Binary Options from zero
1/5What is a binary option?
All or nothing. A binary option pays $1 if the underlying is above the strike at expiry, and $0 otherwise. No partial credit. No scaling with distance.
Compare that with a vanilla call, whose payoff is max(S - K, 0) — a hockey stick that grows dollar for dollar past the strike. The binary is a step function: a cliff from zero to one, right at K.
Drag the slider below. Watch the green step function versus the blue hockey stick. The binary does not care how far past the strike you are — only whether you crossed it.
A vanilla call is like getting paid per degree above 70°F. A binary is like a bet: does the temperature hit 70°F or not? You either collect or you do not. The weather does not pay you extra for hitting 90°F.
Price = Probability
A binary trading at $0.65 means the market says there is a 65% chance of finishing in the money. This is the key insight. The price IS the probability.
Unlike a vanilla call, where the price reflects the expected magnitude of gains, a binary call's price is just the discounted probability of crossing the strike. Under Black-Scholes:
Move the probability slider. Notice how the binary price moves in lockstep — they are the same number (up to discounting). A vanilla call has no such direct mapping. Its price reflects both the probability of exercise AND the expected payoff magnitude.
Greeks behave differently
Binary delta is a spike, not a smooth curve. Binary gamma is extreme. Everything about the risk profile is sharper and more concentrated around the strike.
Vanilla delta transitions smoothly from 0 to 1 as spot moves through the strike — the classic S-curve. Binary delta peaks sharply at the strike and falls to zero on both sides.
Drag spot toward the strike and watch the green spike tower over the smooth blue curve. Binary delta is the derivative of a step function — mathematically, it wants to be a Dirac delta. In practice, with finite time to expiry, it is a sharp peak whose height grows as time shrinks.
Building vanillas from binaries
A vanilla call is an infinite ladder of binary calls. Stack one binary at every strike above K, and the stepped payoff converges to the smooth hockey stick.
This is not just a mathematical curiosity. It is the foundation of static replication and the reason HIP-4 threshold ladders can approximate vanilla option payoffs. Each binary contributes one “rung” of the staircase.
Drag the slider to add more rungs. Watch the staircase tighten against the diagonal.
In the limit of infinitely many rungs, the staircase IS the call payoff. Mathematically: C(K) = ∫K∞ D(x) dx, where D(x) is the binary call at strike x. The call is the integral of its binaries.
Hedging binaries
Near expiry, near the strike, binary delta goes to infinity. This is why binaries are the hardest instrument to hedge, and why market-makers charge wide spreads on short-dated ATM binaries.
The hedging problem is simple to state: if you sold a binary call, you need to hold delta shares of the underlying to be hedged. But as expiry approaches and spot sits near the strike, the delta you need oscillates wildly with every tick. A $0.01 move in spot can flip your position from “hedge with nothing” to “hedge with everything.”
Drag the time-to-expiry slider toward 1 day. Watch the delta spike become a wall.
This is why pin risk is the defining risk of binary options. And it is why binaries are always hedged with call spreads (a tight vanilla spread approximates a binary), not with delta-hedging alone.
Where to go next:
Static replication — how a ladder of binaries approximates a vanilla call
Pin risk — the defining risk of binary options near expiry
Black-Scholes — the standard pricing model for both vanillas and binaries