hard · Quantitative Finance derivatives

Ignoring funding and transaction costs, which statement about the trader's P&L is correct?

1. The total hedged P&L is exactly zero in expectation but has positive variance, since gamma P&L is path-dependent even when realized equals implied volatility
2. The hedged P&L is exactly zero along every path, because the daily gamma-theta balance holds pointwise when realized volatility equals implied volatility
3. The hedged P&L equals the integral of one-half gamma times the gap between squared realized and implied volatility, which is zero only in expectation, not pathwise
4. The hedged P&L is positive because the long-gamma position always profits from the convexity of the option payoff regardless of the volatility match

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