Black–Scholes Simulations Require the Lognormal Drift Correction

Summary

The post investigates a persistent difference between simulated returns on an option account and a delta-hedged account when the Black–Scholes model is run at very high volatility. It asks whether the discrepancy comes from confusing the volatility of log returns with the volatility of ordinary returns, while acknowledging that a calculation mistake is also possible.

The accepted response points to the required drift correction in the lognormal price process: the exponent includes a negative half-variance term as well as the random volatility shock. Omitting that adjustment changes the expected underlying price and can bias simulated account returns, especially when volatility is large. The exchange offers a concise diagnosis rather than a full derivation or independent verification of the simulation, so the reader should still check the account mechanics and assumptions in their implementation.

Key ideas

• Black–Scholes price simulations use a lognormal process with a negative half-variance adjustment in the exponent.
• Leaving out the adjustment changes the expected underlying price and can bias simulated returns.
• The difference between log-return volatility and ordinary-return behavior becomes more material at high volatility.
• The suggested cause is a likely explanation, but the post does not inspect the simulation or rule out other errors.

Tags

# Effect of massive volatility on BS formula


# Effect of massive volatility on BS formula












I am experimenting with very high volatility on the standard Black-Scholes formula. I set risk free to zero, time to expiry to 1, volatility to 1 (=100%), and underlying to 1. Then I simulate the profit and loss on (i) the delta hedge account and (2) the option account between t=1 and t=0 and compute compare the expected return on both. There is a persistent bias in the result – the expected return on the option account is consistently higher.

I’m wondering if this is caused by the difference between volatility of the log returns and actual returns. There is little difference at low volatility, but it is huge at such massive vols. It could be calculation error of course, but I don’t think so because everything works fine at typical volatilities.

## Answer by Mark Joshi (score 1, accepted)

https://quant.stackexchange.com/a/17152

i would guess it is the difference between

$$\exp(-0.5\sigma^2 T + \sqrt{T}\sigma Z)$$

and

$$\exp(\sqrt{T}\sigma Z)$$

The first is correct, the second is wrong.

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