Delta heding & PnL

Question

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

Answer 1

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