Binary Options
A binary option (also called a digital option) pays a fixed amount if the underlying is above the strike at expiry, and zero otherwise. No slope, no scaling with distance from the strike. Just a yes-or-no payout.
This is the building block behind HIP-4 threshold markets and the mechanic underlying prediction markets like Polymarket and Kalshi.
Binary vs Vanilla Payoff
The vanilla call grows with distance above the strike. The binary is flat: above the line you get 1, below you get 0.
A vanilla call can be decomposed into a continuous strip of binary calls. Each binary represents one infinitesimal "slice" of the call's payoff at a specific strike. This is the mathematical foundation behind using HIP-4 threshold ladders to approximate vanilla option payoffs.
Pricing
A binary call's price is the market's implied probability that the underlying finishes above the strike. That's it. A binary trading at 0.70 means the market assigns a 70% chance to "yes, it settles ITM."
This is different from a vanilla call, whose price is the expected payoff and depends on how far past the strike the underlying is expected to go, not just whether it crosses. Binaries strip out the distance and price only the threshold-crossing probability.
Greeks
Binary option Greeks behave very differently from vanilla Greeks. The interactive chart below lets you compare them side by side.
Delta
Binary delta is the most important difference. Instead of the smooth 0-to-1 transition of vanilla delta, binary delta forms a sharp spike centered at the strike.
The hedging problem
Near expiry, binary delta at the strike can spike to extreme values. A small move in the underlying causes a large change in the option's value. This makes delta-hedging a binary option near the strike at expiry extremely difficult and expensive, which is one reason vanilla options are preferred for most trading purposes.
Gamma
Vanilla gamma is always positive (delta always increases toward the strike). Binary gamma flips sign: positive below the strike, negative above.
This means a binary option writer who is ATM faces gamma risk that changes direction as spot crosses the strike, a fundamentally different risk profile from vanilla gamma.
Theta
Vanilla theta is almost always negative (options lose value over time). Binary theta depends on moneyness:
An ITM binary gains value as time passes, because the probability of staying ITM increases as uncertainty decreases. This is the opposite of vanilla behavior.
Vega
Binary vega flips sign at the strike:
- OTM binary: positive vega. Higher vol increases the chance of crossing the strike.
- ITM binary: negative vega. Higher vol increases the chance of falling back below the strike.
Vanilla vega is always positive. For a binary, more volatility is only good if you are on the wrong side of the strike.
Types of binaries
Not all binary options work the same way. The settlement trigger and observation window vary across products.
The distinction matters for pricing and hedging. European digitals (HIP-4) depend only on the terminal value, so they can be priced with Black-Scholes and fit cleanly into scenario-based portfolio margin. One-touch contracts depend on the entire price path, which requires different models and makes PM integration harder.
Prediction markets as binaries
Platforms like Polymarket and Kalshi are binary options markets. A contract that pays 100k on Dec 31" is a binary call with strike $100k.
Most prediction markets use one-touch settlement, not European. HIP-4 thresholds are European digitals: they settle at a specific expiry time against the Hyperliquid oracle, and because they are native HyperCore assets, they can sit in the same portfolio as perps and options for PM scoring.
Connection to vanilla options
Binaries and vanillas are mathematically connected: a vanilla call can be decomposed into a sum of binaries at successive strikes. This is the basis of static replication and the reason a ladder of HIP-4 thresholds can approximate a vanilla call's payoff.
The practical implication: if you're short a 100k vanilla call, being long a ladder of binaries at 100k, 110k, 120k, ... gives you a passive hedge that tracks the call's payoff shape. No rebalancing required, just hold to expiry.
See Static Replication for the full treatment including the Breeden-Litzenberger and Carr-Madan frameworks.
Related:
- Static Replication - How a ladder of binaries approximates a vanilla call
- Pin Risk - Why binaries near expiry are the hardest instrument to hedge
- Scenario Grid - How portfolio margin reprices binaries under shocks
- Exercise Styles - European, American, and exotic exercise types
- Black-Scholes - The standard pricing model
- Delta - Vanilla delta behavior
- Vanilla Options on HIP-4 - How binary thresholds fit into the writer stack