Advanced Equity Derivatives: Volatility and Correlation

Chapter 1Exotic Derivatives

Strictly speaking, an exotic derivative is any derivative that is not a plain vanilla call or put. In this chapter we review the payoff and properties of the most widespread equity derivative exotics.

1.1 Single-Asset Exotics

1.1.1 Digital Options

A European digital or binary option pays off $1 if the underlying asset price is above the strike K at maturity T, and 0 otherwise:

In its American version, which is more uncommon, the option pays off $1 as soon as the strike level is hit.

The Black-Scholes price formula for a digital option is simply:

where F is the forward price of S for maturity T, r is the continuous interest rate, and σ is the volatility parameter. When there is an implied volatility smile this formula is inaccurate and a corrective term must be added (see Section 2-1.3).

Digital options are not easy to dynamically hedge because their delta can become very large near maturity. Exotic traders tend to overhedge them with a tight call spread whose range may be determined according to several possible empirical rules, such as:

• Daily volatility rule: Set the range to match a typical stock price move over one day. For example, if the annual volatility of the underlying stock is 32% annually; that is, 32%/√252 ≈ 2% daily, a digital option ...