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

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)!