Pricing I — The Conditional-Expectation Framework

A conditional market lets you trade an underlying asset conditioned on whether a real-world event happens. This doc explains the single idea underneath all of it: a conditional perpetual's price is an expectation of the underlying across the event's possible outcomes. Once you see that, the whole family of books — the underlying perpetual, the two conditional perpetuals, and the two prediction binaries — falls into place as a coherent set of prices that should agree with each other.

We build the framework up from one equation, explain what each piece means, show how the legs are supposed to line up, and finish with a fully worked numerical example using a generic event.

Everything here describes how fair prices relate to each other. These are no-arbitrage relationships, not rules the matching engine enforces. They hold in practice because traders arbitrage them, not because the exchange pins them.

The core idea: a conditional forward is an expectation across outcomes

Start with the underlying. An underlying perpetual (the underlying perp) is a standard perpetual future on some asset — call it asset-PERP. Its fair price is the forward, written F. F is the price the market expects the asset to be worth, all things considered.

Now introduce an event: a yes/no question about the world that will resolve at some future point. For example, "a scheduled economic decision goes a particular way," or "the asset closes above a threshold K." The event resolves YES, NO, or — rarely — turns out to be unresolvable.

Here is the key observation. The asset's price depends on how that event resolves. If you could split the world into two branches — the branch where the event resolves YES and the branch where it resolves NO — the asset would generally be worth a different amount in each. A conditional market gives you a separate instrument for each branch:

• A conditional perpetual (YES) — the asset if the event resolves YES. It trades on its own order book at its own price. If the event resolves YES, it cash-settles to the underlying's price; if the event resolves NO (or is unresolvable), it is voided: profit-and-loss is zero and your margin is returned. (Voided is not a loss to $0 — it simply ceases to exist with no cash flow.)
• A conditional perpetual (NO) — the mirror: the asset if the event resolves NO.

We write the conditional perp (YES) price as CY and the conditional perp (NO) price as CN.

What should CY be worth? It pays out only in the YES branch. So its fair price is the asset's expected price given that the event resolves YES — its conditional expectation:

CY = E[underlying | YES]

Likewise:

CN = E[underlying | NO]

That is the whole idea in one line. A conditional forward is the expected value of the underlying, computed inside one branch of the world. It is "the asset, in the universe where this event went a certain way."

This is why a conditional perp is not priced at F. F is the expectation averaged over both branches. Each conditional perp is the expectation inside one branch — and those two branch-expectations sit on either side of F.

F = p·CY + (1 − p)·CN explained

The three prices — F, CY, CN — are not independent. They are tied together by the law of total expectation, which says: the overall expectation of the asset equals the probability-weighted average of its expectations in each branch.

Let p be the probability the event resolves YES (so 1 − p is the probability of NO). Then:

F = p·CY + (1 − p)·CN

Read it out loud: the underlying's forward price is the YES-conditional price weighted by the chance of YES, plus the NO-conditional price weighted by the chance of NO. This is the forward identity, the spine of the entire framework.

It says the underlying perp is already a blend of the two conditional worlds. The asset's price today implicitly contains the market's view on the event: it is the average of "the asset if YES" and "the asset if NO," weighted by how likely each is.

A few consequences fall straight out of this identity.

The conditional prices straddle F. Because F is a weighted average of CY and CN, one of them sits above F and the other below (unless the event has no effect on the asset, in which case all three coincide). The gap between them is the impact, written Δ:

Δ = CY − CN

Δ measures how much the asset's expected price differs between the two outcomes — how informative the event is about the asset. A small Δ means the event barely moves the asset; a large Δ means the outcome matters a lot.

The conditional prices are placed asymmetrically around F. Given F, p, and Δ, the forward identity pins down exactly where CY and CN sit:

CY = F + (1 − p)·Δ CN = F − p·Δ

These are not modeling choices — they are the only placement consistent with the forward identity. Notice the asymmetry: the YES leg sits (1 − p)·Δ above F, and the NO leg sits p·Δ below F. The rarer outcome moves further from F. If YES is unlikely (small p), CY sits far above F, because it rarely pays — and when it does, it has to carry the whole conditional move. The conditional perps lever the impact by the inverse rarity of their own branch.

You can check the identity holds: p·CY + (1 − p)·CN = p·(F + (1 − p)Δ) + (1 − p)·(F − pΔ) = F + p(1−p)Δ − (1−p)pΔ = F. The Δ terms cancel exactly, for any p.

Why this is enforced by arbitrage. Consider buying one conditional perp (YES) and one conditional perp (NO) together. Assuming the event resolves YES or NO, exactly one of them is in its winning branch and cash-settles to the underlying price, while the other voids with margin returned — so the pair settles to the underlying and behaves as a synthetic underlying position. By one-price, the pair must cost the same as the underlying forward F. If the two conditional prices summed (probability-weighted) to more than F, a trader could sell the pair, buy the underlying, and lock in the difference. That arbitrage is what holds the identity together. The exchange does not enforce it; the market does. (The same void caveat that applies to the binary box applies here: through a void both conditional perps void, the synthetic breaks, and neither leg settles to the underlying. The forward identity is a YES/NO relationship.)

What p (the implied probability) means

p is the probability the event resolves YES. You can read it off the family in two independent ways, and the framework expects them to agree.

From the prediction binaries. Alongside the two conditional perps, every conditional market includes two prediction binaries:

• A prediction binary (YES), price BY — a token priced between $0 and $1 that pays $1 if the event resolves YES and $0 if it resolves NO.
• A prediction binary (NO), price BN — the mirror, paying $1 on NO and $0 on YES.

A prediction binary is a pure bet on the event with no asset exposure. Its fair price is the market-implied probability of its outcome. A binary that pays $1 on YES is worth exactly the chance of YES:

BY ≈ p, BN ≈ 1 − p

So you can read p directly off the YES binary's price. If BY trades at $0.30, the market is implying a 30% chance of YES. This is a market-implied probability — the price traders are collectively willing to pay — not a claim about the true objective odds.

From the conditional perps. You can also back p out of the two conditional forwards. Rearranging the forward identity:

p = (F − CN) / (CY − CN) = (F − CN) / Δ

This is the implied probability read off the conditional perps. It asks: where does F sit between CN and CY? If F is close to CY, the YES branch dominates the blend, so p is high; if F is close to CN, p is low.

The cross-check. A coherent family has these two readings agree: the binary-implied p (from BY) should match the conditional-perp-implied p (from the forward identity). When they disagree, the family is internally inconsistent and at least one of the three prices — F, CY, or CN — is stale. This cross-check collapses five separate prices into one internally-consistent picture and flags which leg has drifted when the two probability readings disagree.

A note on what p does and doesn't carry. p is the event probability — and an external prediction market on the same event can hand it to you directly. But p tells you nothing about Δ. Two events with the same p can have wildly different impacts on the asset: a coin-flip event that doesn't affect the asset has p = 0.5 and Δ = 0, while a coin-flip event that swings the asset hard has p = 0.5 and a large Δ. The probability is the marginal of the event; the impact is the dependence between the event and the asset. They are independent dials.

The prediction-binary pair summing to ~$1

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

BY + BN ≈ $1

The logic is simple. At resolution, exactly one of the two binaries pays $1 and the other pays $0. If the event resolves YES, BY pays $1 and BN pays $0; if NO, BN pays $1 and BY pays $0. So a portfolio of one YES binary and one NO binary always pays exactly $1 at resolution, regardless of outcome. By one-price, that portfolio must cost $1 today — hence the two binary prices should sum to one dollar.

This is consistent with the probability reading: if BY ≈ p and BN ≈ 1 − p, then BY + BN ≈ p + (1 − p) = $1.

The one exception: void. There is a third outcome besides YES and NO. If the event turns out to be unresolvable, the market is voided, and on a void both prediction binaries pay $0. So the binary box holds through a YES or a NO resolution, but it breaks to $0 through a void. Both binaries paying $0 — not $0.50 each, and not $1 to one side — is the defining payoff of a void. The binary box identity is a YES/NO statement; it carries an implicit "assuming the event resolves" caveat.

To summarize the binary payoffs across all three outcomes:

• YES: BY pays $1, BN pays $0.
• NO: BY pays $0, BN pays $1.
• Void: BY pays $0, BN pays $0.

Coherence between the legs

Putting it together, a single conditional market is five linked order books — the underlying perp, two conditional perps, two prediction binaries — and a coherent family is governed by just three numbers:

• F — the underlying forward (shared by every book in the family).
• p — the event probability (also expressed as the YES binary price).
• Δ — the impact, the conditional spread CY − CN.

Everything else is determined by these three:

• CY = F + (1 − p)·Δ (the YES conditional perp)
• CN = F − p·Δ (the NO conditional perp)
• BY ≈ p (the YES binary)
• BN ≈ 1 − p (the NO binary)

The five prices are therefore not free. They live on a small, three-parameter surface, tied together by two identities:

• Forward identity: F = p·CY + (1 − p)·CN
• Binary box: BY + BN ≈ $1

When all of this lines up, the family is coherent: the underlying is the probability-weighted blend of the two conditional worlds, the binaries sum to a dollar, and the probability you read off the binaries matches the one you back out of the conditional perps. A trader pricing the family estimates the three free numbers, places the five marks consistently, and the family agrees with itself.

When it doesn't line up, there is a no-arbitrage relationship being violated, and someone can trade against it. Coherence is a property of fair prices, maintained by arbitrage across the family. That is exactly why these identities are worth knowing: they tell you what a healthy family looks like, and they tell an arbitrageur where to look when one drifts.

A subtlety worth stating plainly. Each conditional perp is fair at its conditional mean, not at F. If a market maker naively quoted both conditional perps at the underlying forward F, a sharper counterparty would simply lift whichever leg is favorably mispriced — buy the conditional whose true conditional mean is above F, leaving the maker short a positive-expected-value instrument. The stable, "selection-proof" placement is to quote each leg at its own conditional expectation, CY = E[underlying | YES] and CN = E[underlying | NO]. That placement still satisfies the forward identity, but it is individually fair on each leg, so there is no single leg a sniper can pick off.

What the family pins down: the binaries reveal the event's probability p (its marginal), and the conditional perps reveal the two conditional means (the first moments of the asset in each branch). Together they fix where all five prices sit. The remaining structure — higher moments, the detailed conditional distribution of the asset within each branch — lives in the full joint distribution, beyond what these five marks encode. The framework prices first moments and the event marginal; that is precisely the set of quantities the family's order books make tradable.

A word on what drives Δ. The impact is large when the event is highly informative about the asset — when the asset's moves co-vary strongly with the event. Structurally, Δ grows with two quantities: the underlying's volatility (σ) over the event horizon — how much the asset moves — and the correlation (ρ) between the underlying and the conditioning event — how much its moves co-vary with the outcome. An event that barely correlates with the asset, or an asset that barely moves, produces a small Δ. This doc lays out that relationship structurally; estimating Δ for a specific market is a modeling step that builds on these same drivers.

A worked numerical example

Take a generic event over a generic asset. Suppose:

• The asset's underlying perp trades at a forward of F = $100.
• The market believes the event has a 40% chance of resolving YES, so p = 0.40.
• The asset is expected to be worth more in the YES branch than the NO branch, with an impact of Δ = $20. (Illustrative numbers — chosen to make the arithmetic clean, not drawn from any live market.)

Step 1 — place the conditional perps. Using the centers:

• CY = F + (1 − p)·Δ = 100 + (1 − 0.40)·20 = 100 + 0.60·20 = 100 + 12 = $112
• CN = F − p·Δ = 100 − 0.40·20 = 100 − 8 = $92

So the asset is worth an expected $112 in the YES world and $92 in the NO world. Note the asymmetry: YES is the rarer branch (40%), so CY sits further from F (+$12) than CN does (−$8). The rarer outcome moves further.

Step 2 — check the forward identity. The underlying should be the probability-weighted blend:

p·CY + (1 − p)·CN = 0.40·112 + 0.60·92 = 44.80 + 55.20 = $100 = F. ✓

The two conditional worlds blend back exactly to the underlying forward. The market on the plain asset already embeds the event: $100 is "40% chance of a $112 world, 60% chance of a $92 world."

Step 3 — place the prediction binaries. The binaries price the event directly:

• BY ≈ p = $0.40
• BN ≈ 1 − p = $0.60

And the binary box checks out: BY + BN = 0.40 + 0.60 = $1.00. ✓

Step 4 — cross-check the implied probability. Back p out of the conditional perps:

p = (F − CN) / (CY − CN) = (100 − 92) / (112 − 92) = 8 / 20 = 0.40. ✓

The probability implied by the conditional perps (0.40) matches the probability priced by the YES binary ($0.40). The family is coherent.

Step 5 — see what happens when the probability moves. Suppose news shifts the market's view of the event from 40% to 60% YES (p: 0.40 → 0.60), while the underlying forward F and the impact Δ stay put. Re-place the legs:

• CY = 100 + (1 − 0.60)·20 = 100 + 8 = $108
• CN = 100 − 0.60·20 = 100 − 12 = $88
• BY ≈ $0.60, BN ≈ $0.40

As YES became more likely, the YES binary rose ($0.40 → $0.60) and the YES conditional perp fell ($112 → $108). That second move is the asymmetry at work: when YES is more likely it carries more of the weight in the blend, so it no longer has to sit as far above F. Meanwhile the underlying forward is unchanged — it was always the blend, and the blend still lands on $100: 0.60·108 + 0.40·88 = 64.80 + 35.20 = $100. ✓ A shift in the event probability repriced the conditional legs without moving the underlying at all, because the information changed, not the asset's overall fair value.

Step 6 — resolution. At the deadline the event resolves and the books settle:

• If YES: the conditional perp (YES) cash-settles to the underlying's price and the conditional perp (NO) is voided (PnL zero, margin returned). The YES binary pays $1, the NO binary pays $0.
• If NO: the conditional perp (NO) settles to the underlying; the conditional perp (YES) is voided. The NO binary pays $1, the YES binary pays $0.
• If the event is unresolvable (void): both conditional perps void (margin returned), and both binaries pay $0.

That is the full lifecycle of one conditional market, from the moment its three numbers are set to the moment they collapse into a single realized outcome.

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The takeaways. A conditional perpetual is an expectation of the underlying inside one branch of the world. The underlying forward is the probability-weighted blend of the two branches — F = p·CY + (1 − p)·CN. The prediction binaries price the event's probability and sum to about a dollar, except through a void, where both pay $0. And a healthy family agrees with itself: the probability you read off the binaries matches the one implied by the conditional perps. These are relationships between fair prices, held together by arbitrage rather than enforced by the exchange — which is precisely what makes them useful to know.