Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach

CHAPTER 43

PATH-DEPENDENT OPTIONS

Path-dependent options are also referred to as “exotic options”, whose payoffs depend on the path of the underlying asset. In this chapter, we present pricing formulas for some path-dependent options within the Black-Scholes framework.

43.1 Basic Concepts and Facts

Theorem 43.1 (European Barrier Option Price). Let {(X⁽⁰⁾_t, X_t) : 0 ≤ t ≤ T} be the Black-Scholes market given in Definition 42.2. Let

(43.1a)

(43.1b)

(43.1c)

and

(43.1d)

where S = X₀ is the initial price of the risky asset, K is the strike, T is the expiration time, H is the barrier, N(·) is as defined in Equation (42.1c), and

Then

(a) If S > H and the payoff of a down-and-in call is

then the price at time 0 of the down-and-in call is

(43.2)

(b) If S < H and the payoff of an up-and-in call ...