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 *S _{t}* 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 *rS _{t}dt* drives exponential growth and the stochastic term

*σS* contributes random shocks. Within the stochastic term, it is the Wiener Process

_{t}dW_{t}^{[4]}

*W*that causes the random variability. Wiener Processes integrate Gaussian white noise and have the following basic properties:

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

As an additional characteristic of white noise non-overlapping increments of *W _{t}* 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[*W _{t}*] is zero, the expected value of

*S*is still geometric growth unperturbed by random shocks:

_{t}\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 *dW _{t}* over timespan

*Δt*have a variance of

*Δt*. Therefore, the aggregate increment

*ΔW*can be expressed as:

_{t}\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 *ϵ _{t}* are simply draws from a standard normal distribution. It is thus straightforward to obtain GBM paths starting from some initial value

*S*and recursive applications of Equation (5).

_{t}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 It**ô** 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 *dS _{t}^{2}* 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, *dt ^{2}* approaches zero more quickly. Similarly for the product

*dt⋅dW*. However,

_{t}*dW*is proportional in expectation to

_{t}^{2}*dt*because the variance of

*W*is

_{t}*t*. So

*dW*is the only term that impacts the result for

_{t}^{2}*dS*, contributing a deterministic drift of

_{t}^{2}*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 *σ ^{2}/2*. Since E[

*σdW*] is zero, the expected return

_{t}*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 *S _{t}*(0)=

*S*.

_{0}## 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 σ^{2}:

\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*[*X*^{2}] 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 *S*_{0} 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* + *σ*^{2}/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