Swaption Theta, Gamma, and Forward Swap Roll-Down

Summary

The document examines whether a swaption’s theta can be estimated from gamma and implied volatility, and why a model’s reported theta may differ from that estimate. Under the normal model, the response identifies theta as approximately negative one half of gamma times volatility squared, making it a useful theoretical rule of thumb for the volatility contribution to time decay.

The stated explanation for the mismatch is that the estimate omits roll-down in the underlying forward swap rate, while the pricing model includes it. Thus, the gamma-volatility approximation does not capture every source of a swaption’s change in value over time. The exchange does not provide a full PnL decomposition or discuss other model conventions, so the relationship should be treated as a simplified framework whose accuracy depends on what the model includes.

Key ideas

• In the normal model, swaption theta is approximately negative one half of gamma multiplied by volatility squared.
• The gamma-based theta estimate isolates a theoretical volatility-related contribution to time decay.
• A pricing model may include changes from the underlying forward swap’s roll-down.
• Differences between the approximation and model theta can arise when that roll-down effect is omitted from the estimate.

Tags

# Gamma and Theta of a swaption


# Gamma and Theta of a swaption












For a swaption, I had 2 questions:

- how would I guage the PnL based on RV vs IV on a swaption?

I'm guessing its 0.5 x gamma x (RV^2-IV^2)(or realized variance - implied variance)

Not 100% sure on this.

- I understand to calculate the theta of a swaption it's 0.5 x gamma x IV^2 (daily bp vol)

However for say, the 1y1y swaption, when modeling them I find that the actual theta does not equal the output of the above formula.

This, I'm guessing, is due to the vol shift (i.e the jan19/24+1y swaption has a higher Implied vol(118.2 vs 118) in the model, vs the jan18/24+1y swaption), while vega remaining the same. Again not confident so posting here to clarify.

FYI the gamma is 70, vega is 7500, and theta is 3800 for the 19jan24+1y swaption.

When calculating for theta: (0.5*(118.2/sqrt(252))^2*70) = 1940.5 - which is evidently quite different from Model output.

Thanks.

## Answer by user35980 (score 2)

https://quant.stackexchange.com/a/75286

In the normal model framework $\Theta=-\Gamma\sigma^2/2$ is indeed a good theoretical rule-of-thumb. However, you're ignoring the impact of the roll-down of your underlying 1y1y fwd swap in this theta calculation, while your model probably isn't.

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