Derivatives Portfolio & Risk Fixed Income Corporate Finance Probability & Measure Stochastic Models Market Microstructure Guide

Stochastic Models

Random Walk lab Symmetric Random Walk Rescaled Random Walk Standard deviation Central Limit Theorem

Brownian Motion

Brownian lab Defining the Wiener process Increment explorer Joint distributions and covariance Brownian transformations and invariance Quadratic variation Brownian bridge Running maximum and minimum Reflection principle First passage times Hitting probabilities

Geometric Brownian Motion

Lab Definition: ABM vs GBM Euler–Maruyama Mean reversion Jumps Correlated motions

Lab Itô integral Itô's lemma Martingales Girsanov

Lab Variance reduction Feynman–Kac

Connected Guides

Random walks and Brownian motion
Itô's lemma
Filtrations, adapted processes and stopping times
Conditional expectation and martingales
Properties of Brownian motion
The reflection principle and first passage times
The Brownian bridge
The Itô integral

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Brownian bridge

What does Brownian motion look like if you already know where it ends?

Pin the endpoint and see the mean line b·t/T and the variance t(T − t)/T appear.

Endpoint

Endpoint b = B(T)1.00

-2.52.5

Random seed: 3

Drag the handle at the right edge of the chart, or use the slider.

Mean at T/2

0.500

Variance at T/2

0.250

300 Brownian bridges

Dashed: the mean line b·t/T. Solid: mean ± one standard deviation √(t(T − t)/T), pinched to zero at both ends.

Variance of the bridge against time

t(T − t)/TEmpirical variance