Geometric Brownian Motion - Properties of GBM

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

\begin{alignat}{2} d\log(S) & = f^\prime(S)\,dS + \frac{1}{2}f^{\prime\prime} (S)S^2\sigma^2 \, dt \\ & = \frac{1}{S} \left( \sigma S\,dW_t + \mu S\,dt\right) - \frac{1}{2}\sigma^2\,dt \\ &= \sigma\,dW_t +(\mu-\sigma^2/2)\,dt. \end{alignat}

It follows that .

This result can also be derived by applying the logarithm to the explicit solution of GBM:

\begin{alignat}{2} \log(S_t) &=\log\left(S_0\exp\left(\left(\mu - \frac{\sigma^2}{2} \right)t + \sigma W_t\right)\right)\\& =\log(S_0)+\left(\mu - \frac{\sigma^2}{2} \right)t + \sigma W_t. \end{alignat}

Taking the expectation yields the same result as above: .

Read more about this topic:  Geometric Brownian Motion

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    “A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
    —Ralph Waldo Emerson (1803–1882)