Continuous Delta-Hedge P&L and Realized Versus Hedging Volatility

Summary

The document corrects a proposed expression for the profit and loss of a continuously delta-hedged option. It states that the relevant quantity is an integral over time involving one half of the option’s gamma, the square of the underlying price, and the difference between realized instantaneous variance and the variance used to calculate the hedge’s Black–Scholes Greeks.

This formulation explains why hedge P&L depends on the gap between realized and assumed volatility, weighted by gamma exposure, rather than on volatility multiplied by squared gamma. It describes the total P&L for a long option portfolio under continuous delta hedging. The excerpt supplies a formula but no derivation or empirical evidence, and it does not discuss transaction costs, discrete rebalancing, financing, or other real-world effects that can alter realized hedge performance.

Key ideas

• Continuous delta-hedge P&L depends on the difference between realized variance and hedging variance.
• Gamma and the square of the underlying price weight that variance difference over time.
• The stated relationship uses gamma, not squared gamma.
• The result describes an idealized continuously hedged long-option portfolio and omits practical trading costs.

Tags

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


# 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? $$PnL = \sum_{t=0}^T \sigma(t,S_t) * \Gamma^2 $$

## Answer by Quantuple (score 2, accepted)

https://quant.stackexchange.com/a/37634

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.

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