We're talking about Ito's stochastic equation:

[Ito-2]       dP = μ(t,P)dt + σ(t,P)df

where P(t) is the Price of a stock at time t and dP is the change in Price over some small time interval, dt.
This change, dP, comes in two parts called DRIFT and DIFFUSION:

• μ(t,P)dt is the deterministic part
• σ(t,P)df is the stochastic part, with df a random Brownian Motion with Mean = 0 and Standard Deviation = 1.

>Standard Deviation = 1? Where'd that come from?
If the SD is, say, 10, then we could just replace σ(t,P)df by [10σ(t,P)] [df/10].
This guy, [df/10], would still have Mean = 0 but SD = 1 so we'd just ...

>Redefine your σ and df, eh? Give all the volatility to σ, eh?
Exactly. In fact, the entire stochastic term σ(t,P)df has Expected Value = 0.

>Huh?
The Expected Value is the Mean, so if E means we're taking the Mean, then E[σ(t,P)df] = 0.
In fact, both σ and df are random variables, each having a Mean of zero. However, df has Standard Deviation = 1 whereas σ provides the Standard Deviation for the changes in Price.

>Huh?
It's just as you said. Give all the volatility to σ.
Note, however, that we're talking about the volatility of the changes in price.

Look again at [Ito-2].
Let's find the Expected Value for the change in Price over some time interval dt:

E[dP] = E[μdt] + E[σdf]

But, as we've said, the random component has Mean = 0, that is E[σdf] = 0, so E[dP] = E[μdt] and ...

>So the first term, in Ito-2, will gives the Mean for P and the second term gives nothing.
No! The first term gives the Mean for the changes in P ... and the second term contributes nothing to this change. However, the second term does gives the random component. In our case, that means it provides the volatility, or Standard Deviation for the changes in Price. If the second term were absent, then we'd expect a smooth graph, for P versus t ... and that don't hardly happen with stocks.

>You're telling me! I remember buying Microsoft and ...
Pay attention.

So far we've

• Set up an Ito scheme for changes in stock Price, described by [Ito-2].
• We've identified the first DRIFT term, μ(t,P)dt, as providing the Mean for the changes.
• We've identified the second DIFFUSION term, σ(t,P)df, as providing the Standard Deviation for the changes.

>And now you're going to do something useful, eh?
We'll see.

>And you're going to explain this funny Brownian stuff?
We'll see ...

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