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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.

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$ \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.

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  • $\begingroup$ Thanks for your answer, I couldn't find any mention of the units in the link you gave (nor in many other online tutorials). You say $\sigma$ is quoted as a daily percentage, but a percentage of what? What are the units/dimensions of the quantity that is being converted to a percentage? $\endgroup$ – kotozna Dec 16 '14 at 10:31
  • $\begingroup$ $ \sigma $ is in 1 / square root ( time ). It does not make much logical sense - I agree. It is a percentage of the price, the underlying's standard deviation over a given time interval (e.g. daily). You can arrive at the units of sigma by using the fact that $ E[(dX)^2] = dt $ $\endgroup$ – jaamor Dec 16 '14 at 13:55

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