Estimating Theta Retained After Daily Delta Hedging

Summary

The document considers a seller of a swaption who delta-hedges once per day and asks what share of daily theta decay remains as profit when the underlying rate move is smaller than a stated breakeven move. The proposed fraction is one minus the squared ratio of the actual move to the breakeven move.

The answer confirms this approximation under specific conditions: hedging occurs at the close of each business day, and gamma stays roughly constant across the day's move. The constant-gamma assumption is more reasonable when the option is not too close to expiry and the move is not large. The exchange provides no derivation or empirical validation, so the formula should be treated as an approximation within those assumptions, rather than a general result for all hedge schedules or market moves.

Key ideas

• The proposed retained-theta fraction depends on the squared ratio of the actual move to the breakeven move.
• The approximation assumes a daily delta hedge at the business-day close.
• It also assumes gamma is nearly constant over the underlying move.
• The approximation may be less accurate for short-dated options or large moves.

Tags

# Delta hedging theta pnl


# 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 $( y <x)$. 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

## Answer by dm63 (score 1, accepted)

https://quant.stackexchange.com/a/45808

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.

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