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 ≤ t ≤ T} be the Black-Scholes market given in Definition 42.2. Let
(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
(43.2)
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