Derive the profit and loss of a European option hedged continuously at implied volatility sigma_i while the stock realises a time-varying volatility sigma_t. State the resulting integral and explain how a trader can be right about the average level of volatility and still lose.

Derive the profit and loss of a European option hedged continuously at implied volatility sigma_i while the stock realises a time-varying volatility sigma_t. State the resulting integral and explain how a trader can be right about the average level of volatility and still lose.

Approach: Apply Ito to the Black-Scholes value marked at implied volatility along the true price path, then use the Black-Scholes equation to cancel the theta and rate terms.

The hedged profit and loss is 0.5 times the integral over the option's life of the dollar gamma times (sigma_i^2 - sigma_t^2), discounted to today. Applying Ito's lemma to the Black-Scholes value V(S_t, t; sigma_i) along the real path gives dV = theta*dt + Delta*dS + 0.5*Gamma*sigma_t^2*S_t^2*dt, and the delta hedging removes Delta*dS while the Black-Scholes equation replaces theta with -0.5*Gamma*sigma_i^2*S_t^2 minus the carry terms. What survives is 0.5*Gamma_t*S_t^2*(sigma_i^2 - sigma_t^2)*dt accumulated to expiry. Each day is weighted by that day's dollar gamma, which is largest when the stock sits near the strike and near expiry, so a short option position can sell 25 volatility against a year that realises 20 and still lose if the quiet stretch happened far from the strike and the violent stretch happened on top of it. Path dependence decides the outcome, since the gamma-weighted average of realised variance is what is paid rather than the unweighted one.

Follow-up: Which position has a payoff proportional to the unweighted average variance, and what does the hedger of that position need from the strike ladder?

Key concepts: dollar gamma, delta hedging, path dependence, Ito's lemma, implied volatility.