Digital & Binary Options

You already know a vanilla call: above the strike its payoff ramps up dollar-for-dollar, and the Greeks behave like polite, continuous functions you can hedge. Now meet the digital — also called a binary option — the simplest exotic there is, and the one that quietly powers nearly everything else in this course. A digital doesn’t ramp. It flips: you get a fixed payout if the stock finishes on the right side of a line, and nothing if it doesn’t. That single yes/no payoff is the atom. Autocall coupons, range accruals, the knock-in triggers in a structured note — strip away the marketing and they’re all just bundles of digitals. Learn this one cleanly and the rest of the course is mostly assembly. By the end you’ll price a digital as a probability, build it out of a tight call spread, and understand exactly why its delta tries to go to infinity right when you least want it to.

A coin-flip payoff (cash-or-nothing vs asset-or-nothing)

Analogy. A vanilla call is a dimmer switch: turn the stock up and the payoff brightens smoothly. A digital is a plain light switch — off below the strike, on above it, with nothing in between. There’s no “a little bit on.” You either cleared the line or you didn’t, and the prize is the same either way.

The definitions. There are two flavors, and the difference is what you collect:

• A cash-or-nothing call pays a fixed cash amount Q if the stock finishes above the strike K, and $0 otherwise. The payout is $Q \times \mathbf{1}\{S_T > K\}$ — that indicator is just a fancy way of writing “1 if true, 0 if false.”
• An asset-or-nothing call pays one share of the stock itself (worth $S_T$) if $S_T > K$, and $0 otherwise. Same trigger, but the prize scales with where the stock lands.

(Puts are the mirror image — they pay when $S_T < K$.) Here’s the elegant part that ties digitals back to the vanilla world you already know. A vanilla call’s payoff, $\max(S_T - K, 0)$, decomposes exactly into these two atoms:

$\text{vanilla call} = (\text{asset-or-nothing call}) - K \times (\text{cash-or-nothing call with } Q = 1)$

Read it in words: above the strike a vanilla call hands you the stock ($S_T$, the asset-or-nothing piece) and charges you the strike ($K$ in cash, the cash-or-nothing piece) — and below the strike both pieces are zero. The vanilla you’ve used all along was secretly two digitals stapled together.

Worked example. Take a cash-or-nothing call with payout $Q$ of $100 and strike $K = 50$, on a stock trading near 50 today. Walk the payoff:

Notice the cash-or-nothing column: $100 the instant you clear 50, and stubbornly $100 no matter how high the stock flies. The asset-or-nothing column climbs because you’re being handed the share. That flat versus rising contrast is the entire taxonomy.

Pricing a digital: it’s just a probability

The mechanic. Because a cash-or-nothing call pays a flat Q exactly when $S_T > K$, its value today is dead simple in spirit:

$\text{price} = e^{-rT} \times Q \times \Pr{}^{\mathbb{Q}}(S_T > K)$

That’s just “payout × probability of winning, discounted back.” The only subtlety is which probability. Under Black–Scholes the risk-neutral probability of finishing in the money is $N(d_2)$ — the same $d_2$ that already lives inside the vanilla pricing formula you met earlier. So the closed form is clean:

$\text{cash-or-nothing call} = e^{-rT}\, Q\, N(d_2)$

The asset-or-nothing call, by the same logic, prices to $S_0\, N(d_1)$. And — satisfyingly — subtract $K$ cash-or-nothing units from the asset-or-nothing leg and you recover $S_0 N(d_1) - K e^{-rT} N(d_2)$, the textbook vanilla call. The decomposition isn’t a cute coincidence; it’s baked into the prices.

Worked example. Suppose the payout $Q$ is $100, the discount factor $e^{-rT} = 0.98$ (a touch under a year at a low rate), and your model spits out $N(d_2) = 0.42$ — a 42% risk-neutral chance of finishing above the strike. Then:

$\text{price} = 0.98 \times 100 \times 0.42$

Step through it: $100 \times 0.42 = 42$, i.e. $42 of expected payout, and discounting by $0.98$ gives $42 \times 0.98 = 41.16$, i.e. $41.16. So this digital is worth about $41.16 today. The price reads off as a probability almost literally — strip the discounting and the $100, and you’re staring at “42% likely.”

The call-spread replication (and why it bounds the price)

The construction. Here’s the trick desks actually use. Take a bull call spread: long a call struck at $K - \varepsilon$, short a call struck at $K + \varepsilon$, both at quantity $\tfrac{1}{2\varepsilon}$. Below $K-\varepsilon$ both expire worthless (payoff 0). Above $K+\varepsilon$ the two legs cancel to a flat payoff. Across the $2\varepsilon$ gap, the spread ramps linearly — and because you scaled the quantity by $\tfrac{1}{2\varepsilon}$, that ramp rises by exactly 1 across the gap. Now shrink $\varepsilon \to 0$: the ramp gets narrower and steeper while always rising a total of 1, until in the limit it’s a vertical step of height 1 sitting at $K$. That step is a cash-or-nothing digital with $Q = 1$. Want a $100 payout? Multiply every quantity by 100.

The island below makes this concrete. It’s a spread with a $2-wide gap (long call at 99, short call at 101) at quantity 50 each — so across the $2 gap the payoff rises by $2 \times 50 = 100$, i.e. $100, then flattens. Flip it to Payoff mode and look at the shape: a steep ramp from 99 to 101, then a flat plateau at $100. Now imagine pinching that gap from $2 down to $0.50, scaling the quantity up to keep the plateau at $100 — the ramp gets near-vertical. That’s the digital emerging from the limit.

Worked example — how notional scales. Hold the plateau fixed at $100 and watch the quantity explode as the gap tightens:

Same $100 prize, but to keep it you must trade more and more contracts as the gap narrows — the quantity is $\tfrac{Q}{\text{gap}}$, which runs off to infinity. That blow-up isn’t a footnote; it’s a preview of the delta nightmare in the next section.

This is also how digitals get priced in practice. A desk can’t hedge a true step, so it hedges the digital with a finite call spread — say a $0.50 gap — which slightly overhedges: the real digital is a vertical step, but the spread ramps just a hair early, paying out a touch more around the strike. To cover that extra, the desk quotes the digital a little above the theoretical $e^{-rT} Q\, N(d_2)$. That gap is the bid-side spread, and the call-spread replication is exactly why it exists — and, since the spread is sensitive to the two different strikes’ implied vols, it’s also where the skew sneaks into the price.

Discontinuous delta = a hedging nightmare

The problem. Delta is how much the option’s value moves per $1 move in the stock — for a vanilla it’s a smooth number between 0 and 1. For a digital, the value has to travel the entire gap from 0 to Q across an infinitesimal sliver of price right at the strike. Near expiry, with no time left to smear out the uncertainty, that transition gets razor-sharp: the value is basically a cliff. Delta is the slope of that cliff, so delta spikes toward infinity right at $K$ and collapses back to near-zero on either side. The gamma — the rate of change of delta — does something even worse, swinging from a giant positive to a giant negative as the stock crosses the strike.

Now picture the poor trader who sold this digital and wants to delta-hedge it with stock. To stay neutral they’d need to hold a position proportional to delta — which means as the stock creeps toward the strike near expiry, they must buy a colossal amount of stock for a one-cent wiggle, then dump it all the instant the stock ticks back. The hedge demands trading enormous size on microscopic moves, racking up transaction costs and getting whipsawed to death. This is pin risk in its purest, most violent form: the stock “pinning” near the strike at expiry turns the hedge into an impossible game of slamming the brakes and flooring it on every tick.

When you’d use one

When to use it

Reach for a digital when your view is genuinely about a probability, not a magnitude. Three classic uses:

1. A clean probability bet. “I think there’s better than a 60% chance the stock is above $80.” A digital’s price is roughly that probability, so if your number beats the market’s, the digital is mispriced in your favor. A vanilla call would muddy this with a view on how far above $80 — the digital strips that out.
2. Cheap, defined leverage on a binary event. Earnings, a court ruling, an FDA decision — an event with a yes/no outcome. A digital lets you size the bet exactly (you know the max payout and the max cost up front) without the open-ended, vol-crush-prone behavior of a vanilla.
3. The building block for structured products. This is the big one for this course. A coupon that pays only if the index is above a level on an observation date is a cash-or-nothing digital. Autocallables are stacks of these triggers; range accruals count up digitals day by day. Once you see digitals everywhere, structured notes stop looking like magic and start looking like Lego.

The trade-offs, honestly. The flat payoff cuts both ways: there’s no upside beyond Q — if the stock rockets past the strike you collect the same fixed amount as if it squeaked over, so you give up the vanilla’s open-ended tail. And for the seller, the near-strike delta/gamma blow-up makes a sold digital genuinely dangerous to manage at expiry. Clean to express, brutal to hedge.

Putting it together

A digital (binary) option pays a fixed amount on a yes/no condition — the simplest exotic, and the atom behind autocall coupons, range accruals, and structured-note triggers. It comes in two flavors: a cash-or-nothing call pays a flat Q above the strike, an asset-or-nothing call pays the share itself, and a vanilla call is exactly asset-or-nothing minus K cash-or-nothing units. Pricing is just discounted payout × probability of finishing in the money — under Black–Scholes, $e^{-rT} Q\, N(d_2)$ — so a digital’s value reads off as a probability, tilted by a skew-driven slope once implied vol varies across strikes. You build (and hedge, and price) a digital as the limit of a tight bull call spread, scaled up: as the gap shrinks the quantity blows up toward a vertical step, the finite spread slightly overhedges, and that’s why quoted digitals sit a touch above theory. The cost of that step is a discontinuous delta that spikes toward infinity at the strike near expiry — pin risk so violent that nobody hedges it with stock. Use a digital for a clean probability view, defined binary-event leverage, or as a structured-product building block — just remember the upside is capped at Q and the near-strike risk is brutal for the seller.

Next — barrier options — where the payoff doesn’t just depend on where you finish, but on whether the stock ever touched a level along the way, layering a path-dependent trigger on top of the all-or-nothing logic you just learned.