# Digital call option payoff

This post is based on problems 2. I was asked how to price a digital option in a job interview - and had no idea what to do! A call is only worth exercising using if the underlying price, , is greater than at , as the payoff from exercising is. A digital call option with is similar - it pays off one dollar if at expiration, and pays off zero otherwise:. Suppose you have a model for pricing regular call options.

How can you use to price the digital option? As a starting point, consider buying a call with and selling a call with:. A call is only worth exercising using if the underlying price, , is greater than at , as the payoff from exercising is. A digital call option with is similar - it pays off one dollar if at expiration, and pays off zero otherwise:.

Suppose you have a model for pricing regular call options. How can you use to price the digital option? As a starting point, consider buying a call with and selling a call with:.

This is close to the digital option, but not exactly right. We want to make the slope at steeper, so we need to buy more options. By purchasing a basic binary call option, the trader is simply speculating that the price of the underlying asset will be higher than the current market price when the option expires, typically within next few minutes or several hours. It is entirely up to the trader how much he wishes to invest with each purchase of the binary call option. The minimum and maximum he can put in with each call option varies across brokerages.

If the price of the underlying is above the strike price of the binary call option, the option expires in the money and the trader stands to receive a payout.

Otherwise, the option expires out of the money and he loses his initial investment. In the rare event where the price of the underlying asset is exactly the same as the strike price, the option expires at-the-money and the trader will simply get back his original investment. If the option expires out of the money, the trader loses his initial investment. This is also the maximum he can lose in this trade.