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Arbitrage & Cross-Book Coherence ​

A conditional market is not one order book but five, traded side by side: the underlying perpetual (price F, the forward), the two conditional perpetuals (prices CY and CN, the asset's expected price in the YES and NO branches), and the two prediction binaries (prices BY and BN, each a $0–$1 token paying $1 if its branch fires). These five books are linked by fixed payoff relationships, and those relationships imply that their prices cannot wander independently. A coherent family lives on a small surface governed by three numbers — the forward F, the event probability p (p ∈ [0, 1]), and the impact Δ = CY − CN — and everything else is pinned by two identities.

The point of this document is to explain who keeps the five prices on that surface. The answer is: traders, not the engine. The matching engine runs five ordinary order books. It does not reach across them to force F to equal the probability-weighted blend of CY and CN, and it does not force the two binaries to sum to a dollar. Those relationships hold because, when they drift, a profitable trade opens up — and someone takes it. Coherence is an emergent property of arbitrage, the way the law of one price holds anywhere: not by decree, but because a violation is money left on the table.

Every number below is illustrative — a teaching example with clean arithmetic, never a live or calibrated value. If the conditional-market family, the forward identity, or the cash-out path are unfamiliar, the conditional-expectation and probability docs earlier in this layer introduce them; here we focus narrowly on the arbitrage that ties the books together and on where that arbitrage stops working.

The forward identity as an arbitrage relationship ​

The spine of the family is the forward identity:

F = p·CY + (1 − p)·CN.

In words: the underlying forward is the probability-weighted blend of the two conditional forwards — the asset's expected price given YES, weighted by the chance of YES, plus its expected price given NO, weighted by the chance of NO. It is the law of total expectation written in prices, and it is covered in full in the conditional-expectation doc. Here we read it as something more operational: a relationship that, when broken, hands a trader a trade.

The arbitrage logic runs through a synthetic. Buy one unit of the YES conditional perp and one unit of the NO conditional perp together. Assuming the event resolves YES or NO, exactly one of the two legs is in its winning branch and cash-settles to the underlying's price, while the other voids — profit-and-loss zero, margin returned. So the pair settles to the underlying regardless of which way the event goes: it is a synthetic underlying position. By one-price, a synthetic underlying must be worth the same as the real underlying forward F. That equivalence is exactly the forward identity, and it is what makes the identity an arbitrage relationship rather than a mere accounting convention.

Now suppose the books drift out of line. The cleanest way to see it is through the implied probability the conditional perps carry, p = (F − CN) / (CY − CN), versus where the conditional legs actually sit. Consider a generic family:

  • Underlying forward F = $100.
  • YES conditional perp CY = $112, NO conditional perp CN = $92, so the impact is Δ = CY − CN = $20.
  • The conditional legs imply p = (100 − 92) / (112 − 92) = 8 / 20 = 0.40.

That is coherent: 0.40·112 + 0.60·92 = 44.80 + 55.20 = $100 = F. The blend lands on the forward.

Now let the two conditional perps stay where they are but the underlying perp drift up to F = $103 on its own book — a flurry of underlying flow, say, that the conditional legs have not absorbed. The blend of the conditional legs is still $100, but the underlying now prints $103. The identity is violated by $3: the synthetic (the YES+NO pair, worth $100 in blend terms) is cheap relative to the underlying it settles to ($103).

The trade that restores it is the obvious one. Buy the cheap synthetic and sell the rich underlying: go long the conditional pair (long CY and long CN, the combination that settles to the underlying) and short the underlying perp. You have locked a synthetic-vs-real spread of about $3 per unit that converges as the event resolves and the surviving conditional leg settles to the same underlying you are short. The act of putting that trade on — bidding the conditional legs up, offering the underlying down — is what drags F and the blend back together. No one instructed the engine to do this. A trader did it because the spread was there, and in closing the spread for profit they closed it for everyone.

The reverse case is symmetric: if the underlying perp lags below the conditional blend, sell the synthetic and buy the underlying. Either way, the identity is enforced by the incentive to trade the synthetic against the real, not by anything the matching engine knows about the relationship between the books.

A note on what "riskless-ish" means here, because it matters for the rest of this document. The synthetic-vs-underlying trade is low-risk, not literally riskless. It carries execution risk (you have to fill three legs near the prices you saw), inventory cost on the perp legs and funding on the underlying-perp leg (the conditional perps do not pay funding), the margin to carry the position to convergence, and — the one that genuinely breaks the relationship — void risk, taken up at the end. The relationship is a fair-value relationship: it pulls prices toward each other reliably enough that traders chase it, but it is not a hard equality the engine guarantees instant-by-instant.

The binary box and box arbitrage ​

The two prediction binaries carry their own coherence relationship, the binary box:

BY + BN ≈ $1.

The logic is a one-price argument again, and a tighter one than the forward identity because the binaries are pure $0–$1 instruments with no asset exposure. At resolution exactly one binary pays $1 and the other pays $0: if YES, BY pays $1 and BN pays $0; if NO, the reverse. So a portfolio of one YES binary plus one NO binary pays exactly $1 at resolution, whatever the outcome. A claim that pays a certain dollar must cost a dollar today — hence the two binary prices should sum to one.

This is the same probability statement seen from the side: if BY ≈ p and BN ≈ 1 − p, then BY + BN ≈ p + (1 − p) = $1. The box and the probability reading are two faces of one fact.

Box arbitrage is what holds the sum at a dollar. Suppose the two binaries drift apart:

  • YES binary BY = $0.55, NO binary BN = $0.41. The box sums to $0.96 — four cents short of a dollar.

A pair of binaries that together pay a guaranteed $1 at resolution is on offer for $0.96. The trade is to buy the box: buy one YES binary at $0.55 and one NO binary at $0.41 for a combined $0.96, and hold to resolution, where the pair pays $1. That is roughly four cents of low-risk profit per box, captured by the convergence of the sum back toward a dollar. The reverse — a box trading above a dollar, say BY + BN = $1.04 — invites selling the box: sell both binaries for $1.04 today against the $1 you will owe at resolution. In both directions, the traders chasing the four cents are the mechanism that pins the sum near a dollar. The engine never checks that BY + BN = $1; the box arbitrageurs do.

In practice the sum sits near a dollar rather than exactly on it, and the gap is informative. Spread and fees eat part of any small deviation, so a one- or two-cent gap is just the cost of trading the box rather than free money. A persistent, large gap is a different signal: it usually means one leg is stale or thin and the quote on it is not a real consensus. Treat a big, durable box deviation as a flag to investigate which binary has drifted, not as a fair-value reading to take at face value — and remember that capturing it still requires both legs to be fillable in size.

Consistency between the conditional perps and the binaries ​

So far the forward identity ties the underlying perp to the conditional perps, and the box ties the two binaries to each other. The relationship that closes the loop ties the conditional perps to the binaries — and it runs through the probability they must share.

There are two independent ways to read the event probability off the family. The binary gives it directly: BY ≈ p. The conditional perps give it through the forward identity, rearranged: p = (F − CN) / (CY − CN). In a coherent family these two readings agree, because they are reading the same event from two different books. When they disagree, the family is internally inconsistent, and the disagreement is itself the arbitrage signal.

Worked example. Take the coherent family from before — F = $100, CY = $112, CN = $92 — which implies p = 0.40 from the conditional perps. Suppose the YES binary, on its own thinner book, is trading at BY = $0.55 while the conditional legs still imply 0.40. That fifteen-point gap is not a fair-value spread; it is two books quoting two different probabilities for the same event. One of them is wrong, and a trader who has a view on which is wrong can express it by trading the cheap probability against the rich one — for instance, buying the conditional-perp exposure to YES (which is priced as if YES is a 40% event) against selling the YES binary (priced as if YES is a 55% event), so that the position profits as the two probability readings converge, with the directional event exposure largely offsetting between the two legs.

The cross-check does not tell you which book is right — that judgment is where a market maker earns their spread. It tells you that they disagree and roughly how a trade would be oriented to profit from the convergence. The discipline this imposes runs the other way too: a maker quoting the family should derive all five marks from one consistent (F, p, Δ) triple, precisely so that no internal probability inconsistency exists across their own quotes for a sharper trader to pick off. Coherence across the conditional perps and the binaries is maintained, once again, by the people who would profit from any gap.

A practical caution on the two readings. The binary delivers probability cleanly, in probability units, with no dependence on the conditional levels. The conditional-perp implied probability is a ratio, p = (F − CN) / Δ, that divides by the impact Δ = CY − CN. When that spread is tight, the ratio divides by a small number and any staleness in CY or CN gets amplified into a noisy probability. So a large binary-vs-conditional gap on a family with a thin conditional spread more often means the conditional read is the unreliable one. The convergence trade is real; sizing it requires knowing which book you actually trust.

The forward identity and the box are static one-price relationships — they hold at resolution by construction. The relationship that ties conditional-perp prices to binary prices before resolution, and ties them tightly, is a piece of exchange mechanics: the cash-out bridge.

Here is the mechanic, covered in full in the settlement doc. A conditional perpetual's gain is branch-contingent — it is only real in the branch where the event fires, so it cannot be withdrawn as cash, because it would vanish if the other branch won. To exit a conditional position early, you close it, which crystallizes your realized conditional profit-and-loss into prediction-binary tokens entered at $0 in the firing branch: a gain mints a long position in that branch's binary, a loss mints a short, each at a $0 entry. The mint alone is still not cash — a binary token only pays at resolution — so to realize value today you then sell those tokens on the binary book at its going quote q (which is roughly the event probability). The close-then-sell pairing is the early-exit path; there is no separate mint primitive.

Why this matters for coherence: the cash-out bridge makes a conditional perp and its matching binary into two routes to nearly the same economic exposure, and two routes to the same thing cannot be priced too far apart without opening a trade between them. The conditional perp's branch-contingent profit converts, on close, into binary tokens that monetize at the binary's price q. So the value you can extract from a conditional position is governed by where the binary trades — and conversely, a trader choosing how to take YES exposure can do it on the conditional perp or on the binary, and will route to whichever is cheaper for the exposure they want. That routing pressure is the arbitrage that keeps the conditional and binary prices tied: if the binary gets cheap relative to what the conditional implies, exposure flows to the binary and bids it up; if it gets rich, exposure flows the other way.

A short illustration of the bridge in motion. Suppose you are long a YES conditional perp sitting on a realized gain, and you want out before the event resolves. You close it and the exchange mints you, say, 100 long YES binary tokens at a $0 entry. If the YES binary is bid at q = $0.62, selling those tokens returns about $62 in cash now (≈ q × tokens). Your realizable early-exit value is your conditional gain scaled by the binary's price — which is the event probability. This is the concrete sense in which the conditional and binary prices are bound together: the binary is the venue where a conditional gain becomes cash, so the binary's price is one of the prices that determines what a conditional position is worth before resolution. Break that link and you would be able to extract more from one route than the other; the routing trade closes it.

The bridge also explains a practical limit that the static identities alone would not reveal: early cash-out is sourced from the binary book. You can always close the conditional perp and hold the minted tokens to resolution, but converting them to cash today happens against the binary, at its going quote. The depth of the binary leg is therefore what makes immediate early exit available, and the coherence the bridge enforces is only as firm as that book is liquid. A thin binary book both widens the gap a router would need to see before acting and makes the cash-out itself more expensive — the same liquidity that the box arbitrage needs in order to pin BY + BN to a dollar.

How a mispricing in one book pulls the others back ​

It is worth stepping back and seeing the three relationships as one connected system, because a shock to any single book propagates to the others through the trades described above.

Put a coherent family on the table: F = $100, CY = $112, CN = $92 (so Δ = $20, implied p = 0.40), BY = $0.40, BN = $0.60. Every identity checks. Now perturb exactly one book and trace the repair.

  • The underlying perp jumps. F prints $103 while the conditional legs and binaries lag. The forward identity is now violated — the conditional blend says $100, the perp says $103. The synthetic-vs-underlying trade (long the conditional pair, short the underlying) is the response, and it pulls F back toward $100 or pulls the conditional legs up toward a blend of $103, depending on which book the arbitrageurs judge to be the leader. Either way the two converge.

  • A binary drifts. BY climbs to $0.55 on thin flow while everything else holds. Two relationships now flag at once: the box (BY + BN = 0.55 + 0.60 = $1.15, fifteen cents rich) and the conditional-vs-binary probability check (binary says 0.55, conditional perps say 0.40). Selling the rich box and trading the conditional-vs-binary probability convergence both push BY back down toward $0.40. The same drift trips two independent arbitrages, which is why binary mispricings tend to get corrected quickly when the books are liquid.

  • A conditional leg drifts. CY slips to $108 while F, CN, and the binaries hold. The implied probability from the conditional perps moves to p = (100 − 92) / (108 − 92) = 8 / 16 = 0.50, disagreeing with the binary's 0.40, and the forward identity no longer closes (0.40·108 + 0.60·92 = 98.40 ≠ $100). The convergence trades against the binary and against the synthetic both bid CY back toward $112.

The shape of the system is what matters: the five books are over-determined by the three free numbers, so a move in any one book that the others have not matched shows up as a violation of one or more identities, and each violation is a trade. The trades overlap — a single binary drift can trip both the box and the probability check — which makes the family's coherence robust as long as someone is watching all five books at once. That last clause is the whole mechanism. The engine treats the books as five independent markets; it is the arbitrageurs spanning all five that turn them into one coherent family.

The limits of these relationships ​

These are fair-value relationships. They hold because traders trade them, and that phrasing carries every important caveat with it.

They hold to the extent the books are liquid and watched. A relationship enforced by arbitrage is only as tight as the arbitrage capital pointed at it. On a deep, actively quoted family the identities hold to within spread and fees, because any gap is competed away in moments. On a thin leg — and the conditional and binary legs trade less than the underlying — a gap can persist simply because no one is currently standing there to take it. A persistent deviation is best read not as a free lunch but as a sign that the leg is thin, stale, or carries a cost the naive arbitrage ignores. The relationship is a statement about fair value, and a quote on a book no one is trading is a price, not a consensus.

The arbitrage is low-risk, not riskless. Every trade described here carries execution risk across multiple legs, the funding and margin cost of carrying the position to convergence, and the chance that the gap widens before it closes. The "lock" is a fair-value lock that converges by resolution, not a guaranteed instantaneous profit. Sizing these trades is a real discipline, not a free roll.

They break around a Void. This is the sharpest limit and the one that distinguishes these from textbook box arbitrage. Every relationship in this document carries an implicit "assuming the event resolves YES or NO." There is a third outcome — an unresolvable event, a Void — and through a Void the relationships do not merely loosen, they break:

  • The binary box goes to $0. On a Void, both prediction binaries pay $0 — not $0.50 each, not $1 to one side. A box you bought for $0.96 expecting a guaranteed $1 at resolution pays nothing if the event Voids. The "guaranteed dollar" was guaranteed only across YES and NO. So a box bought outright below a dollar is not riskless arbitrage; it is a low-cost long position that is wiped out by a Void. (Binary tokens minted at a $0 entry through the cash-out bridge are different: entered at $0, they have zero profit-and-loss on a Void regardless. It is outright-purchased binaries that take the full Void loss.)

  • The synthetic breaks. On a Void, both conditional perps Void — margin returned, profit-and-loss zero — so the YES+NO pair no longer settles to the underlying. The synthetic that the forward identity rests on simply ceases to exist in the Void branch, and a synthetic-vs-underlying position is left with the underlying leg unhedged.

This is why void risk is the genuine, irreducible risk in trades that otherwise look riskless. The no-arbitrage relationships are conditional on resolution, and they say nothing about how likely a Void is. An arbitrageur buying a cheap box or a cheap synthetic is, whether they price it or not, short a Void. On events where a Void is a real possibility, the few cents of box or synthetic edge are the premium being paid for taking that void exposure, and treating the trade as truly riskless is the mistake the structure is most likely to punish. The Void mechanics are covered in full in the settlement and resolution doc; the point here is narrower and load-bearing: the coherence relationships are fair-value, resolution-conditional statements, and the boundary where they stop holding is exactly the boundary of a clean YES/NO outcome.

The takeaways. The five books of a conditional-market family are tied together by two identities — the forward identity F = p·CY + (1 − p)·CN and the binary box BY + BN ≈ $1 — and by the cash-out bridge that converts a closed conditional perp into the matching binary. None of these is enforced by the matching engine. Each holds because a violation opens a low-risk trade: the synthetic-vs-underlying trade for the forward identity, box arbitrage for the binaries, and the probability cross-check plus exposure-routing for the conditional-vs-binary link. A mispricing in any one book registers as a violation of one or more identities, and the trades that close those violations are what pull the family back into coherence. The relationships are fair-value and resolution-conditional: they hold to the extent the books are liquid and watched, the arbitrage is low-risk rather than riskless, and around a Void — where the binary box collapses to $0 and the synthetic breaks — they stop holding altogether. Coherence is something the market maintains, which is exactly what makes knowing these relationships valuable.