma ladder to estimate historical P&L when the underlying swap curve moves substantially. It describes the first-order delta estimate and asks how to extend the second-order Taylor term when gamma varies across the curve rather than being a single value.

The answer emphasizes that a Taylor expansion is a local approximation, so its accuracy can deteriorate for large moves. It suggests treating gamma as a profile over the underlying swap rate, fitting a curve to that profile, and numerically differentiating it to estimate speed, the derivative of gamma. That higher-order sensitivity can then be used in a third-order expansion. The response offers a possible extension rather than a complete calculation procedure; it does not specify how to fit the curve, handle multiple curve nodes, or validate the resulting historical estimates. For large shocks, the approximation should be treated cautiously.

Key ideas

• Taylor expansions approximate P&L locally and may be inaccurate for large market moves.
• Option gamma varies with the underlying, so a ladder represents a changing sensitivity profile.
• A fitted gamma profile can be differentiated numerically to estimate speed.
• Speed can provide an additional term in a third-order P&L approximation.

Tags

# Historical PNL using Taylor Expansion for Gamma Ladders


# Historical PNL using Taylor Expansion for Gamma Ladders












I have DV01 and Gamma Ladder for IR Swaptions and the historical market data of the underlying swap curve. Can someone please help me understand how to calculate historical PNL using taylor expansion for large move in swap curve

Delta: 100k

Gamma Ladder:

If the swap curve changed by 60bp, I believe the delta PNL would be 100k*60. I need help in understanding how the 2nd term of taylor expansion i.e. 0.5 * Gamma * (change)^2 works when we have gamma ladder rather than a single gamma number.

Thanks a lot

## Answer by SachaTheBrave (score 1)

https://quant.stackexchange.com/a/55061

By definition, a Taylor expansion is a local approximation, so you shouldn't use using it for large moves. Also you always have for options a 'gamma ladder', as gamma is a function of the underlying. One thing you could try maybe is to fit a curve to you gamma profile, then use a numerical method to calculate the derivative at your current point. This will give you Speed (the derivative of gamma with respect to the underlying, so your swap rate). Then you plug in that Speed number into your Taylor expansion up to the 3rd order.

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.