Why Discrete Delta Hedging Does Not Guarantee Short-Option Profit
Summary
The document examines whether dynamically delta hedging a short option guarantees recovery of the option’s initial extrinsic value when it expires out of the money. It explains that this conclusion holds only in an idealized limit: with zero transaction costs, geometric Brownian motion, and constant volatility equal to implied volatility, increasingly frequent hedge adjustments make hedging cost converge almost surely to the option price.
Real hedging uses nonzero intervals, so costs vary with the underlying’s path. A smooth move away from the strike can make hedging relatively inexpensive, while repeated movements near the strike can make it costly, even if the option ultimately expires out of the money. Transaction costs also make continuous rebalancing impractical, since Brownian motion has unbounded variation. The discussion is a conceptual result under specified assumptions, not a guarantee about realized trading outcomes or a quantified estimate of hedging costs.
Key ideas
- Under ideal assumptions, continuous delta hedging makes hedging cost converge to the option price.
- Discrete rebalancing leaves realized hedging cost uncertain and dependent on the underlying’s path.
- A short option expiring out of the money does not ensure that discrete hedging costs stay below the premium.
- Transaction costs and continuous rebalancing make the idealized limit impractical.
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Full text
# Does a delta hedged short option guarantee profit of extrinsic value at expiration? # Does a delta hedged short option guarantee profit of extrinsic value at expiration? If a trader shorts an option and dynamically delta hedges to ensure the delta is equal to 0 if that option expires out of the money does the trader profit that options extrinsic value at the time of selling it? To me this makes sense as even if Vol rises or the option moves ITM, at expiration all extrinsic value goes to 0 and since the delta is hedged the trader will profit the extrinsic value that they sold. But I feel like I must be missing something, where am I mistaken? ## Answer by RRL (score 2, accepted) https://quant.stackexchange.com/a/12947 Consider your question in the idealized case of zero transaction cost and where the underlying stock price follows geometric Brownian motion with constant volatility -- identical to the implied volatility used to price the option. If the delta hedge is rebalanced over time short time intervals of length $\Delta t,$ then the cost of hedging is a random variable that converges almost surely to the option price in the limit as $\Delta t \rightarrow 0$. In practice, of course, it is impossible to continuously delta hedge. Furthermore, in the presence of transaction costs the hedging cost with continuous rebalancing diverges to infinity -- Brownian motion has unbounded variation. Hence, the hedging cost for any realistic strategy has some distribution around the option price with non-zero variance for $\Delta t > 0$. The actual cost will depend on the path of the underlying price. Even if the option expires OTM, there are paths where the cost can be very low -- underlying price runs steadily in the OTM direction, and there are paths where the cost can be very high -- underlying price oscillates frequently through the strike price close to expiration and finishes OTM. In summary, with delta-hedging over non-zero time steps, the hedging cost (conditioned on the option expiring OTM) is not less than the option price with certainty.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.