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Supposedly, a Black-Scholes riskless hedge will break down if the volatility is non-constant. However, a random walk with any sigma could produce any price history with some non-zero probability. If there exists a price history that will break your riskless hedge, then you never had one to begin with, because a random walk with a constant sigma could have produced that price history and broken your hedge.

You might think you can look at a price history and conclude that the volatility changed on a certain day, but in reality there is always a chance that a random walk with constant sigma produced that price history. If a hedge is riskless given constant volatility, then it can't fail under any price history, even price histories that will cause investors to say that the volatility has changed.

What is the resolution to this apparent contradiction?

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The resolution is that the GBM that is assumed in Black-Scholes is continuous, and the hedge is riskless only if it is rebalanced continuously. Now it is true that a GBM with any vol could produce any price history, but if you hedge at discrete intervals, the sampled path history and its observed point-to-point volatility becomes very important for the performance of your hedge.

Imagine in particular that you hedge a continuously-traded, 20%-vol index daily at 4.30pm and that by some freak occurrence the observed path sampled daily at 4.30pm has an observed volatility of zero (at every 4.30pm point, the underlying price is the same), then that's the vol that your hedging strategy will experience, and this will be very different from the underlying vol assumption of 20%.

That leads to potentially large P&L deviation from 0, even if the "continuously sampled" path exhibits the 20% volatility that was assumed.

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