# Delta hedge value formula

When we delta hedge with implied volatility, and dynamically adjust every day, I believe the PnL theoretically is $$PnL = 0.5 \Gamma S^2 (\sigma_r^2 - \sigma_i^2)dt$$ where $\sigma_r$ is realized volatility.

My question is, how accurate is this? I am trying to do a delta hedge experiment, and I find that my daily PnLs range wildly, yet the values given by above formula remain somewhat small (< 1)?

I compute my daily PnL as $$'\text{change in call price'} + \Delta \cdot '\text{change in spot price}'$$ since the first term gives us what we lost/gained through the call, and the second gives us what we earned shorting the stock. But these two formulas don't match .... however the aggregate results do?

$$PnL = 0.5 \Gamma S^2 (\sigma_r^2 - \sigma_i^2)dt$$ this is an average P&L rather than an exact one. So it should agree with your other formula on average but not each day.
At the order 1 in $dt$ you should rather use $0.5 \Gamma(S, \sigma_i) S^2 ((dS/S)^2 - \sigma_i^2 dt)$ to get the true P&L increment, not the average one (see Mark Joshi's answer).
• You could perhaps say that Mark Joshi's answer is the ex-ante estimate of Tuesday's P&L that you form on Monday close not knowing what the Stock market will do tomorrow, while Quantuple's answer is the ex-post estimate you form on Tuesday close once you know what the stock market return $\frac{dS}{S}$ was today. – Alex C Mar 25 '17 at 18:47