Gamma and Theta of a swaption

Question

For a swaption, I had 2 questions:

1. 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.

2. 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 1

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.