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# Geometric Brownian Motion

Geometric Brownian motion (GBM) is a type of random walk model that produces movements reminiscent of stock price charts. Its data generating process (DGP) describes exponential growth perturbed by normally distributed random shocks.

The value of a GBM process St changes as defined in the following stochastic differential equation (SDE):

\begin{align}
dS_t = r S_t dt + \sigma S_t dW_t
\end{align}
\begin{aligned}
S_t &: \mathrm{value~at~time}~t\\
r &: \mathrm{growth~rate}\\
\sigma &: \mathrm{volatility}\\
W_t &: \mathrm{Wiener~process}
\end{aligned}

In Equation (1) the deterministic term rStdt drives exponential growth and the stochastic term σSt​dWt​ contributes random shocks. Within the stochastic term, it is the Wiener Process[4] Wt that causes the random variability. Wiener Processes integrate Gaussian white noise and have the following basic properties:

\begin{align}
&W_0 = 0\\
&W_{t}  \sim~N(0,t)
\end{align}


As an additional characteristic of white noise non-overlapping increments of Wt are independent.

The SDE in (1) integrates to the following analytical solution in Itô calculus[1]:

\begin{align}
S_t = S_0~e^{(r - \frac{\sigma^2}{2})t + \sigma W_t}
\end{align}

Intuitively, since E[Wt] is zero, the expected value of St is still geometric growth unperturbed by random shocks:

\begin{aligned}
E[S_t] = S_0 ~ e^{rt}
\end{aligned}

## GBM in discrete time-steps

Simulations and other practical applications of GBM use a discrete time-step version of Equation (1). Because of their distribution given in (3), increments dWt over timespan Δt have a variance of Δt. Therefore, the aggregate increment ΔWt can be expressed as:

\begin{aligned}
\Delta W_t = \sqrt{\Delta t} ~ \epsilon_t\\
\epsilon_t \overset{\mathrm{iid}}{\sim} N(0,1)
\end{aligned}

Consequently, the discrete version of Equation (1) is given by:

\begin{align}
\Delta S_t &= r S_t \Delta t + \sigma S_t \sqrt{\Delta t} ~\epsilon_t
\end{align}

In computational statistics, the independently and identically distributed (iid) shocks ϵt are simply draws from a standard normal distribution. It is thus straightforward to obtain GBM paths starting from some initial value St and recursive applications of Equation (5).

Figure 1 shows an example of a geometric Brownian trajectory starting at 100 with 4% annual growth and 10% volatility developed over a 10-year period. Because of the similarity of geometric Brownian motion to stock market prices, economists sometimes use projections from such random paths to price derivatives.

## Mathematical Derivation of GBM Returns Using Itô Calculus

To get the investment return, we rewrite the GBM process from Equation (1) in terms of logarithms. Doing so requires Itô calculus[1] approximating infinitesimals as second order Taylor series expansions:

d(f(x)) = \frac{\partial{f(x)}}{\partial{x}}dx + \frac{\partial^2{f(x)}}{2~\partial{x}}dx^2

Plugging in f(x)=ln(x):

\begin{aligned}
\frac{\partial ln(S_t)}{\partial S_t} &= \frac{1}{S_t}\\
d(ln(S_t)) &= \frac{dS_t}{S_t} - \frac{1}{2}\frac{dS_t^2}{S_t^2}\\
\end{aligned}

The term dSt2 is the quadratic of the SDE given in equation (1):

\begin{aligned}
dS_t^2 = &~dS_t \cdot dS_t\\
dS_t^2 = &~r^2 S_t^2 dt^2\\
&+ 2~r\sigma~dt~dW_t\\
&+ \sigma^2 S_t^2 dW_t^2
\end{aligned}

In calculus, as dt→0, dt2 approaches zero more quickly. Similarly for the product dt⋅dWt. However, dWt2 is proportional in expectation to dt because the variance of Wt is t. So dWt2 is the only term that impacts the result for dSt2, contributing a deterministic drift of O(t). Substituting in the Taylor series expansion:

\begin{aligned}
d(ln(S_t)) &= \frac{dS_t}{S_t} - \frac{1}{2}\frac{dS_t^2}{S_t^2}\\
d~lnS_t &= \frac{dS_t}{S_t} - \frac{1}{2}\sigma^2dt
\end{aligned}

The logarithmic variant of the GBM process equation is then given as:

\begin{align}
d~lnS_t &= (r - \frac{\sigma^2}{2})dt + \sigma dW_t
\end{align}

This proves that, when looking at returns, the expected growth r of GBM paths is diminished by a volatility term σ2/2. Since E[σdWt] is zero, the expected return μ follows as:

\begin{align}
\mu = r - \frac{\sigma^2}{2}
\end{align}

Furthermore, solving SDE (1) is now straightforward by first integrating differential equation (6) over time:

\begin{aligned}
ln(S_t) &= \int (r - \frac{\sigma^2}{2})dt + \sigma dW_t\\
ln (S_t) &= (r - \frac{\sigma^2}{2})t + \sigma W_t
\end{aligned}

Equation (4), the solution to SDE (1), is now obtained by applying the exponential function and setting a start condition of St(0)=S0.

## Derivation of GBM-Variance

Finally, a short derivation of the variance of GBM paths. Generally, the variance of a random variable follows from its first and second moments:

\begin{align}
Var[X]  &=E[X^2] - E[X]^2
\end{align}

From equations (6) and (7) follows that the instantaneous return GBM paths is normally distributed with mean μ and variance σ2:

\begin{align}
d~ln(S_t) &\sim N(\mu, \sigma^2)
\end{align}

Per definition a random variable is lognormally distributed when its logarithm is normally distributed. Now define a lognormally distributed random variable X as:

X = e^{d~ln(S_t)}

The moments E[X] and E[X2] can be taken from the moment generating function for lognormal variables[3]:

E[X^n] = exp(n\mu + \frac{n^2\sigma^2}{2})

The first and second moments are therefor given as:

\begin{align}
E[X] &= e^{\mu + \frac{\sigma^2}{2}}\\
E[X^2] &= e^{2 \mu + 2 \sigma^2}
\end{align}

Plugging equations (10) and (11) into (8) yields:

\begin{align}
Var[X] = e^{2\mu + \sigma^2}(e^{\sigma^2}-1)
\end{align}

In line with the definition of moment generating functions[2] the result from equation (12) is valid for a lognormally distributed random variable having a unit value of one at time zero. For a GBM path starting with a value S0 at time zero, which is integrating up the infinitesimals from equation (9), the moments will be:

\begin{aligned}
E[S_t] &= S_0~e^{(\mu + \frac{\sigma^2}{2})~t}\\
E[S_t^2] &= S_0^2~e^{(2 \mu + 2 \sigma^2)~t}
\end{aligned}

Substituting back r = μ + σ2/2, we get the expected value and variance of GBM paths:

\begin{aligned}
E[S_t]&=S_0e^{r t}\\
Var[S_t]&=S_0^2e^{2r t}(e^{\sigma^2 t} -1)
\end{aligned}

## References

[1] Itô calculus: Wikipedia.org

[2] Moment Generating Functions: Statlect.com

[3] Moments of Lognormal Distribution: Statlect.com

[4] Wiener Process: Wikipedia.org

## Related Publications

Geometric Brownian Motion in German: zinseszins.de

Published: December 19, 2022
Updated: May 6, 2023

Financial Algebra
Financial Algebra