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Sorry if it's a duplicate but i didn't find an answer to my simple question in the other posts.

Let say we short a call option on a stock. $K = 100$, $C = 1$, $S = 100$ and $\Delta = 0.5$. No dividends or transaction fees. We buy 0.5 stock then our portfolio $\Pi = -1 + 50 = 49$. If the stock goes to 101, the call worths theoretically 1.5 so we have $\Pi = -1.5 + 50.5$. Also, $\Delta = 0.6$ now. But due to the convexity, C = 1.6 and $\Pi = -1.6 + 50.5 = 48.9$ our $PnL = -0.1$

If the stock goes by to 100 and the call to 1, is the only way to "erase" the PnL of -0.1 not to hedge by buying 0.6 stock at 101 ? Or even if we don't hedge, the PnL will be realized also at 100 ? If the latter, why ?

Thanks

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    $\begingroup$ Yes, as soon as you buy the additional stock at 101 you have locked in the -0.1. If you do not buy any more stock at 101 then you are hoping the stock returns to 100, in which case you will get back the -0.1. $\endgroup$ – dm63 Feb 9 at 14:03
  • $\begingroup$ ok thanks @dm63 that's what i wanted to be sure of $\endgroup$ – TmSmth Feb 9 at 15:39
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In order to erase the remaining P&L, you should gamma hedge your options. Actually, this is due to the non linearity of the options price.
This can be done by going long or short options with important gamma to make the gamma of your hedge portfolio equal to your options one. This way, the change of your delta will be hedged. If you want to know how many options you should purchase, please have a look on this document:
http://banach.millersville.edu/~bob/math472/GammaHedging.pdf

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  • $\begingroup$ Thanks for the link, i know for the gamma but it was not my question ! $\endgroup$ – TmSmth Feb 9 at 15:38

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