Delta heding & PnL

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

• 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.
– dm63
Feb 9, 2020 at 14:03
• ok thanks @dm63 that's what i wanted to be sure of Feb 9, 2020 at 15:39