# Delta hedging theta pnl

Say I sell a swaption and delta hedge it, and the breakeven daily move in the underlying is $$x$$ bps. Then if on any given day the actual move in the underlying is $$y$$ bps $$( y . Then I, as option seller, get to keep some theta decay as my pnl for that day.

Question is what fraction of theta pnl do I get to keep as a function of $$x$$ and $$y$$. My guess is

$$1 - \frac{y^2}{x^2}$$ It looks correct in the limiting case of $$x$$ = $$y$$. But could someone please correct or confirm

## 1 Answer

I assume we are talking about swaptions here? Then your formula looks correct, under a couple of assumptions; first, from the context of the question, you are assuming that you delta hedge once a day at the close of business. Second, you have implicitly assumed that the gamma is constant over the region of the daily move, which is ok as long as the option isn’t too short dated and the move x is not too large. The formula should be pretty accurate under those assumptions.