Why Zero Arithmetic Drift Can Still Produce Negative Compound Growth
Summary
The document asks why a Black–Scholes option price differs from an expected option payoff calculated from a spreadsheet model. One response identifies a key model mismatch: Black–Scholes assumes lognormal stock prices, while the spreadsheet uses normally distributed price movements. Those distributions produce different probabilities and option payoffs, so their expected values need not match.
Another response explains variance drag. Zero average simple return does not imply zero long-run compound growth, because gains and losses compound asymmetrically. For small returns, the gap between arithmetic and geometric growth is approximately half the return variance. The document suggests adjusting the drift in a simulation to account for that gap. These explanations are brief and do not fully specify the risk-neutral measure or option valuation assumptions, so they clarify the apparent discrepancy without providing a complete pricing derivation.
Key ideas
- Black–Scholes assumes lognormal stock prices, unlike a model with normally distributed price changes.
- Different stock-price distributions can produce different expected option values.
- Zero arithmetic average return can coexist with negative geometric growth due to volatility drag.
- For small returns, the arithmetic-to-geometric growth gap is approximately half the variance.
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# Why does expected price of OTM option not equal to BS price? # Why does expected price of OTM option not equal to BS price? If I assume that stock returns follow normal distribution with drift = 0% and S.D. = 10%. In the long, if I keep investing in this stock for a year with the same capital every year for a consecutive 100 years, my expected return would be 0% per year (equal to the zero drift). Put this another way, If I invest in a sea of stocks that have identical return distribution (drift = 0% and S.D. = 10%) for a year, my average return would be 0% (equal to the zero drift). Do I get it right? In case that I got it right, what if I change from investing in stock to option on that stock. What expected return should I see? I have tried to elaborate my problem in an excel file. The example in the file illustrates return distribution of a stock and an option of the stock which are based on normal distribution. I calculate expected price of stock and option by sumproducting probability and price. As you will see, the option price calculated from BS formula (Cell C8) is not equal to expected value got from the table (Cell C9). I expect that BS price should be equal to my expected price, why is it not the case? I have included the link of my Excel file here Expected Price of Option Any contribution will be greatly appreciated. ## Answer by Mark Joshi (score 2) https://quant.stackexchange.com/a/16364 Black--Scholes takes log-normal stock price movements whereas your model gives normal price movements. ## Answer by demully (score 1) https://quant.stackexchange.com/a/47563 You have a "variance drag" problem ;-) Create a new spreadsheet. Model your 0 drift, 10 vol with "=NORMSINV(RAND()) * 10%" and compound those returns for say 1000 years. You should find your investment's value declining by ~0.5% per annum. Put simply, an investment that halves or doubles with equal probability every period should have a zero long-run expected return. Except a 50:50 between +100% and -50% is a +25% expected return in any single period... The difference between the two (ie the arithmetic and the geometric retuns) is 0.5 * sigma^2. Put this into your model for the drift, to give you a zero long-term drift, and the anomaly should (fingers crossed) disappear.
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