Brownian motion & Itô calculus
Why continuous paths accumulate quadratic variation and change the chain rule.
01State the defining increment properties of Brownian motion.
02Explain why quadratic variation converges to elapsed time.
03Construct the Itô integral for adapted elementary processes.
04Apply Itô's lemma and identify the second-order correction.
Observe the object before formalizing it.
Brownian motion is continuous but not differentiable. Its increments are order √dt, so squared increments are order dt and survive when ordinary second-order terms would disappear.
Independent Gaussian increments encode a time-homogeneous noise clock.
Quadratic variation—not visual roughness—is the mathematical signature that changes calculus.
Adapted integrands prevent future information entering a stochastic integral.
Connect the mathematical object to a pricing question.
Diffusion models, hedging arguments, simulation schemes and measure changes all depend on Brownian scaling and Itô's correction.
European and barrier options
stochastic-volatility models
short-rate models
dynamic hedges
Time is measured in years; Brownian increments have √year units, and diffusion coefficients restore the financial unit of the state variable.
Construct the definition and its invariants.
Brownian increments
Increment variance equals elapsed time and disjoint increments are independent.
Quadratic variation
The squared path increments converge to time even though the ordinary variation diverges.
Itô's lemma
The half-variance curvature term is the surviving second-order contribution.
Full derivation
Why stochastic calculus keeps a second-order term
Expand over a short interval and classify terms by Brownian order: ΔW is √Δt, so (ΔW)² is Δt.
- 01
Write a second-order expansion
For ΔX=aΔt+bΔW, retain the Taylor terms that can contribute at order Δt.
- 02
Apply Brownian scaling
Terms in (Δt)² and ΔtΔW vanish faster than Δt; the squared Brownian increment does not.
- 03
Use quadratic variation
Across refining partitions, the accumulated squared Brownian increments converge to elapsed time.
- 04
Collect drift and diffusion
Substitute the differential identities and separate finite-variation from martingale terms.
Itô's lemma is the ordinary chain rule plus the deterministic contribution created by quadratic variation.
Inputs
Wₜ: standard Brownian motionΠₙ: partition of [0,T][W]ₜ: quadratic variationXₜ: Itô process with drift a and diffusion b
Assumptions and limits
- Brownian paths exclude jumps and microstructure discontinuities.
- Continuous-time identities become discretization choices in code.
- An adapted integrand and square integrability are not optional technicalities.
Fit, compute, then challenge the assumptions.
Use analytical scaling to classify terms, then simulate only to illustrate path behavior and convergence—not to define the theorem.
Standard Brownian motion has no calibration parameter; model diffusion coefficients and correlations are the calibrated objects.
06PYTHON IMPLEMENTATIONOpen the implementation and checks.
- Seeded normal-increment generator
- Path and quadratic-variation calculations kept separate
- Convergence diagnostic over refining partitions
Brownian path and quadratic variation
Show that path increments scale with √dt while their squared sum approaches T.
from __future__ import annotations import numpy as np def brownian_path(steps: int, horizon: float, seed: int) -> tuple[np.ndarray, float]: if steps <= 0 or horizon <= 0: raise ValueError("positive steps and horizon required") rng = np.random.default_rng(seed) increments = np.sqrt(horizon/steps)*rng.standard_normal(steps) path = np.concatenate(([0.0], np.cumsum(increments))) return path, float(np.dot(increments, increments)) path, qv = brownian_path(100_000, 1.0, 7)assert path.shape == (100_001,)assert abs(qv-1.0) < 0.02print(f"terminal={path[-1]:.6f} quadratic_variation={qv:.6f}")Run the thought experiment.
Brownian motion & Itô calculus
Use a fixed seed to separate discretization, sampling and truncation effects.
Exact, Euler and Milstein spot paths using the same shocks.
- exact
- Euler
- Milstein
Use Left/Right or Up/Down arrows to inspect values; Home and End jump to the bounds.
Model → algorithm → evidence
Each numerical approximation has an independent control and a reference diagnostic.
Q / GBMMeasure and SDE fixed
32 stepsDiscretization or transform control
SE=0.211Visible convergence evidence
Carry the abstraction into valuation.
“The noise is not the model; the diffusion coefficient decides what one Brownian shock means in market units.”
time grid and calendar
diffusion and correlation
seed and random stream
discretization scheme
Calibrate model diffusion parameters to market instruments; validate the simulation scheme independently against moments or analytical prices.
RISKdiscretization bias
correlation construction
barrier monitoring
- fix measure and SDE
- choose scheme and grid
- reuse shocks for comparisons
- verify moments and convergence
Production failure modes
- unseeded regression tests
- negative states from a naive scheme
- mis-scaled annual time
09MACRO CONNECTIONOpen the transmission channel.
From uncertainty clock to market dispersion
Brownian time is a modelling clock; volatility and correlation translate it into asset-specific uncertainty rather than economic causality.
transmitssets a modelling clock
transmitsscales shocks into returns
outputleaves discretization residuals
10COMMON PITFALLSOpen the failure checklist.
Treating dW/dt as an ordinary derivative.
Dropping the Itô correction.
Using future information in the integrand.
Concluding model realism from a visually plausible path.
11SOURCES / FURTHER READINGOpen sources and continue the track.
Ch. 1 §§1.2–1.3, printed pp. 9–24 and Ch. 2 §2.1.2, printed pp. 29–34 (PDF pp. 28–43, 49–55)
The formulation was checked against the cited sections; prose, examples and code are original to the Academy.
- Source
- Mathematical Modeling and Computation in Finance
- Author
- Cornelis W. Oosterlee and Lech A. Grzelak
- Ref
- First edition (2020)