# Can we use Black-Scholes to price path dependent options?

I know that we can use the Black-Scholes framework to price vanilla products like a European call or put, where the payoff only depends on the share price at maturity.

But can we use it to price path dependent options - those options where the payoff depends not only on the price of the underlying at maturity, but on the entire price history over the life of the contract?

Specifically, what part of the model/derivation allows or disallows us to price such path dependent options?

• You can certainly "use the Black-Scholes framework" $dS/S = \mu dt + \sigma dW_t$ to price virtually any option. Are you asking about the existence of the analytical formulas for path-dependent options, like Black-Scholes formula for vanillas? Aug 17, 2019 at 13:17
• I am asking whether the price of a path dependent option will satisfy the Black-Scholes PDE. Aug 17, 2019 at 13:20
• Three words for you: Cox- Ross- Rubenstein. You can add in any kick-in, kick-out, barrier-sticking nuance anyone might desire. Just don't expect the luxury of closed form equations if/when you want to test these across time! Sorry. Aug 17, 2019 at 23:22
• Ps to your follow-up comment question, no it won't ;-( There are first-passage-problem equations that are timed either side; and equations for timeless probability of hitting one of two sticky barriers higher and lower. Beyond that, you need CRR, which is inordinately computationally expensive. Aug 17, 2019 at 23:32
• Thanks a lot @demully! This may sound silly, but what part of the Black-Scholes model/derivation tells us that such exotic options are not covered in it? Aug 18, 2019 at 3:42

• up-and-out barrier option will have virtually the same pricing PDE, and zero boundary condition at $$S=0$$ and $$S=B$$ (barrier level). The up-and-in barrier option can be priced by $$C_\text{up-and-in} + C_\text{up-and-out} = C_\text{call}$$