Most quants who walk onto a prediction market for the first time bring an options-pricing toolbox and try to use it. I did this myself for about a month. The toolbox is mostly the wrong shape, and the way I figured out which parts of it were salvageable was by going through Black-Scholes term by term and asking "does this concept have an analog on a binary contract that resolves to {0, 1}." Some terms ported cleanly. Some collapsed to nothing. A few translated into entirely new concepts that the prediction-market world has not given proper names to yet.
This page is the translation table I wish I had when I started.
What Black-Scholes Actually Says
Black-Scholes prices a European call option on a non-dividend-paying stock as:
C = S·N(d₁) - K·e^(-r·T)·N(d₂)
Where d₁ and d₂ depend on S (spot), K (strike), T (time), r (risk-free rate), and σ (volatility). The thing the formula is doing, conceptually, is computing the discounted expected payoff of the option under a risk-neutral measure where the underlying drifts at the risk-free rate. The Greeks (Δ, Γ, ν, Θ, ρ) fall out as partial derivatives of C with respect to each of those inputs.
A binary prediction-market contract pays 0 otherwise. If you naively try to apply Black-Scholes machinery, you immediately notice that there is no underlying S, no strike K, no continuous price path, no σ, and no continuous payoff function. Half the inputs do not exist. The other half exist in a degenerate form. The temptation is to declare the framework inapplicable and walk away.
The temptation is wrong. The high-level structure of Black-Scholes — risk-neutral pricing of a contingent payoff under a known measure, with Greeks as exposures — ports completely. What does not port is the specific machinery used to compute the price, because that machinery assumes a continuous diffusion process for the underlying. Binaries do not have an underlying; they have an event.
What Ports: Risk-Neutral Pricing
The single most important idea in Black-Scholes is that you can price a contingent claim by computing its expected payoff under a risk-neutral measure and discounting at the risk-free rate. This is called the first fundamental theorem of asset pricing and it does not depend on continuous dynamics — it depends on the absence of arbitrage. Binaries satisfy that condition (mostly), so the theorem applies (mostly).
Concretely: a Kalshi binary trading at $0.42 with 18 days to resolution is, under the risk-neutral measure, paying you a 42% probability that the event happens. The market price is the risk-neutral probability, modulo a small adjustment for the risk-free rate over the holding period. That is the same statement Black-Scholes makes about an option, just degenerate to a binary payoff.
The bond-trader version of this is what I have been writing about in /blog/prediction-markets-need-fixed-income-language — a binary contract is mathematically a credit-risky zero-coupon bond, and the implied yield is the risk-neutral discount rate. The math is the same. The vocabulary is just different.
What Does Not Port: The Vol Surface
Black-Scholes assumes the underlying follows a geometric Brownian motion with constant volatility. Real options markets relax that assumption with a vol surface — a 2D function of strike and tenor that captures how the implied vol of options at different strikes deviates from the flat Black-Scholes baseline. The surface is the central object in modern options trading. It is what every options desk re-marks every day and what every vol arb strategy is built around.
There is no vol surface for a binary contract. The contract has no underlying continuous price to be volatile against. There is no σ to fit. The "smile" you might draw across the strikes of a multi-outcome event (more on this in from-options-skew-to-multi-strike-pm-events) is shaped like an options smile but it is not generated by the same mechanism — it comes from the structural overround in multi-outcome markets, not from differential implied vol.
This is the cleanest example of what does not port. If you walked onto a prediction market thinking "I will fit a vol surface and trade the dislocations," you would spend a week building infrastructure for a quantity that does not exist in this market.
What Trivially Ports: Delta
Delta is the derivative of the option price with respect to the underlying spot. For a binary contract, the "underlying" is the resolution outcome itself, which is discrete. Delta on a binary is 1 (you bought YES, the contract resolves YES) or 0 (it resolves NO). There is no continuous exposure to spot because there is no continuous spot.
This sounds like delta does not port — and at the intra-tenor level, it does not. But at the position-sizing level, it ports trivially, because what you really care about is "if the YES probability moves from 0.42 to 0.43, how much does my position value change," and that derivative is well-defined: it is just your position size in YES contracts, multiplied by 0.01. You are linear in the YES probability. That linearity is delta-equivalent for sizing purposes, and it is the same thing every options desk computes when they ask "what is my delta on this position."
What Does Not Port: Gamma
Gamma is the second derivative of the option price with respect to spot — the rate at which delta itself changes as spot moves. On an options book, gamma is the dominant risk for short-tenor options near the strike, and the entire "gamma trading" sub-discipline exists to monetize the convexity gamma creates.
Binaries have no gamma in the options sense because they have no underlying. The convexity that does exist on a binary lives in the time axis — the price moves nonlinearly as resolution approaches, especially when the contract is far from the rails. That convexity is what the cliff-risk-index is measuring: |Δp/Δt| × τ_remaining is a discrete approximation of the second derivative of price with respect to time, and a high CRI is the binary-contract equivalent of a high-gamma position.
The translation is: where an options trader watches gamma, a binary trader watches CRI. They are not literally the same quantity, but they play the same role in the indicator stack — they tell you when the contract is in a regime where small inputs cause disproportionately large price moves.
What Does Not Port: Vega
Vega is the derivative of the option price with respect to implied vol. There is no implied vol on a binary contract to differentiate against, so there is no vega. Vol-arb strategies do not exist on binaries.
The closest analog is "thesis confidence" — the derivative of how a position should be sized with respect to how confident you are in your forecast. But that is not a market-implied number; it is your private input. Vega exists as a market-observable on options because options markets explicitly trade vol; binaries do not. There is no machinery to recover.
What Sort-Of Ports: Theta
Theta is the time decay of an option's value as expiry approaches. For an OTM option, theta is the slow death of optionality. For an ITM option, theta is the slow accrual of intrinsic value.
Binaries have a similar but cleaner version: as τ → 0, the contract converges to either 0.00, and the speed of that convergence depends on how close the price is to either rail. A 50¢ contract with one day to go has enormous "theta" — it must move 50¢ in either direction within 24 hours. A 95¢ contract with one day to go has near-zero theta — it has already mostly converged.
This is the one Greek that I think does port to binaries, and the cleaner expression of it is τ-days plus IY. The relationship between time-to-resolution, current price, and implied annualized return is the binary-contract version of theta. If you understand theta in options, you understand IY in binaries, and the ratio gives you the same intuition about "how much value am I losing per day from holding this position."
The PM-Native Replacement: The Indicator Stack
The cleanest way to summarize the translation is that Black-Scholes' Greeks are replaced, on prediction markets, by the pm-indicator-stack:
- IY plays the role of the discount rate, but specifically as a yield rather than a continuous risk-neutral measure.
- CRI plays the role of gamma, capturing nonlinear price movement near the time axis rather than near a strike.
- EE plays a role with no Black-Scholes analog — it captures the consistency of multi-outcome events as a unit, which options markets do not have an equivalent for because options on the same underlying are not mutually exclusive.
- LAS plays a role that options markets bury inside "open interest" and "average daily volume" metrics — the question of whether the orderbook is thick enough to actually trade.
- CVR plays the role of "information flow speed," which options markets handle implicitly via earnings cycles and macro releases but never give a numerical name to.
The mapping is not one-to-one. The indicator stack is shaped by what is cheap to compute on 47,000 binary contracts in milliseconds, which is a constraint Black-Scholes does not face because options markets have orders of magnitude fewer instruments to scan. The constraint produces a different vocabulary, optimized for a different problem.
A Worked Translation
Take a Kalshi Fed-decision contract: KXFEDDECISION-26JUN YES at $0.42, with τ = 42 days. An options trader's instinct is to price it as a "binary call struck at the FOMC announcement date." Walking through:
- Spot: there is no spot. The "underlying" is the eventual decision, which is discrete.
- Strike: there is no strike. The contract resolves on a discrete event.
- Volatility: there is no σ. There is no continuous price process to be volatile against.
- Risk-neutral discount:
IY = (1/0.42)^(365/42) - 1 ≈ 1290%, which is the annualized discount rate the market is putting on the YES outcome under the risk-neutral measure. - Delta: 1 if YES resolves YES, 0 otherwise. For sizing, your linear exposure to the probability is just position size × 0.01 per cent of probability move.
- Gamma: not defined; replaced by CRI = 2.1 (medium-high), which says the market is actively repricing.
- Theta: replaced by the relationship between IY and τ — as τ shrinks, the same conviction translates to a higher annualized return, but only if the price moves to crystallize it.
- Vega: undefined.
The Black-Scholes machine returns a price of $0.42 by tautology, because that is the market price under the risk-neutral measure. The interesting work is everywhere else — in CRI, in the orderbook depth, in whether the thesis is consistent with reality and opinion data, none of which Black-Scholes has anything to say about.
Where the Bridge Breaks
A few places where treating binaries as "degenerate options" fails harder than I have so far described.
Resolution risk has no options analog. A standard equity option resolves against a price you can verify on Bloomberg. A Polymarket contract resolves against a UMA dispute process that can be wrong, slow, or contested. The risk that the contract pays 1 is real, and there is no Black-Scholes term for it. The replacement is the resolution-risk premium — the implicit discount the market applies to contracts on contested venues, which can be 5-15% on Polymarket compared to economically equivalent Kalshi contracts.
There is no continuous hedging. Black-Scholes derives its formula by assuming you can continuously rebalance a delta hedge in the underlying. Binaries have no underlying to rebalance against. The closest thing is taking offsetting positions in a related market on another venue, which is more like a credit hedge than a delta hedge — and the basis risk between the two legs is not zero.
Implied vol surface fitting is inapplicable. Quants who try to "fit a surface" to multi-strike PM events end up fitting a curve that is dominated by the structural overround of the event, not by anything resembling implied vol. The surface they recover is an artifact of the fitting procedure, not a real market quantity.
Gamma trading is impossible. You cannot scalp gamma on a binary because there is no underlying to scalp. The closest thing is dynamic position sizing as CRI rises and falls — trimming when CRI goes high (the contract is in a convex regime) and adding when CRI goes low. That is a sensible behavior but it is not gamma trading; it is risk management based on a different metric.
The Honest Summary
If you came from options, the things you should keep are: risk-neutral pricing as a conceptual frame, delta as a sizing intuition, theta as an analog for time decay (mediated through IY and τ-days), and the general habit of decomposing a contingent claim into its risk components. The things you should drop are: vol surface fitting, gamma trading, vega-based strategies, and any belief that the Greeks form a complete description of risk on a binary.
Stop thinking like an options trader unless you are also thinking about resolution risk, orderbook depth, sibling-market overround, and the propagation of information across venues — things options theory has nothing to say about because options markets do not have those problems. The replacement is the indicator stack and the valuation funnel, which were built specifically for the constraints prediction markets impose.
I run sf scan --by-iy desc --min-tau 14 every morning and the output is a sorted list of contracts ranked by what an options trader would call "yield-equivalent" but which I just call IY. The framing translation from Black-Scholes to the indicator stack is what makes the screen useful. The framing was hard-won — I had to give up a lot of options-trader instincts to get here — and the sooner you do the same, the sooner the binary-market screen starts looking like the right tool instead of a degenerate version of a bond screen. For the philosophical case in the language of bond traders specifically, see implied-yield-vs-raw-probability-bond-markets.