Properties of GBM
The above solution (for any value of t) is a log-normally distributed random variable with expected value and variance given by
that is the probability density function of a St is:
When deriving further properties of GBM, use can be made of the SDE of which GBM is the solution, or the explicit solution given above can be used. For example, consider the stochastic process log(St). This is an interesting process, because in the Black–Scholes model it is related to the log return of the stock price. Using Itō's lemma with f(S) = log(S) gives
It follows that .
This result can also be derived by applying the logarithm to the explicit solution of GBM:
Taking the expectation yields the same result as above: .
Read more about this topic: Geometric Brownian Motion
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“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (16321704)