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.

**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*

(43.1b)

(43.1c)

*and*

(43.1d)

*where S* = *X*_{0} *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*

(43.2)

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

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