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?
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