# Expected value of delta-hedged portfolio

Consider portfolio in black-scholes world

$$\Pi = \Delta S - V$$, where $$S$$ is the stock price and V is the price of the option.

I have read that if we set $$\Delta = \frac{\partial V}{\partial S}$$ then we obtain $$d\Pi = (...)dt + 0 * dW$$, where $$W$$ is brownian motion. And by no-arbitrage we have $$d\Pi = r \Pi dt$$, where is risk-free interest rate, so that $$\Pi_T = (\Delta_0S_0 - V)\exp(rT)$$.

I came across some lecture notes, that claim that if $$\Pi = \Delta S - V$$ is $$\Delta$$-hedged then value of such portfolio is $$0$$ at time of expiration of the option $$T$$.

But I would be expecting such a portfolio to have a value of $$\Pi_T = (\Delta_0S_0 - V)\exp(rT)$$, could someone help to figure out what is going on?

Thank you