1
$\begingroup$

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?

$\endgroup$
2
  • $\begingroup$ The exact P&L cannot be calculated (and it is a random variable), but I suppose you are asking for its expected value. $\endgroup$
    – nbbo2
    Commented Jan 16, 2023 at 10:57
  • $\begingroup$ @nbbo2 Yes I realize that the exact value will be path dependent. $\endgroup$ Commented Jan 16, 2023 at 12:53

1 Answer 1

1
$\begingroup$

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 $$E(DH(t))= C(0.27,T)-C(0.24,T)-C(0.27,T-t)+C(0.24,T-t)$$

I have assumed in the above that dividends and interest rates are both zero, for simplicity.

$\endgroup$
2
  • $\begingroup$ Thanks! One thing I'm confused about though - if E(DH(t)) is equal to the delta hedging PnL, then if the realized vol is 0, it too should be 0 (since the stock didn't move at all and there was no hedging activity). But if I plug an IV of 24 & RV of 0, E(DH(t)) becomes equal to -C(0.24,T) + C(0.24,T-t), which makes it seem like E(DH(t)) is the total expected PnL rather than the delta hedging PnL. What am I missing? $\endgroup$ Commented Jan 18, 2023 at 13:16
  • $\begingroup$ Ah ok. The E(DH(t)) is equal to the realized delta hedging pnl at 27 vol minus the time decay calculated at 24 vol. I should have made that clear. $\endgroup$
    – dm63
    Commented Jan 19, 2023 at 5:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.