Scalar tokens

Scalar tokens give linear long or short exposure to an asset’s price inside a single branch of a conditional market.

They are created from conditional USDC balances in that branch and pay out between $0 and $1 of conditional USDC per long/short token at resolution depending on the asset's price.

Long and short tokens in a branch

Within each branch, you can turn conditional USDC into two types of scalar tokens:

• Long tokens, which gain value when the asset’s settlement price in that branch is higher.

• Short tokens, which gain value when the asset’s settlement price in that branch is lower.

These tokens always come in matched pairs when they are minted.

From 1 unit of conditional USDC in a branch you can mint 1 long token and 1 short token in that branch.

At any settlement price, the combined payoff of that long–short pair is always 1 USDC.

This ensures the position is fully collateralized across all possible settlement prices within the market's upper and lower price bounds.

Scalar bounds and payoffs

Each branch has a scalar lower bound $\text{lowerBoundPrice}$ and upper bound $\text{upperBoundPrice}$ defined in the asset’s price units, for example 60,000 USD and 110,000 USD for BTCUSD.

The width of the bounds is:

$\text{boundsWidth} = \text{upperBoundPrice} - \text{lowerBoundPrice}$

At settlement, the oracle reports the settlement price $\text{settlementPrice}$ of the asset in the realized branch and the payoffs per token are:

• Long payoff:

  • $\text{Long payoff} = \max\left(0, \min\left(1, \frac{\text{settlementPrice} - \text{lowerBoundPrice}}{\text{boundsWidth}}\right)\right) \text{ USDC}$

• Short payoff:

  • $\text{Short payoff} = \max\left(0, \min\left(1, \frac{\text{upperBoundPrice} - \text{settlementPrice}}{\text{boundsWidth}}\right)\right) \text{ USDC}$

The payoffs are clamped at the bounds.

If settlement lands at or below the lower bound, long tokens in that branch pay 0 and short tokens pay 1 USDC.

If settlement lands at or above the upper bound, long tokens pay 1 USDC and short tokens pay 0.

For traders (using $\text{tokenPurchasePrice}$ to represent the price between $0 and $1 at which the trader purchased the short/long token) this means:

• The maximum gain from holding a long/short token is $1 - \text{tokenPurchasePrice}$ USDC and the worst case loss is $\text{tokenPurchasePrice}$ USDC.

The risk is limited and fully covered by the conditional USDC that was used to mint the long–short pair.

How bounds are chosen

Bounds are the same across all branches for a given event–asset pair and are fixed when the market is created.

They are chosen by analysing the asset’s historical price movements over similar time frame to the market’s duration.

The goal is to pick bounds such that the probability that the asset’s price will move outside them during the market’s lifetime is low, on the order of only a few percent.

The expected impact of the event itself is also taken into account, so that the bounds comfortably cover the range of plausible prices across all branches while still remaining tight enough to be useful.

The width of the bounds directly affects how much leverage scalar tokens provide.

Wider bounds give more room for the price to move, so each long or short token’s payoff moves more slowly with the price, which degrades capital efficiency and hence reduces leverage.

Narrower bounds compress the price range, which increases the sensitivity of long and short token payoffs to price changes and raises leverage, but if bounds are too narrow the asset price would often end up outside them, leading to the market settling at the clamped endpoints, degrading the trading experience.

In practice, bounds are chosen to balance these two forces:

• Wide enough that most markets settle inside the range.

• Narrow enough that scalar tokens offer sufficient leverage.

Scalar tokens and asset exposure

Scalar tokens determine exposure to the underlying asset in a branch. The wider the bounds, the smaller the exposure per scalar (long or short) token.

If the lower bound BTCUSD price is $\text{lowerBoundPrice}$ and the upper bound BTCUSD price is $\text{upperBoundPrice}$ USD, then 1 long token moves by:

$\frac{1}{\text{boundsWidth}} \text{ USDC}$

for every 1 USD change in the asset’s price inside the bounds.

To obtain 1 unit of asset exposure in a branch (for example, 1 BTC of conditional exposure), you need to hold:

$\text{boundsWidth}$

long tokens or short tokens in that branch.

For example, if a BTCUSD conditional has bounds at 60,000 USD and 110,000 USD, then:

• The bounds are 50,000 USD wide.

• A change of 1 USD in the BTCUSD conditional price changes each long token’s value by $\frac{1}{50{,}000}$ USDC.

• Holding 50,000 long tokens in that branch gives 1 BTC of conditional long exposure.

• Holding 50,000 short tokens in that branch gives 1 BTC of conditional short exposure.

This scalar‑to‑asset ratio is constant throughout the life of the market, because the bounds never change. Hence holding 50,000 long tokens will give you a consistent 1 BTC exposure in that branch, until the market resolves.

It tells you how many long or short tokens you need to hold per unit of asset exposure in that branch.

From conditional USDC to long or short exposure

When you open a position in a branch, your conditional USDC in that branch is transformed into long and short tokens and then into net long or net short exposure:

1. Mint: Your conditional USDC mints matched long and short scalar tokens in the branch.

2. Trade: To go long, you sell the short tokens you just minted to buy more long tokens, so you end up holding only long tokens. To go short, you do the opposite: you sell the long tokens you just minted to buy more short tokens, so you end up holding only short tokens.

3. Position: Your net holdings of long or short tokens define how many units of the asset you are long or short in that branch, based on the scalar-to-asset ratio.

Because every long or short token comes from conditional USDC and every long–short pair always pays 1 USDC in total, your exposure remains fully collateralized without requiring liquidations.