Expected PnL Through Time for a Delta-Hedged Option

Summary

The document derives expected profit and loss at an intermediate time for a continuously delta-hedged option in a Black–Scholes setting. It separates the option’s mark-to-market value from the cumulative expected gain or loss generated by delta hedging. The option is valued at the realized volatility for the remaining term and compared with its original implied-volatility value; expected hedge PnL bridges that interim difference to the initial volatility-value spread.

The worked setup considers a purchased call with implied volatility below the assumed constant realized volatility and asks about the expected outcome before expiry. The answer uses Black–Scholes call values at the current remaining maturity and at the original maturity to express the expected hedge contribution. This is an expectation under idealized assumptions, not a pathwise PnL forecast. It assumes constant volatility, continuous hedging, no transaction costs, and zero dividends and interest rates, so real trading outcomes can differ.

Key ideas

• Interim expected PnL includes both the option’s current value and realized delta-hedging gains.
• Value the option at the assumed realized volatility for the remaining time to expiry.
• The expected hedge contribution is obtained by comparing the current volatility-value difference with its initial value.
• The derivation assumes Black–Scholes conditions, continuous hedging, and zero rates and dividends.

Tags

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


# 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?

## Answer by dm63 (score 1, accepted)

https://quant.stackexchange.com/a/74381

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.