Symmetric Random Walk | Quant Sandbox

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Stochastic Models

Stochastic Learning StudioBeta

1.1Symmetric Random Walk 1.2Rescaled Random Walk 1.3Standard deviation 1.4Central Limit Theorem 2.1Defining the Wiener process 2.2Increment explorer 2.3Joint distributions and covariance 3.1Running maximum and minimum 3.2Reflection principle 3.3First passage times 3.4Hitting probabilities

Brownian MotionBeta

Geometric Brownian MotionBeta

Itô ProcessBeta

Monte CarloBeta

Curriculum/Chapter 1 · Discrete to Continuous/1.1

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1.1

Symmetric Random Walk

How do many random ±1 steps add up?

Goal: observe how the endpoint distribution and variance change with the number of steps.

Simulation controls

Number of steps100

20500

Simulated paths200

41000

Probability of an upward step p50%

0.10.9

Random seed7

19999

Path animation

Animation speed5.6 steps/s

60500

100 / 100 steps

Current-step statistics

Empirical mean

0.25

Theoretical mean

0.00

Empirical variance

89.00

Theoretical variance

100.00

All 200 simulated paths are drawn on the chart, and the same paths feed the statistics.

Sample paths

100 / 100 steps

200 paths are shown as a translucent cloud; instead of reading individual paths, the widening cloud shows how the distribution spreads.

Position distribution at current step

-32Position26