Making a market and trading math

"Make me a market on the number of floors in the tallest building in this city." Trading interviews ask this kind of question because it is the job in miniature. A trader is asked for prices all day on things whose true value nobody knows for sure, and has to give a buy price and a sell price in seconds, then adjust when someone trades. Nobody cares whether you know the building. They care how you price uncertainty and how you react to information.

The second half of the lesson is the arithmetic a desk does without thinking: basis points, compounding, present value and a bond's yield. Traders do it all day without a calculator, and an interviewer will hand you a problem mid-conversation to see whether you can too.

Making a market

The parts of a good quote

A two-way price has four parts.

• A fair estimate. Reason to it briefly and say it aloud. It is the center of your market.
• A bid, where you will buy, and an offer, where you will sell. Always quote both. A trader who refuses to make a price has no job.
• Width that matches your uncertainty. If you could easily be off by 10, quote roughly 10 either side. If you can calculate the answer, be tight. The sum of two dice has an expected value of 7, so 6.8 bid, 7.2 offered is a fine market. An obscure fact needs a wide one.
• Size. Say how many units the price is good for. A bigger size deserves a wider price, because more is at stake if you are wrong.

The width also protects you from the person you are trading with. Someone asking you for a price may know more than you do. If they do, they will buy when you are too low and sell when you are too high, and you lose on every trade. Traders call this adverse selection. A wider market gives you a cushion against it.

Skewing after a fill

Say someone buys from you at your offer. Two things just happened.

1. You learned something. They chose to buy, so they may think the answer is higher than you do.
2. You now hold a position. You sold, so you are short. If the true value is higher, you lose.

So you skew your market: move both the bid and the offer up. A higher bid makes you more likely to buy back what you sold. A higher offer makes you less likely to sell more. If you think the trade taught you something, you can also tighten a little, because you are less unsure than before.

Here is a full sequence on the building question.

1. Your estimate is 60 floors, but it is rough. You quote 50 bid, 70 offered, for one unit. Center 60, width 20.
2. The interviewer buys at 70. You are short one, and you have learned they may think it is above 70.
3. New market: 62 bid, 76 offered. Center 69, width 14.
4. They buy again, at 76. You are short two.
5. New market: 72 bid, 82 offered. Center 77, width 10.
6. Now they sell to you at 72. You buy one back and are short one.

Say the true answer is 70. You sold at 70 and 76 and bought at 72, so you took in 70 + 76 minus 72 = 74. Your remaining short settles at 70, a cost of 70. Your profit is 74 minus 70 = 4. Your opening estimate was 10 too low, and you still made money, because the skew made you charge more each time the buyer came back.

Notice that the price walked toward where the buyer seemed to think the answer was without leaping all the way. You moved enough to protect yourself, not so far that you gave up your own view.

What if they keep buying?

"I've bought from you three times. What now?" Keep moving up, and ask yourself whether you were wrong about fair value or whether you are being tested. It is fine to say "you clearly think it's higher, so I'm lifting my estimate." What you should not do is keep quoting the same price and keep selling. That is the trader who ends up short a lot of something at the wrong price.

The same thing on a real desk

An FX dealer does exactly this all day. Say a dealer quotes a liquid currency pair at an illustrative 1.1050 bid, 1.1052 offered, in busy hours. The last decimal place, 0.0001, is called a pip, so the market is two pips wide. On 10 million euros, one pip is worth 10,000,000 × 0.0001 = $1,000, so a dealer who buys at the bid and sells at the offer earns about $2,000 on that size.

The width moves with uncertainty. It widens in quiet hours, for bigger sizes, and in the minutes around a major data release, when nobody knows where the price will land. When a client buys a large amount of euros, the dealer is short euros and nudges both prices up, say to 1.1051 and 1.1053. Now the next client who wants to sell euros finds that dealer's bid the best in the market, and the dealer's short position gets smaller. How the FX market works has more on quotes.

The same logic is why inventory is a risk on any desk. A position the flow left behind loses money if the market moves against it, and the longer it sits, the more chance of that. Skewing the price to attract trades that flatten the book is the trader's first tool; hedging is the second.