I know there is a relationship as below:[$]PnL=(Option price sold-Estimated Option value)+ \[\int_0 ^T (sigma_h(t)^2-sigma_r(t)^2) S^2*Gamma_h*dt\][$]given in PARAMETER RISK IN THE BLACK AND SCHOLES Model where sigma_h is the sigma used to calculate delta, sigma_r is the real volatility. Assume we sold the option at sigma_h, and then use it to hedge. If sigma_r is bigger than sigma_h, the integration term is negative, meaning a loss. However, if we can use a bigger sigma_h2 than sigma_r, the integration term can be zero or positive. In such a case, Estimated Option value should be calculated with sigma_h2, otherwise the first term is negative. In particular, when option Gamma and underlying price is big, it is tempting to use a sigma_h close to sigma_r so the second term can be small. So I wonder if there is an optimal sigma_h to minimize the PnL after we sell the option at a low volatility?I have tried to use historical data to do simulation and find that the mean PnL and its standard deviation is quite dependent on the path. There is no obvious advantage of using a sigma_h close to sigma_r or just keep a constant sigma_h.