Brownian Motion

This page illustrates the difference between the Generalized Wiener Process (GWP), also known as Arithmetic Brownian Motion, and Geometric Brownian Motion (GBM). GWP shows up in models for P/L, interest rates, or temperature spreads where values can freely cross zero, while GBM is the backbone of the Black–Scholes–Merton equity-pricing framework where prices must remain positive.

html`<style>
  .plot-title-small {
    font-size: 13px;
    font-weight: 600;
  }
  .plot-subtitle-small {
    font-size: 11px;
    font-weight: 400;
  }
</style>`

plotTitle = text => html`<span class="plot-title-small">${text}</span>`;
plotSubtitle = text => html`<br><span class="plot-subtitle-small">${text}</span>`;

Simulation Parameters

Adjust the parameters below to observe how they affect the paths of both processes. As a rule of thumb for applications: equity underlyings often use \(\mu\) between [-0.05, 0.10] yearly with \(\sigma\) in the 15–40% range, FX pairs tend to have smaller drifts but comparable volatilities, and rates or P/L series modeled with GWP typically sit closer to zero drift with volatility quoted in basis points.

Simulation horizon: Slider inputs are annualized. Each chart simulates \(T = 2\) years of evolution with \(N = 500\) steps (\(\Delta t = T/N \approx 0.004\)), so changing \(\mu\) or \(\sigma\) alters the drift and diffusion applied at every discrete step.

viewof mu = Inputs.range([-2, 2], {step: 0.1, label: "Drift (μ)", value: 0.1})
viewof sigma = Inputs.range([0, 2], {step: 0.05, label: "Volatility (σ)", value: 0.15})

viewof s0 = Inputs.range([1, 200], {step: 1, label: "Initial Price (S0)", value: 2})

viewof regen = {
  // Use htl.html to create a button with custom styling, matching the project's conventions.
  const button = htl.html`<button style="background: #2f71d5; color: white; border: 1px solid #2f71d5; border-radius: 6px; padding: 0.35rem 0.9rem; font-weight: 500; font-size: 0.9rem; cursor: pointer; width: 100%;">New Path</button>`;
  
  let count = 0;
  button.onclick = () => {
    button.value = ++count;
    button.dispatchEvent(new Event("input", {bubbles: true}));
  };
  button.value = count;
  
  return button;
}

T = 2
N = 500
dt = T / N
times = d3.range(0, N + 1).map(i => i * dt)

// Generate random increments (fixed unless regenerated)
// We use a dependent cell on 'regen' to trigger re-calculation
randomNormals = {
  regen; // dependency
  return d3.range(N).map(d3.randomNormal(0, 1));
}

// Calculate paths
data = {
  const arithmetic_path = [s0];
  const gbm = [s0];
  
  let currentGWP = s0;
  let currentGBM = s0;
  
  for (let i = 0; i < N; i++) {
    const z = randomNormals[i];
    const dw = Math.sqrt(dt) * z;
    
    // GWP: dX = mu*dt + sigma*dz
    currentGWP += mu * dt + sigma * dw;
    arithmetic_path.push(currentGWP);
    
    // GBM: S_t = S_0 * exp((mu - 0.5*sigma^2)*t + sigma*W_t)
    // Or iteratively: S_{t+dt} = S_t * exp((mu - 0.5*sigma^2)*dt + sigma*dw)
    // Using iterative update for consistency with the path generation
    currentGBM *= Math.exp((mu - 0.5 * sigma * sigma) * dt + sigma * dw);
    gbm.push(currentGBM);
  }
  
  return times.map((t, i) => ({
    t: t,
    arithmetic_path: arithmetic_path[i],
    gbm: gbm[i]
  }));
}

Comparison

The two graphs below show the evolution of the asset price under the two different models using the same random shocks.

Plot.plot({
  title: plotTitle("Arithmetic Brownian Motion (GWP)"),
  subtitle: plotSubtitle("Can become negative"),
  grid: true,
  x: { label: "Time (t)", domain: [0, T] },
  y: { label: "Value" },
  marks: [
    Plot.ruleY([0], {stroke: "red", strokeWidth: 2}),
    Plot.line(data, {x: "t", y: "arithmetic_path", stroke: "blue", strokeWidth: 2}),
    Plot.tip(data, Plot.pointerX({x: "t", y: "arithmetic_path"}))
  ]
})

Graph A: Generalized Wiener Process

The process is described by the following Stochastic Differential Equation (SDE): \[dX_t = \mu dt + \sigma dz\] The solution to this equation gives the value of the process at time \(t\): \[X_t = X_0 + \mu t + \sigma z_t\] Where \(z_t\) is a standard Wiener process.

GWP moments: \(\mathbb{E}[X_t] = X_0 + \mu t\) grows linearly with time, while \(\text{Var}(X_t) = \sigma^2 t\) is unbounded as \(t\) increases. The combination of additive shocks and unbounded support (\(X_t \in (-\infty, \infty)\)) explains why values cross zero so easily.

Plot.plot({
  title: plotTitle("Geometric Brownian Motion (GBM)"),
  subtitle: plotSubtitle("Limited liability (always > 0)"),
  grid: true,
  x: { label: "Time (t)", domain: [0, T] },
  y: { label: "Price" },
  marks: [
    Plot.ruleY([0], {stroke: "red", strokeWidth: 2}),
    Plot.line(data, {x: "t", y: "gbm", stroke: "green", strokeWidth: 2}),
    Plot.tip(data, Plot.pointerX({x: "t", y: "gbm"}))
  ]
})

Graph B: Geometric Brownian Motion

The SDE for a variable \(S_t\) that follows a GBM is: \[dS_t = \mu S_t dt + \sigma S_t dz\] The solution, which can be found using Itô’s Lemma, gives the price at time \(t\): \[S_t = S_0 \exp\left( (\mu - \frac{1}{2}\sigma^2)t + \sigma z_t \right)\] Notice the drift term is not \(\mu\), but \(\mu - \frac{1}{2}\sigma^2\). This term, often called the “convexity correction” or “Itô term,” accounts for the difference between the arithmetic mean return and the geometric mean return, which is crucial for lognormally distributed processes. Because \(S_t\) is lognormal, we can immediately read off moments that matter for pricing: \(\mathbb{E}[S_t] = S_0 e^{\mu t}\) and \(\text{Var}(S_t) = S_0^2 e^{2\mu t}(e^{\sigma^2 t}-1)\), reinforcing why the distribution never reaches or crosses zero.

Experiment and Observe

Experiment: Try setting the Drift (\(\mu\)) to a negative value (e.g., -0.5) and increase the Volatility (\(\sigma\)) (e.g., 0.8).

Observe the difference in behavior:

1. Graph A (GWP): The path behaves like a random walk with a downward trend. Because the noise term \(\sigma dz\) is additive, large negative shocks can easily push the value below zero. This makes GWP unsuitable for modeling stock prices (which cannot be negative) but acceptable for variables like temperature or profit/loss.

2. Graph B (GBM): As the price drops, the absolute magnitude of the random swings (\(\sigma S dz\)) decreases because they are proportional to the current price \(S\). The price asymptotically approaches zero but never crosses it. This “limited liability” property makes GBM the standard model for stock prices in the Black-Scholes-Merton framework.

Try another scenario: Increase \(\mu\) to 0.4 while keeping \(\sigma\) moderate (0.25) to see GBM’s compounding accelerate relative to the additive GWP drift. In option-pricing terms, that same higher \(\sigma\) implies a fatter lognormal right tail, so call premia expand even when the drift is unchanged—an intuition that’s easier to grasp once you see the multiplicative shocks on the plot.

A Note on Itô’s Lemma

It is important to understand where the solution to the GBM process comes from. The SDE for GBM is \(dS_t = \mu S_t dt + \sigma S_t dz\). We cannot integrate this as a standard deterministic integral because of the stochastic term \(dz\).

However, we can use Itô’s Lemma, a cornerstone of stochastic calculus, to find the process followed by a function of \(S_t\), such as \(f(S_t) = \ln(S_t)\). Itô’s lemma states that for a function \(f(t, S_t)\):

\[ df = \left( \frac{\partial f}{\partial t} + \mu S_t \frac{\partial f}{\partial S_t} + \frac{1}{2} \sigma^2 S_t^2 \frac{\partial^2 f}{\partial S_t^2} \right) dt + \sigma S_t \frac{\partial f}{\partial S_t} dz \]

For our case, \(f(S_t) = \ln(S_t)\), the partial derivatives are: \(\frac{\partial f}{\partial S_t} = \frac{1}{S_t}\) and \(\frac{\partial^2 f}{\partial S_t^2} = -\frac{1}{S_t^2}\).

Plugging these in gives the process for \(\ln(S_t)\):

\[ d(\ln S_t) = \left( \mu S_t \left(\frac{1}{S_t}\right) + \frac{1}{2} \sigma^2 S_t^2 \left(-\frac{1}{S_t^2}\right) \right) dt + \sigma S_t \left(\frac{1}{S_t}\right) dz \]

\[ d(\ln S_t) = \left(\mu - \frac{1}{2}\sigma^2\right)dt + \sigma dz \]

This is a Generalized Wiener Process. Integrating both sides from \(0\) to \(t\) gives:

\[ \ln(S_t) - \ln(S_0) = \left(\mu - \frac{1}{2}\sigma^2\right)t + \sigma z_t \]

Exponentiating both sides gives us the final solution for \(S_t\):

\[ S_t = S_0 \exp\left( \left(\mu - \frac{1}{2}\sigma^2\right)t + \sigma z_t \right) \]

This transformation is fundamental in derivatives pricing, as it allows us to move from a process with a stochastic component in the drift (multiplicative noise) to one with a constant drift. In the next step toward valuation, we replace the real-world drift \(\mu\) with the risk-free rate \(r\) under the risk-neutral measure, a switch justified precisely because Itô’s Lemma gives us a tractable log-price process.