Trading and quant
Market-making games: quote, update, stay consistent
The vocabulary, the five-step method and worked dice and coin markets, plus what interviewers mark you on when they trade against you.
指南为英文，与面试语言一致。
What happens
The interviewer asks you to "make a market" on something uncertain: the sum of dice, the number of heads, or a quantity you can only estimate. You give a price to buy and a price to sell. They may trade with you or reveal information, and you update.
- Bid: the price you'll buy at. Offer (or ask): the price you'll sell at.
- "6.5 at 7.5" means a bid of 6.5 and an offer of 7.5. The width is 1.
- Your position is what you hold after trades: long if you bought, short if you sold.
The method
- 1
Fair value
Work out the expected value. That's the middle of your market.
- 2
Width
Wide enough not to lose on every trade, tight enough to be useful. Widen when you're unsure or the other side may know more.
- 3
Quote
Say it cleanly: "6.5 at 7.5".
- 4
Update on information
When something is revealed, recompute the fair value and move both prices.
- 5
Update on trades
Ask what a trade tells you. A buyer at your offer may know the value is higher. Your position matters too: if you're short, raising your quotes attracts sellers and reduces your risk.
Worked examples
例题 · Sum of two dice
经代码验证Make me a market on the sum of two dice. Then: the first die shows 5.
查看解题过程
- 1.Each die averages 3.5, so the sum averages 7. Quote 6.5 at 7.5.
- 2.With the first die at 5, the fair value is 5 + 3.5 = 8.5. Move to 8 at 9.
答案6.5 at 7.5, then 8 at 9.
例题 · Higher of two dice
经代码验证Make me a market on the higher of two dice. Then: the first die shows 4.
查看解题过程
- 1.P(max = k) = (2k − 1)/36, so the fair value is (1 + 6 + 15 + 28 + 45 + 66)/36 = 161/36 ≈ 4.47. Quote 4 at 5.
- 2.With a 4 showing, the other die only matters when it beats 4: (4 + 4 + 4 + 4 + 5 + 6)/6 = 4.5. Quote 4 at 5, or tighten to 4.25 at 4.75.
答案About 4.47, then 4.5.
例题 · Heads in ten flips
经代码验证Make me a market on the number of heads in ten coin flips. Then: four of the first five were heads.
查看解题过程
- 1.Each flip adds 0.5 on average: 10 × 0.5 = 5. Quote 4.5 at 5.5.
- 2.Four heads are banked and five flips remain: 4 + 2.5 = 6.5. Move to 6 at 7.
答案4.5 at 5.5, then 6 at 7.
What interviewers mark
- Quick, sensible fair values.
- Consistency: your new market follows from your old one and the news.
- Thinking about why someone trades with you.
- Knowing your position and P&L when asked.
- Staying calm when you're picked off.
Common mistakes
常见错误
- Huge widths to avoid losing: "1 at 13" on two dice is useless and is marked down.
- Not moving after being traded with several times in a row.
- Forgetting your position when asked what you've made or lost.
- Moving only one side of the market after news.
Trading and quant