Scalar Prediction Markets for Finance

Part 2: The Ladder, and What It Costs

Binary markets have a standard workaround. If one threshold cannot express how much, list many: a market for above 100 degrees, another for above 110, another for above 120. Traders assemble the view they want from the rungs.

The cost can be stated as a stylized allocation model, with its assumptions in the open. Suppose N traders each want to rest one unit of orders on the same scalar question, the question is split across k threshold contracts on independent order books, and each trader places their unit on the one contract nearest their view. Then the k books divide the N units between them: average resting interest per book is N over k, and the thinnest book holds at most that.

The scalar market on the same question runs two instruments, one per side of the pair, welded together by the arbitrage in Part 5: buying Calm and selling Stress express the same view, so the pair behaves as one venue for directional interest instead of k separate ones. Two honest limits on the model: it counts resting interest, not measured order book depth, and it applies to thresholds on independent books, which is how binary venues list them today. Designs that share one liquidity pool across many thresholds exist in the academic literature, and against those the comparison is different and comes later.

There is a second cost. A trader who thinks the outcome is underpriced has to pick a rung, which is a harder question than the one they came to answer.

Part 3: Banks Sell This Payoff Every Day

The payoff that answers how much already exists. It is a ramp. Pick a floor a and a cap b. Pay nothing at or below the floor, pay in full at or above the cap, and scale smoothly between:

Land a quarter of the way up the range and the payoff is a quarter. This shape is not new mathematics. Anyone who trades options will recognize it: buy a call struck at the floor, sell a call struck at the cap, and divide by the width, and the combined payoff is exactly f(X) at every landing. The construction is called a bull call spread, and desks have assembled it inside structured notes for decades, selling clients a slice of an index's move between two levels.

Banks proved the payoff but distributed it as a dealer product: priced by the desk that sells it, typically held to maturity, no way to take the other side, no exit except the bid the issuer chooses to show. Exchanges proved that continuous benchmark exposure can trade openly, but through margined futures and options, not through fully funded complementary shares anyone can issue. The payoff is proven twice over. What neither distributed openly is the container this piece describes; open versions have been attempted, and Part 9 covers that record.

The same payoff shape in two containers. The shape is decades old, and open containers have been tried before; Part 9 covers that record and what this one assembles differently.

Part 4: The Primitive, Defined

So the object worth defining is not a new payoff. It is the open, collateralized container for one, and every part of it has a named ancestor: the ramp comes from option desks, complementary minting against escrow comes from prediction market complete sets (the Iowa Electronic Markets sold $1 unit portfolios back and forth against the system in 1988, and several protocols tokenized the same primitive around 2020), settlement on a published print comes from listed futures, calibrated bounds come from exchange listing practice. What follows is not a new payoff class and not a new component. It is those parts chosen together, on purpose, for one benchmark at a time: published parameters, complementary minting, full collateralization, open two sided trading, benchmark settlement, all at once and each because the asset requires it. Three definitions carry the whole design.

Three properties follow immediately. The market is fully collateralized: whatever X does, the two sides of a pair redeem f(X) plus 1 − f(X), which is exactly the escrowed dollar. The venue does not set the trading price: it fixes the claim terms before issuance, the benchmark, the bounds, the calendar, the fallback rules, and leaves valuation entirely to the order book. And issuance is symmetric: anyone can mint, keep the side they believe, and sell the other, so neither side belongs structurally to a dealer. The bounds a and b bound the settlement mapping, not the trader's loss; either side's worst case is what it paid, because the claim is prepaid and nothing is borrowed.

The settlement arithmetic in a worked example, using an illustrative 0 to 40 teaching band. Each endpoint of a real window is averaged over five publication days, so no single print decides anything; here the official index change comes in at plus 10 percent:

A trader who bought Stress at 12 cents roughly doubled. A trader who bought Calm at 88 cents got back most of what they paid. Wrong predictions pay right ones, and the pot was fully funded before either side traded.

Why this shape and not another? Because among all payoffs bounded between $0 and $1 on the band, the ramp is the only one that is linear where it is live: every point of index movement inside the band is worth the same 1/(b − a) dollars, so nothing about the payoff itself favors one region of outcomes over another. Monotonicity means more stress pays the stress side more, without exception. And the ramp is the exact payoff of a bought and sold call pair, so where options trade on the same benchmark, it prices off instruments that already exist. And it has one more property that ties this piece together. Take the ladder from Part 2 and imagine it complete: a threshold contract at every level in the band, a continuum of rungs, each paying $1 if the number finishes above its level. Hold every rung at once, in equal weight. The portfolio pays:

The fraction of the band the number clears is exactly the ramp. Any finite ladder is a staircase approximation of this: four rungs pay in quarters, ten rungs in tenths, and the steps flatten into the ramp as the rungs multiply. A Stress share is the continuous limit of the whole ladder, aggregated into one claim on one book. The ladder and the ramp were never rivals: the scalar market lists, as a single instrument, what the threshold approach can only approximate in fragments.

Part 5: Why the Two Prices Sum to a Dollar

A market like this needs its two prices to stay coherent: Stress at 30 cents should mean Calm near 70. Most designs buy that coherence with machinery, an oracle, a peg, or a subsidized market maker whose cost function keeps prices consistent. The scalar market gets it free, from the escrow itself, and the argument fits in a few lines.

The proof is the trade. If the two bids together exceed a dollar plus fees, mint a pair for exactly one dollar and sell both sides into those bids; the difference is free money, so someone takes it until the bids no longer exceed the bound. If the two asks together fall below a dollar minus fees, buy both sides and merge the pair back into its escrowed dollar; again free money, again taken until it is gone. The two prices cannot drift far from summing to $1, and nothing enforces this but greed.

This is the same economics that keeps an exchange traded fund near its asset value through creation and redemption, with one difference: here the mechanism is open to every participant, not to a licensed few. Two consequences matter for traders. First, because the pair sums to about a dollar, the Stress price summarizes the market's current valuation of the settlement payoff; a traded price folds in risk premia and the cost of capital, so it is a valuation, not a pure probability. Second, ignoring fees and the time value of the escrowed dollar, each trader's entry price defines their own breakeven finish:

Buy Stress at 12 cents in a 0 to 40 band and you profit wherever the number lands above 4.8. The contract itself has no line where one side wins; land mid band and the dollar divides mid band. The only borders in this market are the ones traders draw with their own entries.

The same arbitrage has a second consequence, easy to miss and worth stating as its own claim: mint and merge change who can be your counterparty. In a market of threshold contracts on independent books, the set of parties who can execute against your sell order is exactly the set of buyers of your contract. Here that set is strictly larger:

The mirror holds on the way in. When the two sides' bids sum to more than a dollar, anyone can mint a fresh pair for exactly one dollar and sell into both, pocketing the excess: overlapping demand becomes new supply without waiting for a holder to sell. Merge turns opposite side liquidity into exit liquidity; mint turns overlapping demand into entry liquidity. They are one mechanism run in opposite directions, and together they are why a two claim market behaves as a single venue rather than two adjacent ones.

Part 6: A Few Payoffs, Not Every Belief

There is a school of market design that tries to let traders express every possible belief about a number: every interval, every threshold, the full probability curve. The academic work here is elegant, and it solves a real problem, eliciting a complete distribution. But expressiveness and tradability pull in opposite directions. Each additional claim splits attention and order flow, adds a hedging problem for professional liquidity providers, and a market where a thousand views are expressible tends to end up with a reliable price for none of them.

This is not opinion; it is one of the strongest regularities in market structure. In credit, standardized index contracts account for 90 percent or more of all default swap trading activity measured by transaction count (ISDA, November 2024), and within that, five benchmark indices from the CDX and iTraxx families carried 88.7 percent of index transactions in the first half of 2024; meanwhile, of the 725 single name reference entities that traded at all in the second quarter of 2024, only 26, or 3.6 percent, averaged ten or more transactions a day (same ISDA report). By notional outstanding rather than activity, index and multi name contracts are roughly half the market, so the concentration claim here is specifically about where trading happens. Treasury futures, SOFR futures, S&P 500 futures, WTI crude: in market after market, a handful of benchmark contracts with fixed, published, identical terms absorb the flow, and the expressive bespoke alternatives live at the periphery. Terms that are the same for everyone make positions easiest to price, hedge, and exit, and that is where the trading goes.

The cleanest way to say it: prediction markets optimize for information, financial markets optimize for liquidity. Both are legitimate objectives. They lead to different designs.

The scalar market takes that side of the tradeoff deliberately. It publishes a small family of standardized payoff functions over one official number, starting with the simplest pair, and gives up something real in exchange: a trader with a view about the shape of the distribution, not its location, has no instrument here. Sharper shapes can list later on the same settlement engine, the same escrow rule, and the same published number, because every claim is parameterized by its floor and cap; adding one is a listing, not a redesign. The discipline stays constant: few claims, one book per question, every rule public before a single trade.

The goal is to standardize the few payoffs the market most wants to trade, and make them liquid, not to reconstruct every belief it could hold.

Part 7: Who Shows Up, and Why

A mechanism is not a market until someone wants each seat. Four seats, four reasons.

The stress buyer holds something that hurts when the benchmark lands high: a credit portfolio, a lending book, a business whose revenue thins in a squeeze. What they want is proportional coverage, paid in proportion to how bad the landing is, without margin calls arriving mid crisis. A prepaid ramp is that shape; a binary threshold pays them nothing for a landing just below the line and everything for one just above it, which is not how their losses behave.

The calm seller believes the fear is overpriced. Selling that belief in the dealer market means margin accounts and documentation; here it means minting a pair for a dollar, selling the stress side at the fear price, and holding a claim that climbs toward a dollar as the window stays quiet. Their worst case is fixed at what they put in.

The market maker does not need an opinion. The pair has the exact shape of a normalized call spread, so where liquid options trade on the settlement benchmark itself, a dealer can replicate the payoff outright; where they do not, a dealer can hedge the principal risk with related instruments in the professional market, keeping basis, roll, and liquidity risk on their own book and charging for it in the spread. Hedgeability is what makes quoting rational for a professional with no view; it narrows the spread, it does not guarantee a tight book, and a new market may still pay its first makers to show up. What the collateral does guarantee is narrower and absolute: every quoted dollar of payout already exists in escrow, so no maker is ever pricing anyone's credit.

And the trader with a view, the seat prediction markets proved exists, gets the cleanest expression of how much: one price to disagree with, a payout that scales with how right they are, and a worst case equal to their ticket.

Part 8: Choosing the Floor and Cap From Data

The bounds are not decoration; badly placed, they break the market. Here the machine meets its first real number. The first listed market is on credit stress in US high yield, the market for bonds of riskier corporate borrowers: an independent administrator publishes a stress index daily, and the settlement number is its percentage change over a fixed two year window. To place the band we measured a proxy for that number over every rolling two year window from 2007 to 2026, 208 windows in all, spanning the 2008 crisis, the 2015 energy bust, and the 2020 shock.

Proxy construction, stated in full so the table is reproducible: total return (dividend adjusted) of the iShares iBoxx High Yield Corporate Bond ETF (HYG) divided by the total return of the iShares 3 to 7 Year Treasury Bond ETF (IEI) as the duration comparable Treasury leg, compounded over each window, minus one, then sign inverted so the number rises with credit stress. Windows are 24 months, starting each month from May 2007, ending July 2026: 208 windows, heavily overlapping, so the table describes historical coverage, not 208 independent draws, and the percentiles carry no statistical confidence claim. Data: Yahoo Finance adjusted closes, retrieved July 2026. The listed contract reruns this study on the official settlement series.

Two facts jump out. Quiet windows drift negative: the median finish is minus 6.4 percent, because in most two year periods risky bonds out earn Treasuries and the stress measure declines. And the stress tail is long and one sided: the worst window on record finished 33 points above the median.

The drift finding kills the intuitive design. A floor at zero would have pinned 81 percent of historical windows at the floor, settling Calm at its maximum in ordinary times and leaving the market with nothing to trade about for months. The design rule that falls out: on a benchmark that drifts negative in quiet times, the floor must sit well below zero. Against this distribution the band is minus 15 to plus 30, published with the listing. That floor pins about 16 percent of history; the cap has never been touched, and the worst window on record would settle Stress at $0.92, so the market keeps resolution even for an outcome worse than anything yet observed. Bounds are set by a published calibration policy, a mechanical rule applied to the then current data, and frozen before the window opens, so no one holds discretion over the payoff after trading begins.

Part 9: This Has Been Tried, and What Failed Was Not the Payoff

Scalar claims on open venues are not a new idea, and honesty about the record is the strongest argument for the design. Augur v2 (2020) listed scalar markets with creator chosen bounds, minted as complete sets against escrowed collateral. CME listed CPI futures in 2004. Investment banks ran auction based economic derivatives on inflation and payroll prints in the mid 2000s. None found durable volume. If the payoff container were the product, this design would already be refuted.

But look at what each attempt was missing. Augur resolved markets through an economically bonded reporting and dispute oracle: a designated reporter filed a value backed by stake, token holders could challenge it through escalating bonded rounds, and the final backstop was a sixty day fork in which holders migrated tokens to the outcome they backed. Every scalar market also carried a third outcome: if reporters deemed the market ambiguous, it paid the midpoint of the band regardless of where the number landed. Bounds were set at each creator's discretion with no calibration policy, so nothing prevented the dead market that a floor at zero produces on a drifting benchmark. And the whitepaper says nothing at all about market makers, liquidity, or hedging: nobody was paid to quote, and the listed underlyings offered professionals no way to lay off inventory. The auction products settled cleanly but ran as episodic institutional auctions with no continuous secondary market and no open access.

None of this is a failure of intelligence on Augur's part, and this design owes it a real debt: complementary shares minted against escrow, the mechanism Part 4 builds on, worked exactly as its designers intended. Augur optimized for a different goal. It set out to be a general purpose, decentralized truth machine: anyone creates any market, no operator, outcomes decided by token vote. Every gap above follows from that goal, because a protocol that refuses a trusted data source must vote, a protocol where anyone types any question must handle ambiguity, and a protocol with no listing authority cannot calibrate bounds or sign market makers. Generality was the product, and no individual market got built for.

This design makes the opposite choice: the whole stack is assembled with intent for one specific asset. The floor sits at minus 15 because this benchmark drifts minus 6.4 percent in quiet windows, a fact only a calibration study of this index could surface. Settlement reads a print because this administrator publishes one daily under a public methodology. The venue is an order book with committed market makers because this exposure hedges in the professional credit market. Each piece is load bearing, and each was chosen because the asset requires it: the administrator's published observation is dispositive with no token holder override, publication failures are handled by a deterministic fallback ladder published before trading rather than a discretionary midpoint, bounds freeze by published policy before trading opens, the benchmark is hedgeable, and makers are a launch condition rather than a hope. The engine generalizes; the listing work does not, and is redone deliberately for every new number. These prior systems are the closest prior art, and their record is evidence that scalar settlement and complete set issuance are not sufficient by themselves. Their outcomes motivate, but do not prove, the importance of benchmark selection, objective settlement, calibrated bounds, external hedgeability, and committed liquidity: too much else differed, from gas costs to distribution, for any single cause to be isolated. This design tests whether assembling those conditions together changes the result. That is the falsifiable bet stated below.

The Case

Credit is the right first number. It is continuous, it is what the professional market already trades in size, and the exposure can be hedged in that market, which is what gives market makers a reason to quote the book whether or not they agree with the price.

But the machine does not care that the first number is credit. Inflation prints, policy rates, benchmark returns: any number an authority publishes on a schedule can carry a market like this, through the same three definitions, the same escrow bound, the same band calibration.

Financial markets already trade continuous outcomes, but largely through bespoke bilateral contracts and dealer issued notes. Prediction markets demonstrated that public markets can aggregate beliefs about future events. Between them sits a missing category: standardized continuous claims, traded on open venues, settled against published numerical benchmarks. Whether such markets succeed is an empirical question. The structure described here is the framework against which that question can be tested.

What This Piece Claims, and What It Does Not

Claimed: a definition of the standardized continuous claim and the scalar market that lists it; the escrow arbitrage bound that keeps the pair coherent without an oracle or a subsidy; the identity showing one ramp share is the continuous limit of the equal weighted threshold ladder, which any finite ladder only approximates in staircase steps, so the scalar market lists as one claim what the threshold approach fragments; the separation of payoff innovation, which is old, from market innovation, which is the contribution; and a calibration method with one nonobvious rule, the floor below zero, derived from data.

Not claimed: novelty of the payoff container, or that shipping it is sufficient. Prior open scalar attempts shipped the container and failed for the structural reasons Part 9 names; the claim here is the combination, and it can fail too, which is why the launch conditions are stated as falsifiable. Also not claimed: that binary markets are broken as a class. The flow division argument is scoped to thresholds traded on independent books; academic designs exist that share liquidity across every interval under one subsidized market maker, and against those the relevant comparison is different, hedgeability of the listed claim and who finances the machinery, not depth per book. The ramp is a bull call spread; mint and merge descend from prediction market complete sets; print settlement is how futures have settled for a century. Saying so plainly is the point: the contribution is the assembly, not any part.

What is a scalar prediction market?

A scalar prediction market settles on the number itself, not on a yes or no outcome. Where a binary contract pays a fixed amount for crossing a threshold, a scalar contract pays in proportion to where the number lands. Inflation prints a number, rates move by amounts, and a scalar market prices how much.

How is a scalar market different from a binary prediction market?

A binary market pays for crossing a line, so two outcomes that are nearly identical can settle at opposite ends. A scalar market pays along a range, so a result close to another settles close to it. Almost nothing in finance is a yes or no question.

What is a standardized continuous claim?

A standardized continuous claim is a published payoff function on a published benchmark, fully collateralized and traded in the open. It turns a benchmark methodology into one canonical recurring series instead of a market created ad hoc, with bounds set by a precommitted calibration rule.

How does a scalar prediction market settle?

Settlement reads a published benchmark through a fixed source hierarchy, with no vote and no discretionary outcome. The payoff function is published before trading opens, so the contract resolves by reading a number and applying a rule. Every position is fully collateralized, so settlement pays from posted collateral, not from a counterparty promise.

Can a scalar market be built out of binary contracts?

A ladder of binary contracts can approximate a scalar payoff, but it divides the order book. If traders spread across several threshold contracts, average resting interest per book is the total divided by the number of contracts, and the thinnest book holds no more than that average.

A market built to this design lists at rava.money.