Visualizers
Monte Carlo Simulator
Roll thousands of possible futures for a price using geometric Brownian motion, and watch the range of outcomes fan out.
Simulated price paths
Distribution of terminal prices
Shaded-left bars are paths that finished below the start price. The long right tail is the hallmark of a lognormal (right-skewed) distribution.
Each thin line is one possible future for the price, a random walk built from your drift and volatility. A single run tells you almost nothing; the point of running it hundreds of times is to see the range of where things could land, not to predict any one number.
Notice the outcomes lean right: because prices compound, a handful of big winners drag the mean above the median, so the "average" is rosier than the typical path. And even with a positive expected return, the 5th-percentile and the probability of ending below where you started show there is still a very real chance of loss. This is a teaching model (geometric Brownian motion), not investment advice.
Learn how it works
Five worked examples. Read a couple before you dive in, try to answer first, then reveal the solution.
The fanning cone
Set start price = 100, drift μ = 8%, volatility σ = 20%, horizon = 1 year, and a few hundred paths. Hit run and watch the bundle of lines.
Show solution
Every line is one possible future for the price, drawn one random step at a time. They all start pinned at 100, then slowly spread apart into a cone shape.
- Near day 1 the paths are almost on top of each other, little time has passed, so little can happen.
- By the end of the year they've fanned out wide, because uncertainty piles up the longer you wait.
The cone is the whole point: the future isn't a single number, it's a range of outcomes, and that range grows with time.
Median below the mean
Keep the same settings and look at the two markers on the ending prices: the mean (average) and the median (the middle outcome). Notice the median sits lower than the mean.
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This looks like a mistake but it's real. Prices can't fall below 0, but they can rise without limit, so the outcomes are right-skewed, a long tail of big winners on the right.
- A handful of huge winners drag the average upward.
- The typical path (the median, the middle of the pack) doesn't get that boost, so it lands lower.
This lopsided shape is called lognormal, and it's why the average return can look better than what most people actually experience.
Crank up the volatility
Leave drift at 8% but raise volatility σ from 20% to 40%. Run it again and compare the width of the cone.
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The cone gets dramatically wider. Volatility is the size of the random jostling each step, so doubling it means bigger swings in both directions.
- Far more paths end up very high and very low.
- The chance of an extreme outcome, good or bad, goes way up.
More volatility does not mean more profit, it means more uncertainty. The center of the cone barely moves; only the spread explodes.
Positive return, real chance of loss
With μ = 8% (a positive average return), find the probability of loss stat, the share of paths that end below the 100 you started with.
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Even with the odds tilted in your favor, that number is surprisingly large, often around a third of paths finish underwater.
- A positive average is not a promise. The spread from volatility means plenty of futures still end down.
- The wider the cone (higher σ), the bigger this probability of loss climbs.
The lesson: 'expected to go up' and 'will go up' are not the same thing. Risk lives in the paths that don't cooperate.
More steps, smoother lines
Keep everything fixed and increase the number of steps (say from 50 to 500) while keeping the 1-year horizon.
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Each path turns from a jagged staircase into a smooth, detailed squiggle, you're simply sampling the same year more finely.
- The individual lines look prettier and more realistic.
- But the overall cone width and the ending spread barely change.
Steps control the resolution of each path, not the amount of risk. Volatility and horizon set the spread; steps just decide how many little movements you draw to get there.
What you'll learn
How randomness compounds over time, why price outcomes are skewed (log-normal), and what 'expected return' hides about the range of results.