Quant Academy Math · Probability · Quant Finance
Phase 18 - Lesson 18.2

Capstone 2 - Probability and Brownian Motion

Simulate, estimate, and validate the building blocks of continuous-time finance: distributions, the CLT, and Brownian motion’s defining properties.

⏱ 120 min● Advanced🔗 Prereqs: Phases 7, 8, 13
↖ Phase 18 hub
Builds on: Integrates Probability (Phase 7), Stochastic Processes / Brownian Motion (Phase 8), and Monte Carlo (Phase 13).
Leads to: Brownian motion built and validated here is the driver of every SDE and pricing model in Capstones 3–4.

Learning Objectives

Click a status chip to cycle: Not started → In progress → Studied → Practiced → Needs review → Mastered.

Key Vocabulary

Brownian motion
A process \(W_t\) with \(W_0=0\), independent Gaussian increments \(W_t-W_s\sim\Normal(0,t-s)\), and continuous paths.
Central limit theorem
Normalized sums of i.i.d. finite-variance variables converge in distribution to a normal.
Quadratic variation
The limit \(\sum(W_{t_{i+1}}-W_{t_i})^2\to t\); nonzero, unlike for smooth functions.
Standard error
The standard deviation of an estimator; for a mean of \(n\) draws it is \(\sigma/\sqrt n\).
Independent increments
Non-overlapping increments of \(W\) are independent - the Markov backbone of the process.
Scaling law
Brownian increments scale like \(\sqrt{\Delta t}\), not \(\Delta t\); the root of the \(\sqrt{dt}\) in simulation.

Intuition & Motivation

Intuition
Continuous-time finance is built on one object: Brownian motion. Before you trust a pricing model that uses it, you should be able to simulate it, measure its properties, and confirm they match the definition - increments that are Gaussian with variance equal to elapsed time, independent across non-overlapping intervals, and paths whose quadratic variation is \(t\) rather than zero.

This capstone also cements the two statistical facts that make simulation trustworthy: the central limit theorem (why averages of many draws are approximately normal) and the standard error \(\sigma/\sqrt n\) (how precise your estimate is). Together they let you attach an honest confidence interval to every simulated quantity.

The brief

Build a Brownian-motion simulator and a small estimation harness, then validate four properties against theory, each with a quantified error. Deliverable: code plus a validation report stating, for each property, the predicted value, the estimate, and a confidence interval.

Required theory recap

Simulating Brownian motion correctly

On a grid \(0=t_0\lt \dots\lt t_n=T\) with step \(\Delta t\), generate increments and cumulate:

\[W_{t_{k+1}} = W_{t_k} + \sqrt{\Delta t}\,Z_k,\qquad Z_k\stackrel{iid}{\sim}\Normal(0,1).\] (18.3)
Key Idea
The \(\sqrt{\Delta t}\) (not \(\Delta t\)) is the whole game: variance of an increment equals its time span. Get this scaling wrong and every downstream volatility is off by orders of magnitude.

Quadratic variation: the signature of Brownian motion

For a smooth function, \(\sum(\Delta f)^2\to0\) as the grid refines. For Brownian motion it converges to the elapsed time:

\[\sum_{k=0}^{n-1}\big(W_{t_{k+1}}-W_{t_k}\big)^2 \xrightarrow{n\to\infty} T \quad(\text{in probability}).\] (18.4)
Worked Example - Validating quadratic variation on one path
1
Simulate \(W\) on \([0,1]\) with \(n=10{,}000\) steps, \(\Delta t=10^{-4}\).
2
Compute \(QV=\sum(\Delta W_k)^2\). Theory (18.4) says \(QV\to1\).
3
Each \((\Delta W_k)^2\) has mean \(\Delta t\) and variance \(2\Delta t^2\), so \(QV\) has mean \(n\Delta t=1\) and SD \(\sqrt{2n}\,\Delta t=\sqrt{2/n}\approx0.014\).
4
So a correct simulator returns \(QV\approx1\pm0.014\); if you instead see \(\approx0\), you used \(\Delta t\) increments (smooth path), and if you see a huge number your scaling is wrong.

CLT and honest error bars

To estimate \(\theta=\E[g(W_T)]\) by Monte Carlo, average \(N\) independent path values. The CLT gives an approximate 95% confidence interval:

\[\hat\theta \pm 1.96\,\frac{\hat s}{\sqrt N},\qquad \hat s=\text{sample SD of the } g \text{ values}.\] (18.5)

Reporting \(\hat\theta\) without this interval is the single most common sin in simulation: it hides whether ‘0.502’ is distinguishable from ‘0.5’.

▶ Reference architecture for the BM harness
bm/simulate.py -> paths(T, n, N, seed) -> (times, W[N,n+1]) bm/estimate.py -> mc_mean(vals) -> (theta_hat, se, ci95) bm/validate.py -> checks: E[W_T]~0, Var(W_T)~T, increments independent, QV ~ T; prints predicted vs estimate +/- CI

The simulator takes a seed and returns a clean array (17.3 reproducibility); every validation reports an interval, not a bare point.

Common Mistakes to Avoid
  • Using \(\Delta t\) instead of \(\sqrt{\Delta t}\) for the increment SD - the classic scaling error.
  • Reporting a Monte Carlo estimate with no standard error, so you can’t tell signal from noise.
  • Concluding the estimate is ‘wrong’ when it is within a couple of standard errors of the truth (it isn’t).
  • Expecting quadratic variation to be zero (that’s smooth functions) or exactly \(T\) on one finite path.
  • Treating overlapping increments as independent when checking the independence property.
  • Assuming Brownian paths are differentiable and trying to define an ordinary derivative \(dW/dt\).
Quant Practitioner Tips
  • Always simulate increments as \(\sqrt{\Delta t}\,Z\) and cumulative-sum; validate \(\Var(W_T)\approx T\).
  • Attach a \(\pm1.96\,\hat s/\sqrt N\) interval to every estimate; grow \(N\) until it’s tight enough.
  • Use quadratic variation as a built-in unit test that your path generator is really Brownian.
  • Check independence via the sample correlation of non-overlapping increments (should be ~0).
  • Fix the seed so the validation report is reproducible and re-checkable (17.3).

Interactive: simulate Brownian motion and validate its properties

Implement the simulator and the two validators (variance and quadratic variation). The tests encode the theoretical values with tolerances derived from the standard errors above.

Knowledge Check

Q1 Easy
In simulating \(W\) on a grid with step \(\Delta t\), each increment should be drawn with standard deviation:
\(\Delta t\)
\(\sqrt{\Delta t}\)
\(1\)
\((\Delta t)^2\)
Q2 Medium
The quadratic variation of Brownian motion on \([0,T]\) converges to:
\(0\)
\(T\)
\(\sqrt T\)
\(T^2\)
Q3 Medium
You estimate \(\E[g(W_T)]=0.500\) from \(N=10^4\) paths with sample SD 0.8. A correct report is:
Exactly 0.500
0.500 with no uncertainty
0.500 ± \(1.96\cdot0.8/100=0.0157\)
0.500 ± 0.8

Practical Exercise

You must convince a skeptical reviewer that your path generator really produces standard Brownian motion. List four independent checks you would run, the theoretical value each targets, and the approximate standard error of each check for \(N\) paths on \([0,1]\) with \(n\) steps.

▶ Show full solution

Four checks a correct BM must pass:

  1. Mean: \(\E[W_1]=0\); estimate \(\bar W_1\) over \(N\) paths, SE \(=1/\sqrt N\). Reject if \(|\bar W_1|\gt 1.96/\sqrt N\).
  2. Variance: \(\Var(W_1)=1\); sample variance of \(W_1\), SE \(\approx\sqrt{2/N}\) (variance of a normal’s sample variance).
  3. Independent increments: correlation between \(W_{0.5}-W_0\) and \(W_1-W_{0.5}\) should be ~0; SE of a correlation \(\approx1/\sqrt N\).
  4. Quadratic variation: per path \(\sum(\Delta W)^2\to1\) with SD \(\sqrt{2/n}\); average over paths to tighten.

Each check reports predicted value, estimate, and a \(\pm1.96\,\text{SE}\) band. Passing all four - not just one - is what earns trust; a generator can match the mean while botching the increment scaling.

After the reveal, answer for yourself: Which single check most directly catches a wrong \(\sqrt{\Delta t}\) scaling, and why?

Lesson Summary

Brownian motion is the engine of continuous-time finance, so this capstone builds it and validates its definition: Gaussian increments with the crucial \(\sqrt{\Delta t}\) scaling, \(\Var(W_T)=T\), independent increments, and quadratic variation \(\to T\) (not 0). The central limit theorem and standard error \(\sigma/\sqrt N\) turn every simulated estimate into a value plus an honest confidence interval - the discipline all later pricing capstones inherit.

Retrieval Practice

Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.

▶ Show retrieval prompts & answers
Q: Write the Brownian increment update used in simulation and the scaling that matters.
A: \(W_{t_{k+1}}=W_{t_k}+\sqrt{\Delta t}\,Z_k,\ Z_k\sim\Normal(0,1)\). The \(\sqrt{\Delta t}\) (variance = elapsed time) is essential; using \(\Delta t\) is the classic bug.
Q: What does the quadratic variation of Brownian motion converge to, and why does it matter?
A: It converges to \(T\) (in probability), not 0. This nonzero quadratic variation is exactly what forces the second-order term in Itô’s lemma and distinguishes BM from smooth paths.

Completion Checklist

Confidence / mastery rating
Personal notes

Source References

This lesson synthesizes and paraphrases concepts from the sources below. No copyrighted text, problem sets, or solutions are reproduced. Return to the originals for full depth.

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