Why Discrete Hedging Can Fail Under Black–Scholes
Summary
The document explains why the Black–Scholes delta hedge can have gains or losses when volatility changes, despite the model’s claim of a riskless hedge. The key distinction is that the theoretical result assumes geometric Brownian motion and continuous rebalancing. In practice, traders rebalance at discrete times, so the hedge’s performance depends on the price changes between those observations.
The example contrasts an index with an assumed 20% volatility and a daily sampled price path that shows no movement at the chosen observation time. The hedge responds to the sampled path, which can differ greatly from the underlying continuous path, producing a nonzero profit or loss. The explanation clarifies that a possible path under constant volatility does not guarantee a discrete hedge will be riskless. It is a conceptual illustration rather than a quantitative estimate; it does not derive hedging error formulas or cover transaction costs and other real-world frictions.
Key ideas
- The Black–Scholes riskless hedge assumes continuous price paths and continuous rebalancing.
- A hedge rebalanced at discrete intervals is exposed to price movements between rebalances.
- Observed volatility along sampled prices can differ from the volatility of the continuous path.
- That sampling difference can create gains or losses even when the model assumptions hold.
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Full text
# How can the BS riskless hedge break down when volatility changes, if a random walk can produce any price history? # How can the BS riskless hedge break down when volatility changes, if a random walk can produce any price history? 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? ## Answer by Ivan (score 3, accepted) https://quant.stackexchange.com/a/47221 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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