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Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach by Hong Xie, Chaoqun Ma, Guojun Gan

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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, Xt) : 0 ≤ tT} be the Black-Scholes market given in Definition 42.2. Let

(43.1a) equation

(43.1b) equation

(43.1c) equation

and

(43.1d) equation

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

equation

Then

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

equation

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

(43.2) equation

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

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