Free lesson · Stochastic calc

Before Itô: random walks, Brownian motion and information

Open interactive lessonPractice calculationsExplore labs

Start here · the same idea in everyday words

Unpack a technical sentence

Technical wording: Brownian increments have variance proportional to elapsed time.

Read it as: For standard Brownian motion, uncertainty grows with the length of the interval. Variance grows in proportion to time; standard deviation grows with its square root.

A slower worked example · one step at a time

1. Measure time in years. Standard Brownian motion has increments with mean zero and variance equal to the elapsed years. Its increment unit is the square root of a year.
2. Over a quarter year, the increment variance is .25 years, and its standard deviation is the square root of .25, or .5 square-root-years.
3. A standard normal draw comes from a bell-shaped distribution with mean zero and standard deviation one. Multiply that draw by .5 to simulate an increment over this interval. A draw of −1 gives −.5; a draw of +1 gives +.5. These two selected draws illustrate scaling, not the whole normal distribution.
4. Two independent quarter-year increments have total variance .25+.25=.5 years. The total standard deviation is the square root of .5, about .707107, not .5+.5=1.
5. A Brownian increment is not itself a dollar price move. A price model must specify how it scales and uses that shock.

Suggested preparation: ordinary derivatives, integrals, partial derivatives, and probability.

• Standard Brownian motion starts at zero, has independent Gaussian increments, and an increment over length h has mean zero and variance h.
• Its typical scale is therefore √h, not h.
• The paths are almost surely nowhere differentiable; ordinary velocity intuition fails.
• A filtration is the information available through time.
• An adapted process uses current and past information, and an Itô trading integrand must satisfy the relevant non-anticipation and integrability requirements.
• This is a mathematical version of not trading on future shocks.

Architecture pseudocode

from random import Random
from math import sqrt

def brownian_path(horizon,steps,seed=7):
    if horizon<0 or not isinstance(steps,int) or steps<1: raise ValueError("Valid horizon and positive steps required")
    rng=Random(seed)
    h=horizon/steps
    path=[0.0]
    for _ in range(steps): path.append(path[-1]+sqrt(h)*rng.gauss(0,1))
    return path

print(brownian_path(1,4))

# Replay the slower, synthetic example above. Units are stated in its steps.
def plain_language_example():
    from math import exp, e  # Standard library; no extra dependency.
    return {
        "sd_quarter": .25**.5,
        "negative_draw": (.25**.5)*(-1),
        "combined_variance": .25+.25,
        "combined_sd": (.25+.25)**.5,
    }

print(plain_language_example())

An analogy to remember

Many tiny, independent pushes can accumulate into a persistent random displacement even as each individual push gets smaller. Their variances add; their signed values do not all push in the same direction.

How this becomes a building block

Brownian increments are a noise component inside a diffusion model. Adding a drift and state-dependent scale produces an SDE. A random walk is not itself a complete market model; jumps, microstructure and changing volatility may require other assumptions.

from random import Random
from math import sqrt

def brownian_path(horizon,steps,seed=7):
    if horizon<0 or not isinstance(steps,int) or steps<1: raise ValueError("Valid horizon and positive steps required")
    rng=Random(seed)
    h=horizon/steps
    path=[0.0]
    for _ in range(steps): path.append(path[-1]+sqrt(h)*rng.gauss(0,1))
    return path

print(brownian_path(1,4))

# Replay the slower, synthetic example above. Units are stated in its steps.
def plain_language_example():
    from math import exp, e  # Standard library; no extra dependency.
    return {
        "sd_quarter": .25**.5,
        "negative_draw": (.25**.5)*(-1),
        "combined_variance": .25+.25,
        "combined_sd": (.25+.25)**.5,
    }

print(plain_language_example())