Geometric Brownian Motion

Denote the stock price at time $t$ by $S(t)$ for $t\geq 0$ . ${S(t),t\geq 0}$ is a stochastic process adapted to a filtration $(\mathcal{F}_t)_{t\geq 0}$ . $W(t)$ is the one-dimensional standard Brownian motion. We assume $S(t)$ satisfies the following stochastic differential equation(SDE):

(1) $\begin{equation*}\mathrm{d} S(t) = \alpha S(t) \mathrm{d} t + \sigma S(t) \mathrm{d} W(t),\end{equation*}$

where $\alpha$ is the return rate of the stock, and $\sigma$ represent the volatility of the stock. The left side of the equation represents the change of stock price, and the right side of the equation is the sum of return and noise that are proportional to the stock price $S(t)$ .

The solution to (1) is a geometric Brownian motion. It can be solved by the following way. Solution to ODE $y'=\alpha y$ is $y(0)e^{\alpha t}$ . It is reasonable to guess the solution to (1) is $S(t)=S(0)e^{at+bW(t)}$ with $a,b$ to be determined. We then apply Ito’s formula to $f(x,y)=e^{ax+by}$ .

$\begin{aligned}\mathrm{d} S(t) = &\ f_x(t,S(t))\mathrm{d} t + f_y(t,S(t))\mathrm{d} W(t) \\& + \frac{1}{2}f_{xx}(t,S(t))\mathrm{d} \langle t,t \rangle + \frac{1}{2} f_{xy}(t,S(t)) \mathrm{d} \langle t,W(t) \rangle + \frac{1}{2}f_{yy}(t,S(t)) \mathrm{d}\langle W,W\rangle_t\\= &\ ae^{at+bW(t)} \mathrm{d} t + be^{at+bW(t)} \mathrm{d} W(t) \\ & + \frac{1}{2}f{xx}(t,S(t)) \mathrm{d} 0+\frac{1}{2}f_{xy}(t,S(t)) \mathrm{d} 0+\frac{1}{2} b^2 e^{at+bW(t)} \mathrm{d} t \\= & \ (a+\frac{1}{2}b^2)e^{at+bW(t)} \mathrm{d} t + be^{at+bW(t)} \mathrm{d} W(t) \\= &\ (a+\frac{1}{2}b^2)S(t) \mathrm{d} t + bS(t) \mathrm{d} W(t)\end{aligned}$

By letting $a+\frac{1}{2}b^2 = \alpha$ and $b= \sigma$ , and solving for $a,b$ , we will get

(2) $\begin{equation*}S(t) = S(0)\exp((\alpha-\frac{1}{2}\sigma^2)t + \sigma W(t)).\end{equation*}$

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