# Can we rewrite the pnl of a continuous hedge option as the time average of the volatility weighted by the square gamma?

From what I understand of El Karoui BS Robustness Formula, we can write the PnL of a continuously hedged option as the time average of the volatility weighted by the square gamma, is that right? $$PnL = \sum_{t=0}^T \sigma(t,S_t) * \Gamma^2$$

• More like $$PnL = \int_0^T \frac{1}{2} \Gamma(t,S_t,\sigma) S_t^2 \left( \sigma_r^2 - \sigma^2\right) dt$$ where $\sigma_r^2$ is the realised quadratic variation of log-prices and $\sigma^2$ is the hedging vol with a delta hedge ratio (and here gamma) calculated under BS. – Quantuple Jan 8 '18 at 10:55
• Yes, it makes more sense, thank you. Can you post it as an answer so I can accept it? – astudentofmaths Jan 8 '18 at 11:03

The result you're referring to is actually $$P\&L_{[0,T]} = \int_0^T \frac{1}{2} \Gamma(t,S_t,\sigma) S_t^2 \left( (\sigma_t^r)^2 - \sigma^2\right) dt$$ which is the total P&L of a continuously delta hedged long option portfolio, where $(\sigma_t^r)^2$ is the realised quadratic variation of log-prices over $[t, t+dt[$ and $\sigma^2$ is the hedging vol, that is, the volatility with which the Greeks are calculated, here under BS model.