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

Itô's lemma
Random walks and Brownian motion
The Itô integral
Mean reversion: Ornstein–Uhlenbeck and Vasicek

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Martingales and betting strategies

Every strategy that bets on a fair coin from the past only leaves the mean wealth flat. The histogram shows where the losses hide.

Game

Rounds100

10200

Random seed: 7

Mean final wealth: 99.91 ± 1.49
Ahead of the start: 66.4%
Ruined: 24.7%
Lowest wealth: 0

Optional stopping: a bounded strategy cannot beat a fair game. Stopping at +1 wins most of the time and pays for it with a long left tail.

Final wealth

Mean wealth of 2000 players after each round

Mean