# Delta hedging and PF-value

Imagine buying a call option and shorting the delta. After some time $dt$, the stock price changes, and so does the delta and the call option value. We re-adjust our hedge using this new delta.

Question: What is the (formula for the) theoretical change in value of such a portfolio, from one period to the very next? What is the actual change in value? And how are those two related?

• Theoretically the portfolio appreciates at the risk free rate of interest, like any portfolio with no risk. This was the key insight of Black Merton and Scholes. Once you hedge away the stock market movements you have a riskless portfolio (under the BS assumption that vol is constant). I don't know wha you mean by actual change in value. – Alex C Mar 11 '17 at 23:18
• I don't think the portfolio consisting of long option and short delta units of stock is risk-free. See this discussion quant.stackexchange.com/questions/32171/… – dbluesk Mar 11 '17 at 23:25