Event-Linked Perpetual Futures and Options: A Risk-Theoretic Framework Bridging Prediction Markets and Derivatives Theory

Summary

Why prediction markets are missing a layer

Kalshi and Polymarket let people trade on whether an event will happen, and the price they settle on reads as a probability. By 2026 the two venues together carried hundreds of millions of dollars in monthly volume. Nearly all of it sits in one kind of contract. You buy a binary claim, you wait, and at settlement it pays one or zero.

Stock, currency and crypto traders have far more to work with. Futures, options with and without early exercise, margin that nets one position against another: all of that grew up over decades on top of the plain underlying. Event markets have none of it yet. Suppose a trader wants exposure to a regulatory outcome with no fixed date, or wants upside on an election while capping the loss. The binary contract cannot express either position.

This working paper sketches the missing layer from first principles. We treat the traded probability as an index and ask how a perpetual future, a European or American option and a clearing house's margin rule would work on it.

A price that cannot leave the unit interval

Why not borrow the equity toolkit wholesale? Because a probability is trapped between zero and one, and an ordinary diffusion would push it outside that range. We therefore model the logit of the probability, which can take any real value. The probability itself must be a martingale under the pricing measure, and that single requirement fixes the drift of the logit: zero at one half, changing sign on either side of it. Figure 1 shows simulated paths starting from p0 = 0.42 in both coordinates.

One convenience follows. Once the drift is removed, each period's logit move should be a standard Gaussian draw, so the same residuals drive the simulator and test it. On synthetic paths the Kolmogorov-Smirnov p-value is 0.515 (Table 1).

Funding, option prices and Greeks

A perpetual future never expires. Something has to keep its mark tied to the spot, and on crypto venues that job falls to a periodic funding payment between longs and shorts. We derive the rate in closed form. It is zero at probabilities of 0, 1/2 and 1, and Figure 2 draws the whole surface. Longs receive funding when p > 1/2, with an amplitude that scales as σ².

Options come next. For a European call we obtain a logit-normal analogue of the Black-Scholes formula, priced by Gauss-Hermite quadrature, with explicit Greeks. Figure 4 plots them for a six-month call struck at 0.50. Gamma is negative over most of the range, which looks odd beside equity options; it comes from the bounded support and has no counterpart in Black-Scholes. At p0 = 0.50 the closed-form delta is +0.50988 and gamma is -0.65970, and both agree with finite differences to about 10⁻⁵ (Table 3).

For early exercise we use a Cox-Ross-Rubinstein tree. Against a quadrature benchmark of 0.036164 it gives 0.036789 at 25 steps and 0.036375 at 100 (Table 4).

Margin for a book of event contracts

How much collateral should a clearing house hold against a mixed book? Futures exchanges often answer with SPAN. The method shocks the price and the volatility over a grid of scenarios, revalues every leg in each cell, and charges the worst loss it finds as initial margin. We build the same grid for event contracts, seven probability shocks by four volatility shocks, and Figure 6 shows it for a three-leg book.

Table 5 prices four books on one event with p0 = 0.55. A naked long future needs $10,000. The vertical bull call spread needs $10,016, slightly more. An iron condor, whose legs offset, needs $3,348. At the other end sits a delta-hedged short straddle at $11,420, above the condor even though it collects premium on both legs, because in this grid the volatility shocks outweigh the directional hedge.

Trying it on real bounded series

No public exchange lists perpetual event contracts yet, so version five runs the pricers and Greeks on daily stand-ins that also live between zero and one: VIX/100, MOVE/250, a rescaled SKEW and VVIX/200. These are mean-reverting fear gauges. Their residuals are fat-tailed, and the paper treats the exercise as a stress test. Still, the quadrature and tree pricers agree to numerical zero on this data. Across 400 rolling start dates, the closed-form Greeks explain 0.88 of the variance in one-day option profit and loss, with a slope of +0.855 against the 1.0 the model predicts.

What still needs work

One comparison does not come out as the text says it should. Table 6 sets quadrature prices against an 80k-path Monte Carlo and asks for z-statistics below 2 in absolute value. The first three rows report +3.22, +8.36 and +21.41, with Monte Carlo above quadrature every time. A separate 40k-path run in Figure 7 does agree within simulation error. We would reconcile those two before leaning on either.

Multi-event books are open too. When several probabilities move together the joint distribution matters, and the margin grid and the pricer both need to grow into more dimensions before they can handle it.

Who this is for: Investors and journalists following prediction markets, plus researchers in derivatives pricing.