# 8

# Exotic Derivatives

## 8.1 EXOTIC DERIVATIVES WITH FIXED PAYOFFS

In this section, we are going to introduce and describe exotic derivatives with fixed payoffs. Due to this feature, the price of the underlying asset measured on or at certain date(s) (or, more generally, its path within a certain timeframe) will determine only the *occurrence* of the payoff and not the amount of the payoff itself.

### 8.1.1 European digital options

A European-style *digital option*, sometimes referred to as a *binary option*, *all-or-nothing or cash-or-nothing option*, will pay out a fixed amount if the underlying asset is above (in the case of digital call options) or below (in the case of digital put options) a certain strike at a certain date (option expiry).

The European digital option is perhaps the simplest derivative one can imagine, since it requires just one observation that determines the binary event.

In the example below, we will assume a generic payoff of $1. Any other digital payoff can obviously be obtained by scaling up this payoff as appropriate (e.g., a $100 payoff can be thought of as a portfolio of 100 identical digital options each with $1 payoff).

Figure 8.1 shows the payoff of a European digital call ($1 payoff) on a generic underlying with strike equal to $100.

As the payoff is fixed, the price of the derivative is generally expressed as a percentage amount of the fixed payoff.

Therefore, if we say that the price of this digital is 45%, we mean that we have to pay $0.45 to enter this contract ...