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Path-dependent options in BS framework is intuitive to price with monte-carlo under risk-neutral measure, however it appears that several kinds can be priced with PDEs. I understand how does the story goes for Asian options: the payoff depends on more than just asset price and time, so we introduce a new variable, our model happens to be Markovian again, completeness is still in place - hence just write down Kolmogorov-like equation for the option price for the risk-neutral measure.

In case of barrier options, we often don't even need to enlarge the state space: only introduce the additional boundary conditions at barriers. However, in Wilmott's "Mathematics of Financial Derivatives" and "PWOQF2" the derivation is rather informal and hand-shaky, something like "before hitting the barrier the option's price satisfies the BS equation". It's not that clear to me, though, why is that.

Another book I've checked was "Martingale methods" by Musiela and Rutkowski - there they just compute expectation of present value, having indicators for barrier events - they use joint distribution of max/min of Brownian motion with Brownian motion itself; there everything is formal, but not expressed in the framework of PDEs.

I thus interested in:

  1. Formal derivation of PDE and boundary conditions for barrier options in the BS model. I've also checked out Shreve's 2nd volume, Section 7.3.2 - but yet again the argument in Lemma 7.3.2 there is a bit informal on the one hand, and on the other - the rest of the proof is done via martingale methods.

  2. Can you advise any "classical" book on math finance which follows the PDE approach (rather than martingale/expectation approach) and is mathematically rigor?

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Some more concrete sources on Barrier option in the B&S setting and PDEs


More of a general remark to PDE approaches in finance

Ilya as far as I know the literature on that topic is quite limited. Solving a PDE means solving a PDE - it does not matter in which context. Most economists leave the solving of PDEs if they arise in a pricing context to pure mathematicians. This is why almost no finance book will teach you how to solve one explictly - this is either something you learn in pure math or physics or delgegate.

I think you are might be interested in "interfacing theorems" like Feynman-Kac that establish the link between pricing and PDE.

Also to my knowledge there is no unifying PDE-based approach to pricing derivatives.

Still there are some books that have more extensive sections on PDE-theory e.g. PDE and Martingale Methods in Option Pricing You might also find the following report quite comprehensive. (The focus however is not on finding closed-form solution but rather on numerical schemes)

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