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Probability & Implied Markets
A conditional market is, at heart, a measurement device pointed at a probability. Every conditional-market family carries a prediction binary whose price is the market's view of how likely an event is to resolve YES — and, alongside it, two conditional perpetuals whose prices encode how much the underlying asset is expected to move depending on that outcome. Read together, these books let you back out an implied probability, compare it against your own model, and price the binaries themselves on a consistent footing.
This document is about reading and reasoning with that probability. It assumes you have met the conditional-market family elsewhere in this corpus: one underlying perpetual (price F, the forward) plus four conditional legs — the two conditional perpetuals (prices CY and CN) and the two prediction binaries (prices BY and BN). If those terms are unfamiliar, start with the conditional-markets overview; here we focus narrowly on probability.
We use the standard notation: F is the underlying forward, p is the probability the event resolves YES (p ∈ [0, 1]), Δ is the impact (Δ = CY − CN), and K is a strike or threshold where the event references a price level. Every worked number below is illustrative — a teaching example, not a live or canonical value.
Reading an implied probability from prices
There are two places to read a probability off a conditional-market family, and the discipline of comparing them is most of the skill.
From the prediction binary directly
A prediction binary (YES) is a $0–$1 instrument that pays $1 if the event resolves YES and $0 if it resolves NO. The fair price of a contract that pays a dollar in one state and nothing in the other is simply the probability of that state. So:
The price of the YES prediction binary is the market-implied probability of YES. BY ≈ p, and BN ≈ 1 − p.
If BY trades at $0.62, the market is implying a 62% chance the event resolves YES. No conversion, no model — the price already lives in probability units. This is the single most direct reading on the exchange.
A companion sanity check rides alongside it. The two binaries should sum to about a dollar:
Binary box: BY + BN ≈ $1.
If BY is $0.62 and BN is $0.36, the pair sums to $0.98 — close, with the two-cent gap reflecting spread, fees, and the fact that this relationship is a no-arbitrage relationship between fair prices. The box holds because, when it drifts, traders have an incentive to close the gap. Treat a persistent, large deviation as a signal that one side is stale or thin and a trade to be taken, rather than a fair-value reading.
One important exception. The binary box holds through a YES or NO resolution, but it breaks to $0 on both legs if the event is voided (an unresolvable event). In a void, every prediction binary pays $0, and a position you thought of as "a dollar either way" pays nothing. The probability reading from a binary price is conditional on the event actually resolving; it is silent about void risk, which is a separate consideration covered in the resolution mechanics.
From the conditional perpetuals
The same probability is implied a second way — by the two conditional perpetuals — and reading it there is the cross-check that keeps you honest.
The underlying forward is the probability-weighted blend of the two conditional prices. This is just the law of total expectation: the expected price of the asset is its expected price given YES, weighted by p, plus its expected price given NO, weighted by (1 − p):
Forward identity: F = p·CY + (1 − p)·CN.
Rearranging for p gives the conditional-perp-implied probability:
Implied probability: p = (F − CN) / (CY − CN).
Worked example. Suppose an asset's underlying perp trades at F = $100. The YES conditional perp trades at CY = $108 (the asset is expected to be worth $108 if the event fires YES) and the NO conditional perp at CN = $96 (expected $96 if it fires NO). Then:
- p = (100 − 96) / (108 − 96) = 4 / 12 ≈ 0.33.
The conditional perps are implying a 33% chance of YES. The impact is Δ = CY − CN = $12 — the underlying's expected price differs by $12 between the two outcomes.
Notice what this means geometrically. The YES leg sits (1 − p)·Δ above the forward, and the NO leg sits p·Δ below it:
- CY = F + (1 − p)·Δ = 100 + 0.67·12 = $108.
- CN = F − p·Δ = 100 − 0.33·12 = $96.
The rarer outcome moves further from the forward. Because YES is unlikely here (p ≈ 0.33), the YES leg has to sit a full $8 above F: it rarely pays, so when it does it must carry the whole conditional upside. The NO leg, the likely outcome, sits only $4 below. Conditional perps lever the impact by the inverse rarity of their own branch.
The cross-check
You now have two independent reads on p: one from the binary (BY directly), one from the conditional perps (the implied-probability formula). In a healthy, coherent family they agree. When they disagree, the family is internally inconsistent and at least one of your three inputs — F, the binaries, or the conditional perps — is stale, thin, or mispriced.
Worked example of a disagreement. Suppose the binary says BY = $0.55 but the conditional perps imply p = (F − CN)/(CY − CN) = 0.33. That 22-point gap is a flag. It might mean the binary book is thin and lagging, or the conditional perps are stale, or — most interestingly — the two books genuinely disagree about the event probability and one of them is expressing a view the other has not absorbed yet. The cross-check does not tell you which book is right; it tells you that they disagree and where to look. Reconciling that gap is exactly the judgment a market maker is paid for.
A practical note on which read is cleaner: the binary gives you pure probability with no dependence on the conditional perps' levels, while the conditional-perp implied probability is a ratio that depends on three prices (F, CY, CN) and amplifies any staleness in the spread CY − CN when that spread is small. When the conditional spread is tight, the implied-probability formula divides by a small number and gets noisy. For a quick, robust read, the binary price is usually the better instrument.
Model-implied vs market-implied probability (conceptual)
So far every probability has been market-implied — read off live prices. The other probability that matters is model-implied — the one you compute from a view of the world. Keeping these two straight, and knowing how to combine them, is the conceptual core of pricing on a conditional market.
What each one is
Market-implied probability is what the order book is currently saying: the binary price, or the conditional-perp implied p. It reflects the aggregate, real-money opinion of everyone quoting and trading the family right now. It is observable, it moves continuously, and it already incorporates information you may not have. It is still the market's opinion rather than an objective truth — a quote on a thin book carries less consensus than one on a deep, actively traded book.
Model-implied probability is what your own analysis says the probability should be. For a threshold event ("will the asset be above K at resolution?"), you can derive it from a distributional model of the underlying: given the forward, a volatility estimate, and the horizon, the probability the asset finishes above K is a calculable quantity. For an exogenous event (a scheduled economic decision, another asset's direction), you typically source it from an external market or a structured estimate rather than from the underlying's own distribution.
These two probabilities answer the same question and routinely differ. The discipline is to treat that difference as information, not noise.
Why they differ — and what the difference usually means
When your model says 35% and the book says 55%, the honest first question is which of us knows something the other doesn't? A few common resolutions:
The book is leaning on a directional view. A market-implied probability for a threshold event embeds the market's view of where the underlying is drifting. A simple driftless model deliberately strips that drift out. So a model-vs-book gap on a threshold event is frequently a disguised view on the underlying's direction, not a disagreement about the event mechanics. Warehousing that gap directionally — taking the side of your model and sitting on it — is taking a naked position on the underlying in a costume. If you want to express it, hedge the underlying exposure out and keep only the piece you actually have an edge on.
One book is stale or thin. Conditional legs trade less than the underlying. A market-implied probability from a book that has not traded in a while is a quote, not a consensus. The cross-check between binary-implied and conditional-perp-implied p often reveals which book is lagging.
Genuine information asymmetry. Sometimes the market genuinely knows more — there is news in the order flow you have not seen. A model that ignores live prices will be confidently wrong. This is why pure model-implied probabilities, untethered from the book, are dangerous to quote against.
Blending the two
The practical posture is neither "trust the model" nor "trust the book" but a blend, weighted by how much you trust each source in the current regime. The robust way to blend two probabilities is to combine them in log-odds (logit) space rather than averaging the raw probabilities — log-odds blending behaves well near 0 and 1, where naive averaging distorts. Conceptually:
- Convert each probability to log-odds.
- Take a weighted average of the log-odds, with weights reflecting your confidence in the model versus the book (and, where available, an external market).
- Convert back to a probability.
The weights are a judgment call and a place where edge lives, so we will not prescribe them. The principle is what matters: lean on the book when it is liquid and informative; lean on the model when the book is thin or you have a specific, hedgeable reason to disagree. And when you do disagree, widen — a larger model-vs-market gap is itself a signal of uncertainty and should translate into a wider quoted spread, not a more aggressive bet.
The asymmetry between p and Δ
A crucial conceptual point closes this section. The probability p and the impact Δ are not equally knowable, and they are not where edge lives in the same way.
Δ (the impact) is the modeled quantity — how much the underlying reacts to the event. It is a property of the underlying's distribution and its relationship to the event, and a market maker who models it well has a real, defensible edge. It is also the quantity an external event market can never hand you: a market on "will the Fed cut?" tells you p, but it says nothing about how your underlying asset reacts to a cut. That reaction is Δ, and you have to model it.
p (the probability) should be sourced and hedged, not warehoused. It is often available more cleanly from an external market or the binary book itself, and the disagreements about p tend to be disguised directional bets (see above). The skilled posture is to take your p from the most reliable available source, hedge out the directional exposure it implies, and concentrate your risk-taking on Δ, where modeling actually pays.
In short: price the impact, source the probability. Confusing the two — warehousing a p disagreement as if it were a Δ edge — is the most common way to turn a market-making operation into an accidental directional fund.
Pricing the prediction binaries
Pricing a prediction binary is, in principle, the simplest pricing problem on the exchange: the fair price is the probability. Everything in the previous two sections feeds straight into it. But there are mechanics and subtleties worth making explicit.
The fair price is the probability
A YES prediction binary pays $1 if YES, $0 if NO. Its fair price is therefore p — the probability you have arrived at after the model-vs-market reasoning above. If your blended estimate is p = 0.48, you quote the YES binary around $0.48 and the NO binary around $0.52, then add a spread around each.
This makes the binary the cleanest expression of a probability view on the exchange. It carries no separate position margin — because its maximum loss is bounded and fully reflected up front (a binary can only move within the $0–$1 range, so the worst case is known and pre-accounted), you are not posting maintenance margin against it the way you would a perp. That makes binaries a capital-efficient instrument for expressing a probability directly, without the leverage and liquidation dynamics of the conditional perps.
Quoting both sides coherently
Quote the binaries as a pair anchored to one probability estimate. If you believe p = 0.48, the YES binary centers on $0.48 and the NO binary on $0.52, so that the two centers sum to $1 before spread. Quoting them from independent estimates invites a sniper to lift whichever side you have drifted on. The same defensive logic applies across the whole family: derive all your quotes — both binaries and both conditional perps — from one consistent (F, p, Δ) triple, so that no internal inconsistency exists across your own five prices for someone to pick off. Quoting coherently from a single triple is standard market-making practice and is how you stay on the right side of the arbitrage.
Spread: what drives how wide you quote
The spread around the binary's center is where you price your uncertainty and your adverse-selection risk. Conceptually it should widen with:
- Model-vs-market disagreement. The bigger the gap between your model p and the book p, the less sure you are, and the wider you should quote. Disagreement is uncertainty made visible.
- Volatility. A more volatile underlying means more event uncertainty and more chance of being run over; quote wider.
- Time to resolution. As an event approaches resolution, a binary's payoff sharpens toward $0 or $1 and the cost of being caught on the wrong side rises. A sensible spread schedule widens into the deadline rather than tightening, and it should account for the discrete jump risk of resolution itself — the price can gap from a smooth probability to a hard $0/$1 in a single step.
We deliberately do not give a calibrated spread formula here. The structure above — wider for disagreement, volatility, and proximity to resolution — is the reasoning; the constants are a function of your own risk tolerance and inventory and are not something to lift from a document.
Threshold events and the floor
When the event is a price threshold — "will the asset be above K at resolution?" — the conditional perps inherit a structural constraint that is worth understanding because it also disciplines the binary.
If the event resolves YES, the asset is by definition above K at resolution. So the YES conditional perp, which represents the asset's expected price given YES, should not trade below K — it is a conditional expectation taken over a region that lies entirely above the strike. Pricing the YES leg below K would be pricing an impossible state, and a trader who sees it there has an incentive to bid it back up. This is a fair-value, no-arbitrage property of how the leg trades: a YES leg printing under K is a sign the book is thin or lagging and an opportunity to lift it back toward fair value. The NO leg mirrors this from below.
This has a clean consequence for probability: for a threshold event, the conditional means are tied to where K sits relative to the forward, and the binary's probability and the conditional perps' levels move together as K, the forward, and volatility change. A threshold binary near $0.50 with the strike right at the forward is the symmetric case; as the strike moves away from the forward, the probability slides toward 0 or 1 and the conditional legs spread accordingly.
The impact is driven by correlation and volatility
The impact Δ — and therefore where the conditional perps sit relative to the forward, and the whole shape of the family around the binary's probability — is driven by how informative the event is about the underlying. Two ingredients govern it:
- Volatility (σ): how much the underlying can move over the horizon. A placid asset has a small Δ; a volatile one has a large Δ.
- Correlation (ρ): how much the underlying's moves co-vary with the event or conditioning asset. An event that says nothing about the underlying produces Δ ≈ 0 (the two conditional perps collapse onto the forward, and the conditioning is irrelevant); an event tightly coupled to the underlying produces a large Δ.
So the impact scales, roughly, with correlation times volatility times the price level. That is the structural story, and it is enough to reason correctly: a more volatile underlying or a more event-coupled one produces wider conditional perps and a more informative family. The precise coefficient that turns ρ and σ into a dollar Δ is a modeling choice — it depends on the distributional assumptions, the horizon, and the event type — and it is proprietary to each market maker. We give the relationship structurally, not as a ready-to-trade formula, because the calibration is regime-dependent and is exactly where a market maker's edge lives.
Calibration: what the books identify and how to estimate the inputs
Everything above rests on estimated quantities — a probability, a volatility, a correlation, an impact. Calibration is the process of pinning those estimates to data. The most important thing to understand about calibration here is which quantities the books identify cleanly and which you have to model.
What the books pin down, and what you model
A conditional-market family is a precise instrument for some quantities and a structural one for others.
The books pin down first moments and the event marginal. From the family you can read, cleanly, the event probability p (the binary) and the two conditional means CY and CN (the conditional perps). These are well-identified: the prices map directly onto a probability and two expected values.
The books identify the first moments rather than the full joint distribution or higher moments. The family tells you the average asset price in each branch, not the shape of the distribution within each branch — not the conditional variance, the skew, or the tail behavior. Two families with identical F, p, and Δ can have completely different distributions behind those numbers, so the shape within each branch is a modeling choice rather than a reading from five prices.
This is why the impact Δ — a first-moment, difference-of-means quantity — is the part you can model and validate against the book, while richer claims (precise tail probabilities, exact event-vol structure) are model assertions that sit above what the book reports.
Where calibration inputs come from — and how to handle their uncertainty
The inputs that feed a probability or an impact estimate are each estimated, and each carries its own uncertainty to manage:
Volatility (σ) can come from historical price data, from options-implied volatility, or from a blend. Each source has a different bias, and the "right" volatility is horizon-specific — the volatility over the next hour is not the volatility over the next week, and event days carry isolated jumps in volatility that a smooth historical estimate misses entirely.
Correlation (ρ) is estimated over some lookback window, and it is regime-dependent: it changes with regime, spikes in stress, and is sensitive to the window length you pick. A correlation estimate carries error bars; treat it as an estimate rather than a constant.
Event probability (p), when sourced from external markets, comes from prices that are already vigged — they include the external venue's own spread and margin — and aggregating across venues (a simple average, say) treats vigged prices as clean probabilities and ignores liquidity and recency. Account for these distortions rather than reading the external price straight through.
Event-specific reaction shocks — how much this underlying moves on this kind of event — are typically the input set most by judgment, since the relevant comparable history is thin (how many comparable events are there, really?). Treat these as assumptions with wide uncertainty, and size accordingly.
The posture this implies
The summary is that the impact axis can be modeled and validated against the book, while the probability axis is where disagreement concentrates and demands the most discipline. A model's Δ can be checked against the book's Δ and will often agree closely, because Δ is a first-moment quantity the family identifies cleanly. The probability p is where model and book diverge, and that divergence is usually a directional view in disguise.
This leads to a small number of durable rules of thumb, none of which require a calibrated constant:
Validate Δ against the book; treat agreement as confirmation and disagreement as a flag. When your modeled impact lines up with the book's spread, the impact axis is well-anchored and you can quote it with confidence. When it does not, find out why before sizing into it.
Source p, don't warehouse it. Take the probability from the most reliable available source, hedge out the directional exposure, and avoid betting your p against the market's p unless you have a specific, hedgeable reason. A p disagreement is rarely an edge; it is usually a naked underlying position.
Let disagreement widen your spread. The size of the gap between your model and the market is a measure of your own uncertainty. Quote wider when it is large; do not quote tighter and hope.
Re-estimate often, and refresh inputs across events. Volatility and correlation drift; a calibration from last week can be off today, especially across an event. Refresh σ and ρ on a cadence that matches how fast your regime moves.
Treat a calibrated constant as an estimate, not a law. The coefficients that turn σ and ρ into a dollar Δ, the weights that blend model and book probabilities, the reaction-shock magnitudes — these are starting points fit to the available data. Revisit them, stress-test them, and hold them loosely; this is where a market maker's modeling edge is built.
The framework — read the implied probability, cross-check it two ways, separate the impact you model from the probability you source, and price the binary as the probability plus a disagreement-aware spread — is durable, and it is what the conditional-market design is built to support. The numbers that fill it in are estimates that you refresh as the regime moves. Calibration is the ongoing work, and the discipline of this entire document is to keep its uncertainty in view and priced into your spreads rather than hidden inside a confident-looking formula.