Binary Option

A binary option has about the simplest payoff in finance. Pick a price and a date. If the market is above that price on that date, you get a fixed sum. If it is not, you get nothing. There is no in-between and no partial win: being right by a cent pays the same as being right by ten dollars.

The same contract leads two very different lives.

• Between professionals it is a useful building block. It is the cleanest way to say "the chance of this happening", and it sits inside many larger structured products. It is legal, it is priced properly, and it is traded by people who know what it is worth.
• Sold to the public it turned into something else: websites offering bets that expire in sixty seconds, at payouts that made money for the seller whatever the customer guessed. It is now banned for retail clients in the EU and the UK, and restricted in many other countries.

This page covers both, because seeing why the arithmetic cannot be argued with is the best defence against the second version.

The price is a probability

Under Black–Scholes, a cash-or-nothing call paying 1 is worth the discounted risk-neutral probability of finishing in the money:

• This is exactly the N(d₂) computed by the probability-of-expiring-in-the-money tool. A binary option is that number, made tradeable.
• At the money with a short expiry, N(d₂) is close to 0.5. A fair coin flip should therefore cost about half the payout — and pay roughly double the stake.

Why the retail version could not work

At a genuine 50% win probability, staking 100 to win 80 or lose 100 has an expected value of 0.5 × 80 − 0.5 × 100 = −10 per trade. Trading it repeatedly converges on ruin with mathematical certainty — a worse edge than most casino games, and the reason regulators treated it as a product-design problem rather than a disclosure problem.

The hedging problem: a discontinuous payoff

The reason binaries are genuinely difficult instruments has nothing to do with their retail history. The payoff jumps at the strike, so the Greeks explode there:

• Delta becomes unbounded as expiry approaches with spot at the strike — an infinitesimal price move flips the payoff between everything and nothing. No finite hedge exists.
• Dealers hedge with a call spread instead. Buying a call at K−ε and selling at K+ε replicates the binary with a payoff that is steep but continuous, and finite to hedge. The tighter the spread, the closer the replication and the larger the position — this is the real trade-off, and it is why quoted binaries always carry a spread over theoretical value.
• Pin risk is the same problem at expiry: with spot sitting exactly at the strike, the payoff is genuinely undetermined until the settlement print. Desks manage it by unwinding early rather than by modelling it.

The binary is the negative derivative of the call price with respect to strike — which is another way of saying it is the risk-neutral probability density's cumulative value, and the cleanest statement of why its price is a probability.

Where binaries legitimately appear

• Inside structured products. Every fixed coupon paid conditionally — an autocallable's coupon barrier, a reverse convertible's knock-in — is a digital in disguise.
• FX digitals are a standard institutional market, used to express event views around central-bank meetings.
• Range accruals and one-touch structures are families of digitals across time rather than at a single date.
• Regulated event contracts on some exchanges are binaries with transparent pricing and a real order book — the same payoff, in a venue where the price is set by participants rather than by the counterparty taking the other side.

The regulatory record, briefly

Retail binary options were prohibited for retail clients in the EU and UK from 2018–19 after regulators documented widespread losses, price manipulation on unregulated platforms, and refusal to process withdrawals. Several jurisdictions followed. The prohibition targeted the retail distribution model; the instrument itself remains a normal part of derivatives markets, which is the distinction worth carrying away.

The formulas above are standard textbook formulations, simplified for teaching. They explain the mechanism — they are not a valuation tool, and they will not reproduce a dealer’s price.