Delta hedge value formula

Question

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?

Answer 1

$$PnL = 0.5 \Gamma S^2 (\sigma_r^2 - \sigma_i^2)dt $$ 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.

Answer 2

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).

Also this gives you the replication error i.e. difference between the option price evolution (which you are long) and that of your self-financing delta hedge (= assumes you hedge in a self-financing fashion by transferring cash between stock and cash account with no exogenous infusion/withdrawal of cash).