Derivatives: Markets, Valuation, and Risk Management

CHAPTER 8

Valuing Nonstandard Options Analytically

In Chapter 7, we valued standard European-style options analytically with the Black-Scholes (1973)/Merton (1973) option valuation framework. This chapter continues to focus on options that can be valued analytically within the BSM framework, however, the types of options that we examine are unusual or nonstandard.¹ While we discuss eleven different types of contracts, do not be misled. There are a virtually limitless number of variations of derivative contracts that have or can be structured. Some can be valued analytically. These are the focus of this chapter. Some require the use of numerical methods. These are the focus of the next chapter. As we proceed through this chapter describing the different types of contracts and their analytical valuation equations, it is important to try to imagine possible applications. In many instances, the contracts have sensible return/risk management properties. In other instances, the contracts seem only to be a cleverly structured bet.

ALL-OR-NOTHING OPTIONS

In Chapter 5, we showed that the valuation equations for asset-or-nothing and cash-or-nothing call and put options were impounded within the BSM call and put formulas. Recall that an asset-or-nothing call that pays one unit of the asset at time T if the asset price exceeds the exercise price X. The terminal profit from buying an asset-or-nothing call option is shown in Figure 8.1. Note that, for terminal asset prices below X, the option ...