# What is the profit and loss of a hedged call option portfolio?

Could somebody explain the following to me:

Essentially, an approximative (ignoring higher-order terms) formula for the change in a call-price is computed. If we delta-hedge, then delta is set to 0, giving us the last formula.

Here's what I don't get: The $\Delta$ for the call is not zero. So you can't just set it to 0, right?

It's true that the "overall delta" is 0 .... but that's not equivalent to the $\Delta$ that appears in the first and second equations in the picture. So I don't see how the text can argue that the 'overall delta' is 0, which is true, and then proceed to set the Call's $\Delta$ equal to 0?

More generally, I am looking for the theoretical PnL formula for a simple delta-hedge where 1 call option is bought, and we short delta in the underlying.

• Over a short interval of time the Call profit has a term due to the Delta of the option, but it is cancelled out by a negative term due to the short position in the stock. That is why there is no $\Delta S$ term in your equation only $\Delta t$ and $\Delta S^2$ (or if you want there are two such terms and they cancel out). – Alex C Mar 11 '17 at 1:42