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Pricing II — Spread, Correlation & Volatility
This is the second pricing doc. The first established the decomposition: a conditional-market family reduces to three numbers — the forward F, the probability p, and the impact Δ — and every leg's fair price is a deterministic function of those three. This doc zooms in on the impact, Δ, which is the conditional spread between the two conditional perpetuals. It is the piece of pricing that the underlying's oracle never gives you and that no external event market ever quotes, so it is worth understanding on its own terms.
The goal here is to explain what the conditional spread represents, why correlation and volatility are the forces that set it, how to reason about the fair value of the two conditional legs, and how to apply that reasoning in practice. The framework is a way to think about conditional pricing from first principles — structural intuition you build a quoting view on top of, rather than a single number you read off.
Symbols, defined once and reused: F is the underlying forward (the price shared by every book in the family). p is the probability the event resolves YES (the price of the YES prediction binary, in $0–$1). CY and CN are the prices of the conditional perpetual (YES) and conditional perpetual (NO). Δ is the impact, Δ = CY − CN. σ is the volatility of the underlying over the relevant horizon. ρ is the correlation between the underlying and whatever drives the event. K is a strike or threshold. All worked numbers below are illustrative.
1. The conditional spread (CY − CN): what it represents
A conditional perpetual pays out only if its branch wins. The conditional perp (YES) tracks the underlying in the world where the event resolves YES; if the event resolves NO instead, that position is voided — its profit-and-loss is zero and its margin is returned (net of funding and fees accrued), not settled to $0. The conditional perp (NO) is the mirror. So the two conditional perps are prices on two different futures: "the asset, given YES" and "the asset, given NO."
Each conditional perp is fair at its own conditional mean, not at the underlying forward. The fair price of the YES leg is the expected price of the underlying conditional on YES, written E[underlying | YES]; the fair price of the NO leg is E[underlying | NO]. The forward F is the unconditional expectation — the probability-weighted blend of the two:
F = p·CY + (1 − p)·CN
This is just the law of total expectation: the average of the asset across all outcomes equals the YES-conditional average weighted by how likely YES is, plus the NO-conditional average weighted by how likely NO is. It is a relationship that holds between fair prices. The exchange does not enforce it; it holds because if it were violated, the mispricing could be traded away (long one synthetic, short the other), so in a liquid family it tends to hold.
The conditional spread is the gap between those two conditional means:
Δ = CY − CN = E[underlying | YES] − E[underlying | NO]
In plain terms: Δ is how much the underlying's expected price differs between the two outcomes. It is the size of the move the event is expected to cause in the underlying. If the event resolving YES would push the asset up and NO would leave it lower, Δ is positive and large. If the event tells you essentially nothing about where the asset will be, Δ is near zero.
Two consequences are worth internalizing immediately:
An event with no impact is just a prediction market. If Δ = 0, then CY = CN = F. The two conditional perps collapse onto the underlying forward and carry no information beyond it. All that remains is the prediction binary — a pure bet on the event probability p. The conditional perps only earn their keep when the event actually moves the asset.
The sign of Δ is the sign of the co-movement. A positive Δ says YES-states carry a higher asset price (event and asset move together); a negative Δ says the asset is expected to be lower in the YES world. The conditional-perp pair is, quite literally, a market on the direction and magnitude of the asset's reaction to the event.
Given F, p, and Δ, the two leg prices are pinned uniquely. Writing the conditional perps as offsets from the forward and requiring the blend to equal F forces an asymmetric split:
CY = F + (1 − p)·Δ and CN = F − p·Δ
The asymmetry is structural, not a modeling choice. The rarer branch sits further from F. If YES is unlikely (p small), the YES leg sits far above the forward — it has to, because it rarely pays, so when it does pay it must carry the whole conditional move. The likely branch barely moves off F. A short, illustrative example: take F = $100, Δ = $10, and p = 0.2 (YES unlikely). Then CY = 100 + 0.8·10 = $108 and CN = 100 − 0.2·10 = $98. The unlikely YES leg is $8 above the forward; the likely NO leg is only $2 below it. Flip to p = 0.8 and the picture reverses: CY = $102, CN = $92. The conditional perps lever the impact by the inverse rarity of their own branch.
Reading the same relationship backwards gives the implied probability from the conditional perps:
p (implied) = (F − CN) / (CY − CN) = (F − CN) / Δ
This is a useful internal cross-check: the probability you back out of the two conditional perps should agree with the price of the YES prediction binary. When they disagree, it points to one of your three estimates moving ahead of the others — for a market maker, that divergence is a signal to act on, not a fact about the world.
2. Why correlation and volatility drive the spread
Here is the structural heart of the doc: the conditional spread Δ is, mathematically, the statistical dependence between the event and the asset. F and p are pure marginals — F is the average of the asset, p is the average of the event — and neither contains any information about whether the two are related. All of the coupling between event and asset lives in Δ. This is not an analogy; it falls straight out of the definitions.
Write the event as a 0/1 indicator (1 if YES, 0 if NO). The covariance between the asset's settlement price and that indicator works out to:
Cov(underlying, event) = p·(1 − p)·Δ
Equivalently, Δ is that covariance normalized by the event's own variance, p·(1 − p):
Δ = Cov(underlying, event) / Var(event)
That second form is the regression slope of the asset on the event indicator — the discrete-event analog of a factor beta. Δ is, precisely, how much the asset's expected price changes when the event flips from NO to YES. Once you see Δ this way, why correlation and volatility set it becomes obvious: a covariance is built from how much each thing moves (volatility) and how much they move together (correlation).
To make this concrete, consider the common case where the event lives on one asset (the "driver") and the conditional perp is on a different underlying that reacts to it — for example, an event defined on a major asset's direction, with the conditional perp written on a correlated alt-coin. Model the underlying's move as its beta to the driver's move; the underlying's reaction to the event is its sensitivity times the driver's move. Working that through a standard joint-normal model of the two assets' returns yields a clean qualitative result:
Δ is proportional to ρ · σ · (asset price)
where ρ is the correlation between the underlying and the driver, and σ is the underlying's own volatility over the event horizon. Three things in that proportionality matter:
The driver's own volatility drops out. This is the single most important and most counterintuitive point, and it is the source of a recurring mistake. The size of the move in the asset that hosts the event does not enter the spread on the conditional perp's underlying. What survives is the underlying's own volatility and its correlation to the driver. The intuition: the event is a coin-flip-like threshold on the driver (did it cross the line or not?). Once you condition on which side of the threshold the driver landed, the magnitude of that crossing is averaged out; what remains is how strongly your asset tracks the driver (ρ) and how much your asset moves on its own (σ).
Correlation scales the spread linearly. If the underlying is perfectly correlated with the driver, the event's information transmits fully and Δ is at its largest for a given volatility. If they are uncorrelated, ρ = 0 and Δ = 0 — the event tells you nothing about the asset, and you are back to a pure prediction market. Correlation is the channel; volatility is the amplitude.
Volatility sets the absolute size. A more volatile underlying has a wider conditional gap, because there is simply more room for the two conditional worlds to diverge. Double the underlying's horizon volatility and you roughly double the spread, all else equal.
There is also a constant of proportionality in that relationship — a factor that depends on where the event's threshold sits relative to its typical outcome. For a symmetric, roughly even-odds event (the threshold sits at the middle of the driver's likely range), that constant takes one fixed value. As the threshold moves out into a tail — a rarer YES — the constant grows: a rarer event, when it does fire, implies a larger conditional move, because crossing a far threshold is more informative than crossing a near one. We deliberately do not publish the calibrated value of that constant or the exact functional form across thresholds. The load-bearing intuition is the shape: spread grows with correlation, grows with volatility, and grows as the event becomes more lopsided.
Why this is where the edge lives. The probability p is quoted by external prediction markets and is the same number everyone can see. The spread Δ is not quoted anywhere — there is no market that prices "the covariance of this asset with this event." It has to be modeled from the underlying's volatility and correlation. That is why a market maker's durable edge in conditional markets sits on the Δ axis: you can build a tighter, more accurate conditional spread than the rest of the market because you are the one estimating the dependence from first principles. The probability is a commodity to be sourced and hedged; the spread is the thing you actually have a view on.
3. Reasoning about the fair value of the conditional legs
Putting the pieces together gives a clean way to reason about where the two conditional perps should trade, without needing any number you can't observe or model:
- Source the forward, F. This is the underlying perpetual's price. It is shared by every book in the family and is never your edge — it is free and visible to everyone.
- Source the probability, p. Read it off the YES prediction binary, or off an external event market, or model it for a threshold event. This is the speculative axis; quote it humbly (more on this below).
- Model the spread, Δ. Estimate it from the underlying's volatility and its correlation to the event driver, using the structural relationship above. This is the modeled piece and the place a view actually expresses itself.
- Place the legs. CY = F + (1 − p)·Δ and CN = F − p·Δ. The binaries sit at p and 1 − p.
The reason to build all four conditional legs from one triple (F, p, Δ) rather than quoting each book independently is defensive. If you quote the two conditional perps both at the forward F — ignoring the conditional structure — a sharper counterparty simply trades the leg where your quote is biased and leaves you holding a position that is mispriced by construction. Quoting each leg at its own conditional mean is the selection-proof equilibrium: there is no leg a counterparty can pick off, because every leg is individually fair. Coherence across your own quotes is not a nicety; it is what stops you from being arbitraged by your own family.
A special case worth flagging is the threshold event — where the event is defined on the underlying itself, such as "the asset closes above K." Here the same asset distribution generates all three of F, p, and Δ at once: p is the probability the asset finishes above K, and the two conditional perps are the means of the asset given it finished above (or below) K. The economically important fact is that the fair value of the YES leg of an "above K" event sits above K. If "the asset is above K" resolves YES, then by definition the settlement price is above K, so the conditional mean of that branch — its fair price — is also above K, regardless of how unlikely the event is. The strike acts as an economic floor on the YES leg's fair value — a no-arbitrage property of how the leg should trade. Quote the YES leg above K and the structure is self-consistent; a pricing approach that ignores this and tries to value the YES leg at today's (sub-K) price is pricing an impossible state. The NO leg, by contrast, can sit well below the forward, since the "below K" world pulls its conditional mean down.
One discipline runs through all of this: the spread is modeled, the probability is sourced and hedged. Empirically, when you compare a well-built spread model against a liquid family's order book, the spread tends to agree closely — the Δ axis is structural and stable. The disagreement, when it exists, concentrates almost entirely on p. And a p disagreement is dangerous: warehousing a difference between your probability estimate and the market's is, on a threshold family, a disguised naked bet on the underlying's direction, not a hedged spread. The robust posture is to blend your probability estimate toward the liquid market's, widen your quotes when the two disagree, and never lean on the p gap. The spread is where you have an edge; the probability is where you have an exposure you should mostly neutralize.
4. Reading the model correctly — where the linear picture stops
The relationship "spread ≈ correlation × volatility" is a first-order, linear-Gaussian picture. It is the right mental model and a starting point for a quote, not a finished trading recipe. Knowing where the linear approximation stops applying is what turns it from a guide into a precise tool.
The marks pin down the centers, and you supply the distribution. The books of a family identify the event probability and the two conditional means — first moments. They describe the center of each conditional world rather than its shape: how wide it is, how skewed, how fat its tails. Two different futures can share identical leg prices when their conditional means match. This is exactly why the quote and the margin play different roles. Your exposure when you hold a conditional perp is to the full tail of the winning branch at settlement, not just to the center Δ. The fair quote uses the conditional mean; the width of your quote and the margin you hold reflect the conditional tail. A symmetric Δ over asymmetric tails describes one leg's center well and its tail less well — so the tail is where the width and margin do their work.
Volatility is horizon-specific. The "σ" in the relationship is the underlying's volatility over the exact horizon to resolution, and that horizon is often short and dominated by the event itself. Total variance over an event window splits into ordinary day-to-day diffusion plus the variance the event itself injects:
Var(underlying) = (within-branch diffusion) + p·(1 − p)·Δ²
The second term is the event's own contribution to variance, and it peaks when the event is a coin flip (p near ½) and vanishes as the event becomes a near-certainty (p near 0 or 1). The event-horizon volatility — not a generic calendar volatility — is the one that belongs in the relationship, together with the event's variance injection. The spread is most load-bearing exactly when the event is most uncertain.
Correlation is regime-dependent, and the right posture is asymmetric. The clean ρ·σ relationship uses a single correlation. In stressed markets, cross-asset correlations move toward 1 — assets that normally move somewhat together move in closer lockstep — so a down-move event can make the underlying gap more than the calm-market spread implies. The disciplined response is asymmetric: keep the fair quote at the calm-market spread, and set margin and risk limits off the stressed correlation. Price the center on the prevailing correlation; margin the tail on the stressed one.
The threshold constant moves with the threshold. The proportionality constant between Δ and ρ·σ·price depends on where the event's threshold sits. The symmetric, even-odds value applies to a roughly 50/50 event; for lopsided events the constant grows, since a rare-YES event implies a larger conditional move when it fires. The constant is a function of the threshold, so it is computed per event rather than held fixed across the book.
Linearity is a small-move approximation. The factor relationship is first-order in returns. For small moves over short horizons it is accurate; for large moves the true relationship between the assets is curved rather than straight, so the linear term is a first cut and the curvature is added on top. Events expected to cause big jumps are exactly the ones where the curvature carries more of the weight.
The probability axis carries direction on threshold families. As noted above, on a self-referential threshold family the event is a function of the underlying, so a view on the event probability is a view on the underlying's drift. The decomposition presents two clean axes (p and Δ); for threshold events they are coupled through the same distribution. A "probability edge" on such a family is frequently a directional bet, and treating it as one keeps the two axes honest.
5. Using the framework in practice
Everything in this doc is meant to build judgment and a quoting view, not to hand over a single number. The relationships are real and the structural intuitions are durable; the constants, the volatility and correlation inputs, and the way they are combined are estimated per event and per regime. Use the framework the way it is meant to be used:
The qualitative laws are reliable. Spread grows with correlation. Spread grows with volatility. Spread grows as the event gets lopsided. Spread is zero when event and asset are independent. The sign of the spread is the sign of the co-movement. The rarer branch sits further from the forward. These you can lean on.
The quantitative constants are yours to calibrate. The exact proportionality factor, its variation across thresholds, the precise volatility horizon and source, the correlation estimation window — these are calibration choices, and they are where a market maker's own modeling does its work. We deliberately do not publish them: a single fixed formula presented as universal would misprice tails, lopsided events, and stressed correlations, and the structure is better served by estimating these inputs per market.
Estimate the inputs honestly and quote the uncertainty. The spread is only as good as the volatility and correlation feeding it, and both vary with regime. When your inputs are less certain — a thin sample, a shifting regime, an event with no clean precedent — the disciplined response is to widen your quotes and size down rather than lean on a single point estimate. Width should scale with how much you don't know.
Reconcile, don't override. Where a liquid book exists, treat it as information. If your modeled spread disagrees sharply with the market's, that is a reason to widen and investigate, not automatically a reason to trade against it. And keep the two axes in their lanes: model the spread (your edge), source and hedge the probability (your exposure), take the forward for free.
Margin against the full branch, not just the center. The fair price uses the center; the risk uses the tail. The conditional structure carries within-branch dispersion, correlation that rises in stress, and the branch you are not currently in. The disciplined operator prices fairly off the center and margins off the worst plausible branch — because the quantities that determine survival are exactly the ones beyond the center the linear picture describes.
The conditional spread is the most interesting object in conditional-market pricing: it is where the real structure lives and where the genuine edge sits. Understand the forces that drive it — correlation and volatility — and quote it with the discipline the structure rewards.