Why Black–Scholes Uses Log-Return Volatility
Summary
The document explains why the Black–Scholes model uses the volatility of log returns rather than the standard deviation of absolute price levels or price changes. Its central point is that the model treats the log of the underlying price as normally distributed, so the distribution of the price itself is lognormal. To estimate the probability that the underlying finishes above or below a strike, the model standardizes the log ratio of strike to current price using the expected log return and its volatility over the option’s life.
The answer connects that calculation to the normal cumulative distribution terms in the Black–Scholes formula. A brief follow-up notes that interest-rate options may instead use a normal model, where the volatility input is the standard deviation of the rate itself. The explanation is conceptual and omits derivations of the full pricing formula; the appropriate volatility measure depends on the model’s assumptions and the underlying being modeled.
Key ideas
- Black–Scholes models log prices as normally distributed, which implies a lognormal distribution for prices.
- The model’s volatility input describes log returns over the option horizon.
- Strike probabilities are calculated by standardizing the log ratio of strike to current price.
- Normal models for interest rates can use the standard deviation of the rate level instead.
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# Answer by David Duarte (score 4, accepted)
# Why the volatility of log-returns and not the volatility of the absolute level of the underlying is used in the Black-Scholes model?
If I want to price an option with the B-S model, why do I have to use the standard deviation of the log-returns of the underlying for the sigma parameter and not just the standard deviation of the absolute price level of the underlying?
## Answer by David Duarte (score 4, accepted)
https://quant.stackexchange.com/a/51225
Black Scholes assumes the price series is lognormally distributed, and so $ln(S_t) = X$, where $X$ is normally distributed
Notice that to price the option, you want to get the probability of the underlying price reaching the strike, so basically you want the probability of a jump from the current level to the strike level, ie:
$$X_i = ln(\frac{K}{S})$$
This variable is normally distributed and to convert it to standard normal we'll subtract the mean and divide by the standard deviation,
$$ \frac{X_i - \bar{X}}{std(X)} $$
which in the BS framework would be $\bar{X} = (r - \frac{1}{2}\sigma^2)t$ and $std(X) = \sigma \sqrt{t}$
Replacing this above gives you:
$$d = \frac{ln(K/S) - (r-\frac{1}{2} \sigma^2)t}{\sigma \sqrt{t}}$$
This would allow you to determine the probability of underlying finishing below the strike by using the standard normal distribution, ie, $P[S_t < K] = N(d)$
Because the normal distribution is simetric, $P[S_t \geq K] = 1 - P[S_t < K]$, so you just have to change it to $1 - N(d)$ or $N(-d)$, which is basically the N(d2) in the BS option pricing formula
All this to say, that you are modelling the price returns so you should use the volatility of the returns, and not the absolute price change
## Answer by Randor (score 0)
https://quant.stackexchange.com/a/51251
note that for options on interest rates , often people use the Normal model rather than black-scholes , and for the Normal model, the standard deviation of the rate (rather than stdev of log returns of it), is the input volatility.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.