Pricing Barrier Options with Black-Scholes PDE Boundary Conditions
Summary
The document explores why barrier options in the Black-Scholes framework can be priced with the usual pricing PDE while incorporating barrier behavior through boundary conditions. Unlike path-dependent products such as Asian options, the barrier feature can often be represented without adding a state variable: the option’s value is constrained when the underlying reaches the barrier. The questioner seeks a formal PDE derivation and mathematically rigorous references, noting that some textbook explanations feel informal and that alternative treatments use expectations involving Brownian extrema.
Responses explain that the assumptions supporting the vanilla Black-Scholes pricing equation also apply before the barrier is hit, so the governing PDE remains the same; the barrier changes the boundary conditions. They point to Feynman-Kac as a bridge between pricing expectations and PDEs and mention technical sources and numerical methods. The exchange offers conceptual guidance rather than a complete derivation, and it notes that PDE pricing does not imply one universal PDE method for all derivatives. It does not supply a detailed proof or assess the cited references.
Key ideas
- Barrier features can often be encoded as boundary conditions without enlarging the pricing state space.
- Before the barrier is reached, the option value follows the Black-Scholes pricing PDE under its standard assumptions.
- The barrier payoff and contract type determine the relevant boundary conditions.
- Feynman-Kac connects risk-neutral valuation expectations with pricing PDEs.
- The discussion recommends further references but does not provide a rigorous derivation itself.
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Full text
# PDE pricing of barrier options in BS # 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: - 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. - Can you advise any "classical" book on math finance which follows the PDE approach (rather than martingale/expectation approach) and is mathematically rigor? ## Answer by Probilitator (score 2) https://quant.stackexchange.com/a/10811 Some more concrete sources on Barrier option in the B&S setting and PDEs - PDE methods for pricing barrier options (quite technical) - Pricing Europ ean Barrier Options 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) ## Answer by Yian Pap (score 2) https://quant.stackexchange.com/a/35956 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:)
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