# Calculating the PnL of a delta-hedged option at a point in time

In a BS world (constant volatility, no transaction costs, continuous hedging) If I buy or sell an option and continuously delta-hedge, I know how to calculate the final expected PnL based on implied vs realized volatility, but how do I calculate the PnL at some point in time before expiry of the option?

For example, I buy a 20dte call option at 24% IV and delta-hedge continuously. Realized volatility is 27% (and constant). How would I calculate the expected profit in, say, 8 days time?

• The exact P&L cannot be calculated (and it is a random variable), but I suppose you are asking for its expected value. Commented Jan 16, 2023 at 10:57
• @nbbo2 Yes I realize that the exact value will be path dependent. Commented Jan 16, 2023 at 12:53

If we denote the value of a BS call option by $$C(vol, exp)$$ then we know that at any time $$t$$ the total expected p/l including the option value and the expected realized delta hedging p/l is $$C(0.27, T-t) -C(0.24, T-t) + E(DH(t))$$ where the last term is the expected Value of the delta hedging activity from time 0 up to time $$t$$. This expression is constant, since all we are doing over time is realizing the value of an option valued at 0.27 vol. so we may equate it to its initial value at $$t=0$$ which is $$C(0.27, T) - C(0.24,T)$$ and thus $$E(DH(t))= C(0.27,T)-C(0.24,T)-C(0.27,T-t)+C(0.24,T-t)$$