Puzzle · this month
The honest coin.
Every month we post one problem from the kind of reasoning we do every day. No prizes, but correct answers with clear reasoning get a reply from the research desk.
A contract pays $1 if a fair coin, flipped 100 times, shows at least 60 heads. Flips happen one at a time and are public.
- Before any flips, what is the fair price of the contract? (Two significant figures is fine.)
- After 50 flips you have observed 34 heads. What is the fair price now, to the nearest cent?
- From that state, by how much should the fair price move if the next flip is tails? If it is heads?
- You are the only market maker. Where would you set your bid and ask after 50 flips, and why would your answer change if the flips arrived one per second rather than one per day?
How we think about problems like this
Parts 1–3 are exact calculations: binomial tails and how they change with one more observation. Part 4 is the job. A market maker has to set a spread wide enough to survive the next piece of information but narrow enough that someone actually trades. The arrival rate of information changes everything about that trade-off.
If this is the kind of problem you like, our note on pricing event contracts is a good next read, and our open roles are the one after that.
Last month: the overnight storm
Last month's puzzle asked for the fair value of a contract on whether a tropical storm would make landfall, given two ensemble forecasts that disagreed. The best answers combined the ensembles with weights proportional to their historical skill, then widened the spread to reflect the probability of a model update before morning. The cleanest solution came from a meteorology PhD student who is now interviewing with us.