Case 037Market-making gamesWarm up

In a market on the sum of two dice you sell 3 to interviewer A at 7.5, buy 2 from B at 6.5, sell 4 to A at 7.2 and buy 4 from B at 7.0, at Rs 100 a point. Where is your position, and what is your P and L if it settles at 8?

OptiverAmsterdam · 2023

The situation

At Halvora Securities' trading game you make a two-way market in a contract that settles at the sum of two dice, rolled at the end. Each point is worth Rs 100. Two interviewers, A and B, trade with you in turn, and you must keep quoting between trades without writing anything down.

The trades: you sell 3 lots to A at 7.5, buy 2 from B at 6.5, sell 4 to A at 7.2 and buy 4 from B at 7.0. The dice then come up 3 and 5.

Your task

Track your position and cash after each trade, compute the P and L at a settlement of 8, split it into what came from your prices and what came from the roll, and say how your position should shape your next quote.

Quick check

After the four trades, what is your position?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

You finish short 1 lot with Rs 1,030 of cash, and at a settlement of 8 your P and L is 2.3 points, Rs 230. Your prices earned 3.3 points of edge against the fair value of 7; being short 1 when the dice came up one above fair cost 1.0. Between trades, being short means you lean your quote up so the next trade is more likely a buy.

Step 1What do you keep track of between trades?

Two numbers, and only two: your signed position and your cash. A shopkeeper at the end of the day needs to know how much stock is left on the shelf and how much money is in the till; everything else can be reconstructed. Sells subtract from position and add price times quantity to cash; buys do the opposite. Keeping both running totals in your head, out loud, is the skill the game is testing, because every quote you make next depends on the position you are carrying.

Trade by trade, Halvora's position goes to minus 3, minus 1, minus 5 and minus 1 while cash builds to 10.3 points, and settling the short lot at 8 leaves a P and L of 2.3 points, Rs 230.

Step 2How do you work out the P and L?

Run the ledger. Sell 3 at 7.5: position minus 3, cash plus 22.5. Buy 2 at 6.5: minus 1, cash 9.5. Sell 4 at 7.2: minus 5, cash 38.3. Buy 4 at 7.0: minus 1, cash 10.3. At settlement you buy back the short lot at 8. P and L is cash plus position times the settlement price: 10.3 minus 8 is 2.3 points, or Rs 230. Check it a second way by counterparty: you sold A seven lots at an average of 7.33 and bought six from B at an average of 6.83; the six matched lots earn about 3.0 points between them, and the seventh, sold to A at 7.33 and settled at 8, costs about 0.7.

The relationship

\text{P\&L} = \underbrace{\sum_i q_i\,(7 - p_i)}_{\text{edge vs fair}} + \underbrace{\text{position} \times (8 - 7)}_{\text{the roll}} = 3.3 + (-1)(1) = 2.3 \text{ points}

q_i	signed quantity of trade i, positive for a buy
p_i	price of trade i
7	fair value of the sum of two dice
8	the settlement

What it says in wordsYour result splits into the edge you charged on every trade against fair value, plus the luck of whatever position you held when the dice landed.

Step 3How much was skill and how much was the roll?

Measure every trade against the fair value of 7. Selling 3 at 7.5 earned 1.5, buying 2 at 6.5 earned 1.0, selling 4 at 7.2 earned 0.8, and buying 4 at 7.0 earned nothing. That is 3.3 points of edge from your prices, and the roll then cost 1.0 because you were short one lot when the dice landed a point above fair. Interviewers want this split, because it shows you judge your trading by the edge you charged, which you control, and not by the settlement, which you do not. Had you been flat, the P and L would have been the full 3.3 whatever the dice showed.

Trade	Position after	Cash after	Edge vs 7
Sell 3 to A at 7.5	-3	+22.5	+1.5
Buy 2 from B at 6.5	-1	+9.5	+1.0
Sell 4 to A at 7.2	-5	+38.3	+0.8
Buy 4 from B at 7.0	-1	+10.3	+0.0
Total	-1	+10.3	+3.3

In points: the four trades leave Halvora short 1 with 10.3 points of cash and 3.3 points of edge charged against a fair value of 7.

Step 4How should the position change your next quote?

Short 1 lot, you would rather buy than sell. Lean the market up a little, for example from 6.6 at 7.4 to 6.8 at 7.6, so the next counterparty is more likely to sell to you and bring you back towards flat. Watch the pattern too: A has bought from you twice and B has sold to you twice. In a game where the dice are not yet rolled neither can know more than you, so their flows are noise; in a game with hidden information, a counterparty who keeps trading the same way is telling you something, and you should move your fair value, not only your lean.

Where candidates lose it

The usual loss is forgetting the sign of the position, or ending flat in your head when you are short 1. Then the settlement adjustment is missed and the P and L comes out as 10.3 points instead of 2.3.

The second is judging the trading by the settlement: calling it a loss because the dice came up 8 while you were short. The edge you charged was 3.3 points; the roll is luck, and the interviewer is checking whether you can tell the two apart.

What the interviewer asks next

The dice come up 6 instead. What is the P and L, and did your trading get better?
A asks to buy 10 lots at your offer. Where do you quote, and why?
How would you track position if you were also trading a second contract on the product of the dice?

Asked at Optiver, Prop Trading, Amsterdam, 2023 (Wall Street Oasis): some difficult trading games where you had to profit making a market whilst remembering your position and the position of two interviewers

Company names and figures are illustrative.

1The situation

2Your task