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?

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    $\begingroup$ 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? $\endgroup$ – starovoitovs Aug 17 '19 at 13:17
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    $\begingroup$ I am asking whether the price of a path dependent option will satisfy the Black-Scholes PDE. $\endgroup$ – Dhruv Gupta Aug 17 '19 at 13:20
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    $\begingroup$ 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. $\endgroup$ – demully Aug 17 '19 at 23:22
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    $\begingroup$ 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. $\endgroup$ – demully Aug 17 '19 at 23:32
  • $\begingroup$ 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? $\endgroup$ – Dhruv Gupta Aug 18 '19 at 3:42

It depends very much on the individual option you are pricing.

Sometimes you can get a Black-Scholes PDE with some extended state and boundary conditions.

  • 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 $$C_\text{up-and-in} + C_\text{up-and-out} = C_\text{call}$$

  • PDE for Asian options will include running average

  • lookback options will include running minimum / maximum

Chapter 7 in "Stochastic Calculus for Finance 2" by Shreve can give you a good insight.


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