Time Units of Volatility and Wiener Process Increments in Black–Scholes

Summary

The document clarifies dimensional consistency in the Black–Scholes stock price stochastic differential equation. If price is measured in currency and time in days, the drift coefficient has units of inverse time. To keep the diffusion term dimensionally consistent, volatility has units of inverse square root of time, while the Wiener increment has units of square root of time.

The explanation invokes the defining scaling of a Wiener process: the expected squared increment is proportional to the time interval. This makes the product of volatility, stock price, and Wiener increment have the same units as a price change. Volatility is commonly reported as an annual or daily percentage, which expresses a rate over the chosen time convention. The note also cautions that the model parameter corresponds to, but is not exactly, a standard deviation. It is a dimensional explanation rather than a derivation of option prices or a discussion of estimation.

Key ideas

• In the Black–Scholes diffusion term, volatility carries inverse square root of time units.
• A Wiener process increment carries square root of time units.
• The squared Wiener increment scales with the time interval, preserving dimensional consistency.
• Volatility is commonly quoted on an annual or daily convention and corresponds to, but is not exactly, standard deviation.

Tags

# What are the units of the variables appearing in a standard stochastic differential equation for a Wiener process?


# What are the units of the variables appearing in a standard stochastic differential equation for a Wiener process?












The Black Scholes model assumes the following form for the Wiener process describing the evolution of the stock price `S`:

$dS=\mu S dt + \sigma S dX$

Clearly $S$ and $dt$ have units of dollars (say) and days (say), respectively. That means $\mu$ has units of "per day". What are the units of the other variables: $\sigma$ and $dX$ ?

At no point in my textbook or any other derivation I've seen is a normalisation performed, so I assume these variables retain some meaningful units. I can't find a textbook that mentions the units, and would like to set the record straight.

## Answer by jaamor (score 2, accepted)

https://quant.stackexchange.com/a/15886

$ \sigma S $ is in units of dollars per square root of a unit of time.

$ \sigma $ is usually quoted as an annual or daily percentage.

$ dX ^2 $ is in units of time, as $ E[(dX)^2] = dt $.

Here is an online tutorial which you may find helpful.

EDIT by kotozna: $\sigma$ has dimensions 1/(square root of time) and $dX$ has dimensions square root of time. Note that $\sigma$ corresponds to but is not exactly the standard deviation.

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.