Negative-gamma delta hedging (for a call option writer): how will the stock price affect the portfolio profit?

Question

Suppose a (European) call option writer is hedging their risk by taking a long position in stocks (holding $\delta_C$ shares). The value of the portfolio is $V(S)=\delta_CS-C$. Then is the gamma of the portfolio $\Gamma_P=-\Gamma_C<0$? If so, is the plot of the profit of the portfolio against stock price a concave down graph below the $x-$axis (the reflection of the yellow curve in the following graph by the $x-$axis)? Then wouldn't it imply that such a hedging will always generate a loss (negative profit) no matter how the stock price changes (if changes in other greek variables are small enough)?

I am asking this question since I am not sure whether or not the results I derived above are correct, and I find this strategy taken by an option writer to be a bit absurd if my conclusion (i.e. the portfolio will never be able to generate gain from change in stock price) is correct, since they still get a chance to gain when stock price decreases if they do not hedge at all. My sincerest gratitude for any help (or references)!

Answer 1

Yes the p/l profile is the reflection of the yellow line so it appears that the option writer always loses money. However the option writer benefits from time decay, denoted by the Greek letter theta. The theta equals the expected loss from the yellow line, basically. Consult almost any book on options or Google the black scholes equation.