Digital (Binary) Options

A digital option (also called a binary) is the simplest payoff to describe and one of the hardest to trade. It's all-or-nothing: if a condition is true at expiry, it pays a fixed amount; if not, it pays zero. There is no in-between.

The classic version is a cash-or-nothing call: it pays a fixed sum, say $1, if the underlying finishes above the strike $K$, and nothing if it finishes below. Unlike a normal call, it does not pay more when the price finishes further in the money. Finishing one cent above the strike pays exactly the same as finishing 50 dollars above it. That flat, cliff-edge shape is where all the trouble comes from.

The payoff and its price

The payoff of a cash-or-nothing call paying $Q$ is a step:

$$\text{payoff} = \begin{cases} Q & \text{if } S_T > K \\ 0 & \text{if } S_T \le K \end{cases}$$

where $S_T$ is the underlying's price at expiry and $K$ is the strike. Because the payoff is just a fixed amount times the event "we finished in the money," its fair value is that amount, discounted, times the probability of the event:

$$V = Q\, e^{-rT}\, \mathbb{Q}(S_T > K) = Q\, e^{-rT} N(d_2).$$

Here $e^{-rT}$ discounts the future cash back to today ($r$ is the interest rate, $T$ the time to expiry), and $N(d_2)$ is the risk-neutral probability of finishing above the strike, the same $N(d_2)$ that appears inside the Black-Scholes formula. So a digital is, quite literally, a bet on a probability.

The one trick that runs the desk

Here is the insight that makes digitals tradeable. A digital is the strike-derivative of an ordinary call:

$$V_{\text{digital}} = -\,\frac{\partial C}{\partial K}.$$

In plain terms, if you take a normal call and nudge its strike, the value changes, and that change is the digital. And a derivative is just a difference over a tiny gap, so you can build a digital out of two vanilla calls: buy a call at $K - \varepsilon$, sell one at $K + \varepsilon$, and scale the whole thing up by $1/(2\varepsilon)$. That's a call spread squeezed very tight, exactly the dashed ramp in the picture. This is how desks quote, price, and hedge digitals in practice.

Why it's a hedging nightmare

Look again at the step. Right at the strike, near expiry, the payoff goes from 0 to $Q$ over almost no price range. That means its delta and gamma explode: an inch of spot movement flips the option from "worthless" to "fully paid," so the number of shares you'd need to hedge spikes toward infinity and then collapses. No finite share position can track a cliff.

So real traders never try. They deliberately over-hedge with a wider call spread than the exact replication demands, pricing the digital as if it were, say, a spread 1 point wide instead of a hair wide. That wider spread always pays at least as much as the digital (it super-replicates it), so the trader can never be caught short, they just give up a little edge as a safety buffer.

Worked example

A cash-or-nothing call pays $1 if a stock finishes above 100 in one year. Suppose the risk-neutral chance of that is $N(d_2) = 0.55$, rates are zero, so $e^{-rT} = 1$. The fair value is

$$V = 1 \times 1 \times 0.55 = 0.55.$$

You'd pay 55 cents for a coin that pays a dollar 55% of the time, sensible. Now replicate it with a call spread 2 points wide: long a call struck at 99, short a call struck at 101, sized so the 2-point width delivers $1. As expiry nears with spot sitting at exactly 100, that spread is worth about half its max, roughly the same 0.55, but its hedge is stable, because you're trading a gentle ramp instead of a cliff.

Where it also bites

• Skew sensitivity. Because the price is $-\partial C/\partial K$, a digital depends directly on the slope of the volatility smile, not just the level of vol. Ignore the skew and you misprice it.
• Discrete settlement. "Above the strike" at exactly which instant? The fixing convention matters, and disputes at the boundary are real.
• It looks like a fair coin, but isn't. The payout is fixed, so retail "binary options" often bury a house edge in the price. The math above is the fair value; check what you're actually being charged.

For the broader family of non-standard payoffs and the closed forms behind them, see Exotic Options and Barrier Options.