Completeness is the contrast, not the passport

A toy that prices exactly, on purpose

The Cox-Ross-Rubinstein binomial model is a complete discrete market: a stock moves up or down each period, a money-market account pays a fixed rate, and under those two rules alone every contingent claim has a unique no-arbitrage price and a unique replicating hedge. Take a concrete tree: stock at 100, up to 110, down to 90, one period, risk-free gross return 1.05. Check the sandwich first, because a market that fails it is not a market: 0.90 < 1.05 < 1.10, admissible.

Price a call struck at 100. It pays 10 in the up state, 0 in the down state. Solve for the replicating portfolio, Δ shares plus B cash: Δ·110 + 1.05B = 10 and Δ·90 + 1.05B = 0. Subtract and Δ = 0.5, then B ≈ -42.86. The hedge costs 0.5·100 - 42.86 ≈ 7.14 today.

Now price it the other way, by risk-neutral expectation. The weight is q = (1.05 - 0.90)/(1.10 - 0.90) = 0.75, and the discounted expected payoff is (0.75·10 + 0.25·0)/1.05 ≈ 7.14. Same number, twice, by two methods that share nothing except the market. That agreement is not a coincidence to admire. It is what completeness buys: enough structure that hedge and measure are forced to agree.

Put-call parity, a free audit

Price the matching put on the same tree and you get 2.38. Parity says C - P should equal S₀ - K/(1+r): 7.14 - 2.38 = 4.76, and 100 - 100/1.05 = 4.76. They match to the cent, computed by two independent replications. That is not a fact to admire either. It is a test to automate: any pricing service that returns a call and a put on the same underlier, strike, and maturity must satisfy parity within tolerance, or one of exactly three things is wrong — a silent discount-convention swap, two quotes computed under different information sets, or an actual arbitrage. None of the three needs a statistician. A complete-market toy is valuable precisely because it hands the harness identities to check for free; an incomplete live book offers far fewer of them, which is a reason for more care, not less.

Breaking the sandwich on purpose

Raise the rate in the same tree to r = 0.12. The sandwich now reads 0.90 < 1.12, but 1.12 is not less than 1.10: the risk-free account beats the best possible stock outcome. Short a share for 100, lend the proceeds at 1.12, and at maturity you hold 112 while owing at most 110 — a certain 2 with no capital at risk. Compute the implied q anyway: (1.12 - 0.90)/(1.10 - 0.90) = 1.10. A probability of 1.10. The formula does not object; a downstream pricer will happily discount against it and print a call value of 9.82, a float that looks exactly like a price. The gate is one line: any node whose implied q falls outside [0,1] refuses before the expectation is even evaluated. Cheap, total, and it catches a whole family of upstream bugs — a mis-parsed rate, a stale volatility, a swept parameter with no constraint — without needing to know which one produced this particular node.

The wrong coin costs real currency

Keep the admissible tree, q = 0.75. Suppose a research simulator carries a physical up-probability of p = 0.80 — a perfectly reasonable estimate of how often the name has actually rallied — and someone prices the call with it, because the code has one probability field and nobody wrote down which measure it holds. That gives (0.80·10)/1.05 ≈ 7.62 against the true 7.14: 6.7% rich, and systematically so. Every call in the book marks up, every put marks down, and the desk's inventory looks like it is making money the moment it is written. The point worth sitting with: q and p are not two estimates of the same thing. There is no sense in which q is a better or worse forecast of the up-move; it is the weight that makes discounted tradables fair, and it exists to price, not to predict. Using it to forecast is the mirror-image error.

One more state, and the number becomes a band

Everything above depends on a structural fact quiet enough to miss: two successor states, two instruments to hedge with. Two equations, two unknowns, one solution. Change nothing but the state count. Add a middle outcome to the one-period tree — up, middle, down — and try to replicate the same call with the same two instruments (stock and cash). Three equations, two unknowns: generally no solution. The claim is no longer attainable. The market went incomplete by adding one perfectly ordinary state, nothing exotic.

What survives is a range of admissible risk-neutral measures, and each gives a different price. Push the range to its edges: one extreme reproduces the old measure and the old price, 7.14. The other extreme gives 4.76. Every value between is supported by some admissible measure:

Sit with that. One extra state — not jumps, not liquidity, not funding, not a second untradable factor — converts a unique 7.14 into a band a third as wide as the price itself. And the old unique answer survives only as the band's upper edge, the value you get from the degenerate measure that pretends the middle state cannot happen. That is what pasting a binomial float onto a live book actually does. It does not average the possibilities; it silently selects the most expensive admissible measure and calls the selection a calculation.

What this has to do with CEH-001, and with a model that "converged"

CEH-001 is not a two-outcome, one-stock, constant-rate claim. Live books carry jump risk, liquidity, funding, name-specific events, and dual frames that already disagree near 1.50 and 2.10 before completeness ever enters the conversation. A model or optimizer that converges cleanly on a simulator built like the two-state toy has proven the simulator was complete. It has proven nothing about the live book, which is why a card claiming completeness has to enumerate states and enumerate the instruments that span them — a written argument, not a paragraph expressing confidence, and not a training curve that stopped moving.

Averaging CEH-001's dual clouds to 1.80 because a tree returned one clean float is the same midpoint laundering the earlier chapters already deny, arriving this time with a multiperiod alibi. The honest output of an incomplete book is a band. CRR earns its place in this book by being the one object precise enough to show you, in exact numbers, how much a live book is missing.

Book: Chapter 10 in Rulers Before Models.