You must show a two sided market on the sum of two fair dice to a counterparty who has already seen one die and trades only when the trade has positive expected value for him. What is the tightest market you can show that never loses money against him?

You must show a two sided market on the sum of two fair dice to a counterparty who has already seen one die and trades only when the trade has positive expected value for him. What is the tightest market you can show that never loses money against him?

Approach: Work out his fair value given the die he has seen, then ask which of your prices he trades on and what the sum is worth conditional on exactly those trades.

4.5 at 9.5. Seeing a die d, his fair value for the sum is d + 3.5, which takes the values 4.5 through 9.5. He lifts an offer at a only when d + 3.5 > a, and then the conditional expectation E[sum | he buys] is strictly above a for any a below 9.5. Quoting 6 at 8, he buys only when d is 5 or 6, so the sum is worth 9 in expectation and you lose 1 on every trade. Only an offer of 9.5 and a bid of 4.5 leave him with no edge, so that is the tightest safe market. The whole spread here is adverse selection cost, and any tighter quote on a real desk is priced off the share of uninformed flow.

Follow-up: If he is informed with probability 0.5 and otherwise trades at random, what is the tightest market with zero expected profit?

Key concepts: adverse selection, conditional expectation, bid ask spread, informed flow.