Geometric Brownian Motion

1. The Philosophy of Returns

One of the fundamental principles in finance, often attributed to Paul Wilmott, states that "the level of price doesn't matter, only returns matter." While we model asset prices, they rarely appear directly in our equations. Instead, our focus is consistently on returns.

Simple vs. Continuous Compounding

For compound returns, we begin with the discrete compounding formula:

In this expression:

• $S_0$ is the initial investment.
• $t$ is the total duration, usually in years.
• $r$ is the annual return rate.
• $n$ is the number of compound periods (days, weeks, or seasons).

When we take the compound period to be infinitesimally small ($n\to \infty$), we obtain the continuous compounding formula, which is fundamental to derivatives pricing:

The exponential form above is equivalent to working with log returns. Define the log return:

for the continuous model the cumulative log return is simply $r_t$. Because log returns sum nicely over time (i.e. they are time‑additive), they become the natural object for introducing randomness. By contrast, simple returns – the familiar $\frac{S_t-S_{t-1}}{S_{t-1}}$ – are additive across assets rather than across time. A portfolio's simple return is just the weighted average of its components (${\sum }_i{w}_i{r}_i$), which is why most portfolio and risk models operate with simple returns.

An important limitation of simple returns is their multiplicative nature across time:

This multiplicative property means that the product of simple returns does not follow a normal distribution, even when the individual returns are normally distributed. This non-normality can complicate statistical analysis and modeling.

Time-series analyses, stochastic calculus, and any argument that lets the time step shrink instead favor log returns. One of the key advantages of continuous compounding is its additive property in the exponential term. We can demonstrate this mathematically:

This property makes continuous compounding particularly useful for mathematical analysis and modeling. Divide $S_0$ on both sides and take the natural log, we obtain the log return at period $T$:

What's more, we can decompose the gross return into additive log returns:

This is the additive feature of log returns. Note that $r_{{}_T}[T]\ne r_{{}_T}$; the former is an accumulative return, while the latter is a single-period return. In contrast to simple returns, $r_{{}_T}[T]$ follows a normal distribution because it is a linear combination of $T$ normal distributions of $r$.

Fortunately the two notions converge as the compounding period grows without bound, so nothing magical is lost by choosing one framework over the other. Conceptually the advantage is simple: asset *prices* evolve by multiplying factor after factor, which makes them cumbersome to add or average.

Returns, on the other hand, accumulate additively, and when compounding is continuous, those additive pieces are just the log increments.

Comparison Experiment: Portfolio Returns

To see the difference between asset-additive simple returns and time-additive log returns, let's simulate a portfolio consisting of two assets over one year (252 trading days):

• Asset A (60% weight): $\mu =10\%$, $\sigma =20\%$
• Asset B (40% weight): $\mu =5\%$, $\sigma =10\%$

We calculate the cumulative portfolio return using two methods. The Simple Return method correctly calculates the daily portfolio return as the weighted average of the assets' simple returns, then compounds them over time. The Log Return method incorrectly assumes log returns are asset-additive, calculating the daily portfolio log return as the weighted average of the assets' log returns.

As shown in the chart, the two methods diverge over time. This discrepancy highlights a crucial rule in quantitative finance: simple returns are used for cross-sectional portfolio aggregation, while log returns are preferred for longitudinal time-series aggregation.

2. Asset Return Modeling

Real assets fluctuate around that drift. To reflect this we add a stochastic term to the per step return, producing a model that directly governs price changes through time. This is the essence of asset return modeling.

If we prefer to express the return in terms of annual parameters and number of subperiods~$T$, the simple return can be written equivalently as

The $\sqrt{\delta t}=\sqrt{\frac{1}{n}}$term arises from a fundamental property of standard deviation: when summing independent random variables, their variances add linearly, but standard deviations scale with the square root of time.

Why $\sqrt{\delta t}$ and not $\delta t$?

Suppose we want to model a random process over a total time $T=1$ by breaking it into $n$ small steps of size $\delta t=1/n$. We want the total variance at $T=1$ to be a constant ${\sigma }^2$.

Because the steps are independent, the total variance is the sum of the variances of each step:

To achieve a total variance of ${\sigma }^2$, the variance of a single step must be $\frac{{\sigma }^2}{n}={\sigma }^2\delta t$. Since standard deviation is the square root of variance, the standard deviation of each step must be $\sigma \sqrt{\delta t}$.

If we mistakenly scaled the random shock by $\delta t$ instead, the variance of each step would be ${\sigma }^2(\delta t{)}^2$. The total variance after $n$ steps would be $n\times {\sigma }^2(1/n{)}^2={\sigma }^2/n$. As we take smaller steps ($n\to \infty$), the total variance collapses to zero! The randomness completely disappears.

Volatility Dominates Drift in the Short Term

For $0<\delta t<1$, it follows that $\delta t<\sqrt{\delta t}$. As $\delta t$ approaches $0$, the term $\epsilon \sigma \sqrt{\delta t}$ becomes increasingly dominant. This indicates that in the short term, random fluctuations significantly outweigh the overall growth (drift).

3. Arithmetic Brownian Motion (ABM) & Donsker's Theorem

Before arriving at Geometric Brownian Motion, it is instructive to consider a simpler model: Arithmetic Brownian Motion (ABM). In ABM, we assume that the change in the asset price itself (rather than the return) is driven by a constant drift and volatility:

The term $d{W}_t$ represents the increment of a standard Brownian motion, but increment is always discrete in practice. So where does this continuous-time random process come from? The answer lies in Donsker's Theorem (also known as the functional central limit theorem).

Donsker's Theorem

Donsker's Theorme states that you obtain a Wiener process (Standard Brownian Motion) by letting a random walk (e.g. flipping a coin to move up or down) take infinitely many infinitesimal independent steps, with variance accumulating at a linear rate in time.

More precisely, let $X_i$ be iid with mean zero and variance one, and define the partial sums $S_n={\sum }_{i=1}^n{X}_i$. For each $t\in [0,1]$ set$W_n(t)=\frac{1}{\sqrt{n}}{S}_{\lfloor nt\rfloor }$. Donsker's theorem asserts that the sequence of processes $W_n$ converges in distribution in the Skorokhod space $D[0,1]$ to a standard Wiener process $W(t)$. Compare this with the classical central limit theorem, which only gives convergence of the one-dimensional marginals $S_n/\sqrt{n}\Rightarrow N(0,1)$ for fixed $n$. Donsker upgrades the CLT to the entire function-valued path.

Quadratic Variation

Another fundamental property of Brownian motion is its Quadratic Variation. While the total length (first variation) of a Brownian path over any interval is infinite (it is infinitely jagged), we can state convergence using a partition

where the mesh is small:

In differential notation, this is written as:

This is not just a curiosity — it is the reason stochastic calculus is fundamentally different from ordinary calculus. In classical calculus, when you differentiate a function $g(f(t))$, Taylor expansion gives $dg=g^{\prime }df+\frac{1}{2}{g}^{\prime \prime }(df{)}^2+\cdots$. For a smooth function, $(df{)}^2$ is of order $(dt{)}^2$ and vanishes — the chain rule works as usual. But for Brownian motion, $(d{W}_t{)}^2=dt$ does not vanish. The second-order term survives and contributes an extra drift:

This is Itô's Lemma — the stochastic chain rule. It is the direct consequence of $[W{]}_t=t$. In ABM, $d{S}_t=\mu dt+\sigma d{W}_t$, Itô's lemma is what tells you how any function of $S_t$ evolves. In particular, applying it to $\log S_t$ is exactly how ABM is transformed into Geometric Brownian Motion — where the drift picks up an extra $-\frac{1}{2}{\sigma }^2$ correction term, ensuring prices stay positive.

Why QV Matters: Smooth vs. Brownian

The chart below makes the distinction concrete. As you refine the partition, the quadratic variation of a smooth curve collapses to zero — confirming classical calculus applies. For a Brownian path, it converges to a non-zero constant ($\approx 1$ for $t=1$), meaning the Itô correction never disappears no matter how fine the grid.

Finally, we can take talk about ABM, while mathematically elegant, ABM has a fatal flaw for modeling stock prices: because the changes are independent of the current price level, the simulated paths can easily become negative. Furthermore, it implies that a $1 change is equally likely whether the stock is at $10 or $1000, which contradicts the proportional nature of financial returns.

4. The Brownian Motion and Stochastic Differential Equation (SDE)

To resolve the issues of ABM, we model the returns rather than the absolute price changes. By rearranging the discrete return model, we can express the change in the asset price level over a small step:

To transition to continuous time, we take the limit as $\delta t\to 0$. Let's break down how the mathematical concepts are redefined in this limit:

• The discrete change in price $S_{t+\delta t}-S_t$ becomes the infinitesimal differential $d{S}_t$.
• The discrete time step $\delta t$ becomes the infinitesimal time differential $dt$.
• The random shock term $\sqrt{\delta t}{\epsilon }_t$ undergoes the most profound transformation. Since ${\epsilon }_t\sim N(0,1)$, the term $\sqrt{\delta t}{\epsilon }_t$ is normally distributed with mean 0 and variance $\delta t$. In continuous time, this is formally defined as the increment of a Wiener process, denoted as $d{W}_t$.

Applying these continuous-time definitions, we arrive at the Stochastic Differential Equation (SDE) for Geometric Brownian Motion. This equation is the absolute foundation of the Black-Scholes-Merton framework:

Here, $d{W}_t$ represents the continuous injection of randomness into the system, satisfying $\mathbb{E}[d{W}_t]=0$ and $\text{Var}(d{W}_t)=dt$.

The Riemann-Stieltjes Integral

The Riemann-Stieltjes integral ${\int }_0^{1}f(t)dg(t)$ generalises ordinary integration by replacing the uniform measure $dt$ with increments of another function $g$. It is defined as the limit of the sum:

The key geometric insight: if you plot $f$ as a function of $g$ (a parametric curve in the $(g,f)$-plane), the RS integral is simply the area under that parametric curve. Each orange bar below has base $\Delta g_k=g(t_{k+1})-g(t_k)$ and height $f(t_k)$ — their total area converges to $\int fdg$ as $n\to \infty$.

For this to work, $g$ must have finite total variation — otherwise the "base" of each bar can flip sign unpredictably and the sum diverges. Brownian motion has infinite total variation, which is exactly why Riemann-Stieltjes fails for stochastic integrals and forces us to use Itô calculus.

The Itô Integral: Lifting BM onto the Integrand Plane

Now replace the smooth integrator $g(t)=t^2$ with a Brownian path $W_t$, and take the simplest possible integrand: $f(x)=x$ — a 45° plane in the$(W,f)$-space. The Itô integral is:

Reading the 3D chart below: the red curve is the BM path lying flat on the floor (the integrator). The grey grid is the plane $z=W$ (the integrand $f(W)=W$). Cyan drop-lines lift each partition node straight up to meet that plane—showing the height $W_{t_k}$ at each left endpoint. The orange fence panels lock that height constant across each interval$[t_k,t_{k+1}]$, exactly as in the RS fence—except the base now zigzags instead of curving smoothly.

Classical calculus would give $\int xdx=\frac{1}{2}{x}^2$, predicting$\frac{1}{2}{W}_1^2$. But the true Itô result is$\frac{1}{2}{W}_1^2-\frac{1}{2}$. The missing $-\frac{1}{2}$ is exactly $-\frac{1}{2}[W{]}_1=-\frac{1}{2}\cdot 1$ — half the quadratic variation we measured earlier. Watch it appear in the readout as you slide $n$ up.

Itô vs. Stratonovich Integrals

When trying to solve the SDE $d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t$, standard Riemann-Stieltjes calculus fails because the paths of Brownian motion have infinite variation. We must define a new type of stochastic integral.

The value of a stochastic integral depends crucially on where we evaluate the integrand within each time step $[t_i,t_{i+1}]$:

• Ito Integral: Evaluates the integrand at the left endpoint $t_i$. This is non-anticipatory (you only use information available at the start of the interval), making it the standard choice in finance for modeling trading strategies.
• Stratonovich Integral: Evaluates the integrand at the midpoint $(t_i+t_{i+1})/2$. This preserves the standard chain rule of ordinary calculus, making it popular in physics, but it implies "looking into the future," which violates the no-arbitrage principle in finance.