Discrete-time random walks as the foundation for continuous models
A simple symmetric random walk is , where each or with equal probability. Despite its simplicity, it captures the essence of randomness in finance and is the building block for Brownian motion.
For a walk starting at :
The expected position never changes (it's a martingale), but the spread grows as .
Three symmetric random walks, 100 steps, with the one standard deviation envelope
Each of the three jagged paths (blue, purple, green) is a fair-coin walk , one step of or at a time. The dashed line is the expected position : every walk leaves it and crosses back through it. The amber curves are , one standard deviation, since ; they widen fast at first and then flatten, reaching at . Here two walks end inside the envelope (at and ) and one ends outside (at ).
By Pólya's theorem, a symmetric random walk in 1D and 2D returns to the origin with probability 1 (recurrent), but in 3D and higher escapes to infinity (transient).
A gambler starts with $k and bets $1 per round on a fair game, stopping at $0 (ruin) or $n (target):
For a biased game (): where .
Interview tip: Gambler's ruin is one of the most asked probability questions at trading firms. Know both formulas cold.
Check your understanding
A gambler sits down with $20 and bets $1 per round on a fair coin. She stops when she is broke or when she has won $30 net. What is the probability she reaches her target rather than going broke?
Starting at 0, what is the probability a symmetric walk reaches before ?
This is gambler's ruin with (distance from lower barrier) and (total gap):
Rescale steps to size at intervals ; as , the walk converges to Brownian motion (Donsker's theorem). Every BM property has a discrete random walk analog.