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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!

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  • $\begingroup$ Ofcourse the stock can go to K, then to K-10 for example, an large amount of times..obviously you can't replicate the option with this naive strategy. $\endgroup$ Commented Jun 20, 2024 at 19:55
  • $\begingroup$ Maybe replication isn’t the right word, I’m considering purely pnl, and what it looks like at expiry. $\endgroup$ Commented Jun 20, 2024 at 22:08

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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

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    $\begingroup$ +1. That is indeed the paper to read for the OPs question. And the Ito variation is called Ito-Tanaka-Meyer formula if I'm not mistaken. $\endgroup$ Commented Jul 1, 2024 at 13:33
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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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