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Geometric Brownian Motion

The usual model for the time-evolution of an asset price $S(t)$ is given by the geometric Brownian motion, represented by the following stochastic differential equation:

\begin{eqnarray*} dS(t) = \mu S(t) dt + \sigma S(t) dB(t) \end{eqnarray*}

Note that the coefficients $\mu$ and $\sigma$, representing the drift and volatility of the asset, respectively, are both constant in this model. In more sophisticated models they can be made to be functions of $t$, $S(t)$ and other stochastic processes.

The solution $S(t)$ can be found by the application of Ito's Lemma to the stochastic differential equation.

Dividing through by $S(t)$ in the above equation leads to:

\begin{eqnarray*} \frac{dS(t)}{S(t)} = \mu dt + \sigma dB(t) \end{eqnarray*}

Notice that the left hand side of this equation looks similar to the derivative of $\log S(t)$. Applying Ito's Lemma to $\log S(t)$ gives:

\begin{eqnarray*} d(log S(t)) = (log S(t))' \mu S(t) dt + (log S(t))' \sigma S(t) dB(t) + \frac{1}{2}(log S(t))'' \sigma^2 S(t)^2 dt \end{eqnarray*}

This becomes:

\begin{eqnarray*} d(log S(t)) = \mu dt + \sigma dB(t) - \frac{1}{2}\sigma^2 dt = \left(\mu - \frac{1}{2} \sigma^2 \right)dt + \sigma dB(t) \end{eqnarray*}

This is an Ito drift-diffusion process. It is a standard Brownian motion with a drift term. Since the above formula is simply shorthand for an integral formula, we can write this as:

\begin{eqnarray*} log(S(t)) - log(S(0)) = \left(\mu - \frac{1}{2} \sigma^2 \right)t + \sigma B(t) \end{eqnarray*}

Finally, taking the exponential of this equation gives:

\begin{eqnarray*} S(t) = S(0) \exp \left(\left(\mu - \frac{1}{2}\sigma^2\right)t + \sigma B(t)\right) \end{eqnarray*}

This is the solution the stochastic differential equation. In fact it is one of the only analytical solutions that can be obtained from stochastic differential equations.

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