Why Strike-Based Stop-Loss Hedging Does Not Replicate Options
Summary
The document considers whether shorting an out-of-the-money option and hedging only when the underlying reaches its strike can reproduce the option payoff or cap losses at the premium received. It describes a stop-loss/start-gain approach: the hedge is triggered at the strike, but the strategy is not self-financing. Carr and Jarrow's analysis is cited to explain that concentrating the hedge near the strike still incurs replication cost, even as the trigger region becomes very small; a local-time term at the strike is needed in the mathematical treatment.
A second answer notes that after a short call's strike is crossed, continued movement and fluctuation above the strike expose the position to delta and gamma changes. Discrete rehedging can therefore leave the hedge imperfect, while continuous hedging is an idealized assumption. The discussion challenges the idea that losses are simply equal to the premium collected, but it does not provide a complete derivation, quantify losses, or specify all option and market assumptions. The cited work is presented as a useful lead for deeper study.
Key ideas
- A hedge triggered only at the strike is a stop-loss/start-gain strategy.
- Strike-triggered option replication is not self-financing and incurs a replication cost.
- The analysis uses a local-time term at the strike to represent the missing cost mathematically.
- After the strike is crossed, changing delta and gamma can make a discrete hedge imperfect.
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Full text
# BSM replication with expiry delta # BSM replication with expiry delta I’ve been thinking about this problem and I’m missing something. Assuming a BSM world, I sell an OTM option at strike K. I then proceed to delta hedge it at the strike K each time K is touched. Why will this not work, and will my losses be equal to the premium I received? With an ITM option I see why this wouldn’t work. And if the price touches say from below and then drops back down again this doesn’t work. But in all other cases I’m unsure why it wouldn’t work? Or am I along the right tracks thinking about the first two scenarios? Any help is greatly appreciated. Thanks! ## Answer by NBF (score 1, accepted) https://quant.stackexchange.com/a/79909 Peter Carr and Robert Jarrow have a paper on Stop-Loss Start-Gain strategies for replication of options.They show that they are not self-financing. If you put a small delta around the strike, buy high and sell low, even as delta->0, you will have a cost - the replication cost. Technically, you would have to use a variation on Ito which includes the Local time at strike K. Good paper. Think this may help. The Stop-Loss Start-Gain Paradox and Option Valuation ## Answer by KaiSqDist (score 0) https://quant.stackexchange.com/a/79811 Assuming you shorted an OTM call and the spot hits strike price $K$, you would go long on delta-times-stock. If the spot continues fluctuating above the strike, you are exposed to the delta and the gamma, which means your hedge could be imperfect due to the changing delta and the discrete nature of hedging (unless you hedge continuously of course). I am not quite sure what you mean by in the second paragraph - maybe you can add more details? And please specify if you are looking at call or put.
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