Markov Chains in Quant Interviews

Markov chains are the workhorse behind a huge family of quant interview questions — dice games, random walks, waiting times, board games, credit-rating migrations. The good news: interviews use a small, repeatable toolkit. If you can set up a transition matrix, find a stationary distribution, and solve an absorption problem, you can handle nearly everything asked.

The four ideas that cover the ground

• The Markov property. The future depends only on the current state. The interview skill is choosing a state space that MAKES the problem Markov — e.g. for "expected flips until HTH", the state is your current progress into the pattern, not the whole history.
• First-step analysis. Condition on the first transition and write recursions for hitting times/probabilities: $h_i = 1 + \sum_j P_{ij} h_j$. This one template solves most "expected number of steps until X" questions.
• Stationary distributions. Solve $\pi P = \pi$ with $\sum_i \pi_i = 1$. For a quick sanity check on symmetric chains, stationary mass is proportional to degree.
• Absorbing chains. Gambler's-ruin-shaped questions: probability of hitting A before B, expected time to absorption.

Worked example

A mouse moves between rooms A↔B↔C in a line, each minute moving to a uniformly random adjacent room. Long-run fraction of time in B? The chain's stationary distribution weights each room by its number of doors: A and C have 1, B has 2, total 4 — so $\pi_B = 2/4 = 1/2$. Interviewers love this degree shortcut because the honest linear-algebra route takes five times longer; knowing both (and when the shortcut is valid — random walk on a graph) reads as real fluency.

Where it shows up by firm type

Trading firms wrap Markov chains in dice/coin games and racing puzzles (state = progress). Quant research loops ask for stationary distributions, mixing intuition, and occasionally reversibility. ML-leaning firms connect chains to PageRank and MCMC. The stochastic-process question bank maps the whole family.

Practice with solutions

• Stationary distribution of a conveyor-belt chain — the full πP = π workflow.
• Absorption probability on a finite random walk — first-step analysis, worked.
• Gambler's ruin on a fair coin — THE canonical absorbing chain.
• HHT before THH — pattern-race chains, a trading-interview staple.