Exotic and Path-dependent Options

Abstract

Introduction If the simple derivatives that we have so far considered, which are almost all variations on vanilla European or American calls and puts, were the only derivative securities that needed to be modelled, this book would not be worth writing, nor would the subject have much mathematical interest. This part of the book is devoted to an introductory tour d'horizon of some of the huge variety of seemingly complex derivative instruments that have been created and are traded; each poses a challenge both to the mathematical modeller and to the people who trade and hedge them in practice. In this chapter we give an overview of some of the more common exotic and path-dependent options. In subsequent chapters we consider partial differential equation models for their valuation. In Chapter 13, in particular, we introduce a very simple but general framework for valuing many different path-dependent options. With this goal in mind, it is useful here to consider a classification of the varieties of exotic and path-dependent options. A path-dependent option is an option whose payoff at exercise or expiry depends, in some non-trivial way, on the past history of the underlying asset price as well as its spot price at exercise or expiry. We have already considered one type of path-dependent option in detail: the American option. This is clearly path-dependent since there is usually a finite probability of the option being exercised before expiry and thus ceasing to exist.