Binary Options and Their Hedging Problem

A binary option pays a fixed amount or nothing at all, and that step shaped payoff makes it nearly impossible to delta hedge near the strike.

A binary option, also called a digital option, pays a fixed amount if the underlying finishes above a strike and nothing otherwise, in contrast to a vanilla option whose payoff grows linearly the further it finishes in the money. That single design choice, a fixed payout rather than a linear one, makes binaries genuinely difficult to hedge close to the strike as expiry approaches, in a way that has no real equivalent among vanilla calls and puts.

Take a cash or nothing binary call struck at $100 that pays $10 if the stock finishes above $100, with the stock at $102 and 30 days to expiry. Using a standard option pricing model with a 25 percent implied volatility assumption and a 5 percent annualized risk free rate, the fair value works out to about $6.15. That price reflects a risk neutral probability of finishing above $100 of a little under 62 percent, discounted slightly to account for the 30 day wait before the payout is received.

If the stock finishes at $105, the binary pays its full $10, a profit of $3.85 after the $6.15 paid. If the stock instead finishes at $99, one dollar below the strike, the binary pays nothing at all, a total loss of the full $6.15 premium. A one dollar move in the stock right around the strike is the difference between the maximum payout and a total loss, which is not how a vanilla call or put behaves near its own strike, where a one dollar move only changes the payoff by about one dollar.

A vanilla option's delta rises smoothly from zero to one as the stock moves through the strike. A binary option's delta does not rise smoothly. As expiry approaches and the stock sits close to the strike, the delta becomes an extremely tall, narrow spike centered on the strike, because the payoff itself is a step function rather than a straight line, and gamma near that point becomes correspondingly large with no upper bound as time to expiry shrinks toward zero.

Traders describe binaries as carrying pin risk with no functioning hedge, in contrast to the more familiar pin risk on a vanilla option near expiry, where at least the payoff transitions continuously across the strike rather than jumping from nothing to the full payout in an instant.