# PDE pricing of barrier options in BS

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?

• Wilmott's PDE for Barriers is same as for any other option; as he says "The details of the barrier feature come in through the specification of the boundary conditions." You can do it via FDM. BTW, you can post on wilmott site, and may find him responding on it. If you do, please post a link to that thread here. Commented Apr 29, 2014 at 23:00

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)

• Thanks for your answer. I actually didn't mean solution of PDEs, especially an analytic one, just a PDE formulation. At least one advantage it gives is useful formulas for Greeks. My point is that the PDE for barriers in BS is "derived" using arguments like "value of option satisfies BS before hitting the barrier", so obviously we need to solve BS equation with an additional boundary condition on the barrier. As usual, it is this obvious step that can make the whole result being incorrect - so I just wondered whether there is a detailed explanation of this.
– SBF
Commented Apr 4, 2014 at 10:01
• Regarding PDE approach, I'd say Wilmott follows it everywhere in his books. Actually that's the same as martingale approach + Markovian structure, but without mentioning the latter two things too often (as Shreve does, in contrast) and using instead $\Delta$-hedging-like arguments, which of course leads to the same PDE as the martingale approach does. So I'd be interested in a book with a similar approach, but slightly more formal on the PDE side (not necessarily on a stochastic side). Maybe there are some known textbooks of that kind, if not - nevermind.
– SBF
Commented Apr 4, 2014 at 10:06
• @Ilya - This may be of a nice place to start: Finite Difference Methods in Financial Engineering by D. Duffy Commented Apr 9, 2014 at 9:25
• @llya Could you please clarify why do you say that Wilmott's approach is "the same as martingale approach + Markovian structure, but without mentioning the latter two things"? Commented Apr 29, 2014 at 22:55

That's an old post by now, but since I somehow came across it here's my take. The reason why Wilmott is correct is because all the hypotheses made in order to formulate the pricing PDE for vanillas (I assume you're comfortable with that), still hold in the case of barrier options. So why is it that you doubt that the same PDE can be used? So yes, it's only the boundary conditions that differ and yes, it is indeed obvious that the PDE is satisfied by barrier options when you use the same BS assumptions.

As for probilitator's implying that the actual solving of the PDE's is usually left to pure mathematicians, I'm not sure about that! This should be engineers, (or at least applied mathematicians) as they are the ones who like to (and thus specialize) in solving things in practice, pure mathematicians usually prefer to stay in their abstract world:)