Optimal stopping asks when to commit under uncertainty — and since trading is a stream of commit-or-wait decisions, quant loops return to it constantly. Three setups cover the interview canon.
1. The secretary problem (the 37% rule)
See $n$ candidates in random order, accept or reject each irrevocably, maximize the chance of picking the best. Optimal: observe the first $n/e \approx 37\%$ without accepting, then take the first candidate better than everything seen. Success probability: $1/e \approx 37\%$, independent of $n$. Fully worked here. Interview follow-ups: what if you want to maximize the VALUE rather than P(best)? What if you can recall past candidates? Each variant changes the rule — knowing THAT is the point.
2. Finite-horizon backward induction
Dice games with rerolls, "three offers, take one" — solve from the end: the value of the last stage sets the threshold for the stage before. The three-roll dice game is the canonical drill (thresholds 5, then 4.25). If you can narrate "value of continuing vs value of stopping" cleanly, every finite problem in this family falls.
3. Threshold strategies on continuous draws
Uniform draws with a stop decision — "observe up to $n$ uniforms, keep the last one you accept" — where the optimal rule is a declining sequence of thresholds. Worked here. The structural lesson interviewers probe: optimal rules in this family are ALWAYS thresholds ("accept if above x"), never randomized — because the value of continuing is a fixed number you compare against.
Why firms care
Market-making games embed stopping constantly: when to fade a quote, when to cut a position, when to stop trading a signal that might have decayed. The trading games exercise the same muscle live. And in options terms, every stopping problem is an American-option exercise decision — a bridge senior interviewers like you to notice out loud.
Practice with solutions
- The secretary problem
- Three-roll dice game
- Optimal stopping for the max of uniforms
- The expectation bank — 75 free problems in the parent family.
More topic guides
- Bayes' Theorem in Quant Interviews
- Coin Flip Questions in Quant Interviews
- Dice Questions in Quant Interviews
- Gambler's Ruin in Quant Interviews
- The Kelly Criterion in Quant Interviews
- Markov Chains in Quant Interviews
- Martingales in Quant Interviews
- The Monty Hall Problem — and the Variants Interviews Actually Ask
- Random Walks in Quant Interviews
- All guides & explainers
Frequently asked questions
What is the 37% rule?
In the secretary problem, reject the first n/e ≈ 37% of candidates, then accept the first one better than all seen so far. It picks the single best candidate with probability 1/e ≈ 37%, regardless of n.
How do you solve finite optimal stopping problems?
Backward induction: compute the value of the final stage, then at each earlier stage stop if the current offer beats the expected value of continuing. In the classic three-roll dice game the thresholds are: keep ≥5 on roll one, keep ≥5 on roll two (value 4.25 to continue), keep anything on roll three.
Why are optimal strategies thresholds?
Because the value of continuing is a fixed number: accept exactly when the current observation exceeds it. This 'threshold structure' answer is itself a common interview question.
How does optimal stopping relate to trading?
Position exits, quote fades, and signal-decay decisions are stopping problems, and every American option is an optimal-exercise problem — the interview versions are the same math with dice and uniforms.
Practice the real thing
QuantVault has 2,800+ quant interview problems with full solutions, intuition, and hints, firm-by-firm interview funnels, and an auto-graded coding judge. Start free.