FreeMarket makingEasy

Make Me a Market on a Coin Game

Asked atOptiverIMCSIG

I flip a fair coin repeatedly until it comes up heads. You are paid the number of flips it took, in dollars.

Make me a two-sided market on the value of this game. I may buy or sell at your quote, and I get to choose which.

Solution

Step 1 — value the game. The number of flips $N$ until the first head is geometric with success probability $1/2$ , so

E[N] = \sum_{k=1}^{\infty} k \left(\tfrac{1}{2}\right)^{k} = 2.

Quick derivation to say out loud: condition on the first flip. With probability $1/2$ you are done in one flip; otherwise you have used one flip and are back where you started. So $E = \tfrac{1}{2}(1) + \tfrac{1}{2}(1 + E)$ , giving $E = 2$ .

Step 2 — quote around it. Fair value is $2.00, so quote symmetrically: $1.90 at $2.10. You buy at 1.90, you sell at 2.10.

Step 3 — justify the width, which is the actual question. The payoff is unbounded and right-skewed: $\text{Var}(N) = (1-p)/p^2 = 2$ , and the tail is fat enough that a single flip sequence can cost you far more than the edge on a fair-value trade. A 20-cent width on a $2 game is roughly 10% — wide enough to survive adverse selection from a counterparty who chooses the side, tight enough to be a real quote.

The trap. Candidates quote something like $1.99 at $2.01 to look confident. That is the wrong instinct: your counterparty picks the direction, so a market that tight is free optionality for them. Equally, quoting $1 at $3 gets you told the market is not competitive.

If asked to tighten, tighten — but ask for a size limit first. Width and size are the same conversation.

Question

Solution

Solution