Reflection and Absorbing Barriers in Barrier Option Pricing

Summary

The question seeks an integral-based route to barrier option prices, motivated by the reflection principle and a guessed reflected probability density. The answer points to a derivation for Black–Scholes barrier options that uses reflection together with Girsanov’s theorem, offering a way to reach pricing formulas without directly solving a complicated partial differential equation. It also notes that the cited work discusses extensions beyond its main derivation.

A key correction is that an up-and-out call is modeled with an absorbing barrier: paths that reach the barrier are knocked out. A reflecting barrier would instead describe paths bouncing away from the boundary, so it does not represent that option feature. The brief answers do not derive the integral, explain the power term the question asks about, or establish the formulas’ assumptions in detail. The suggested method is therefore a pointer to a fuller derivation rather than a self-contained pricing procedure.

Key ideas

• Reflection and Girsanov’s theorem can be used to derive Black–Scholes barrier option prices.
• An up-and-out call uses an absorbing barrier because paths reaching it are eliminated.
• A reflecting barrier represents a different path behavior and does not price an up-and-out call.
• The answers refer readers to a separate derivation and do not work through the requested integral.

Tags

# simple, intuitive barrier option derivation


# simple, intuitive barrier option derivation












Is there a simple integral that gives barrier option prices without having to deal with messy, hard PDEs and change of variables

I understand there is a reflection principle such that the simulation stock prices bounces off the reflection at the barrier $b$

Taking the derivative with respect to $x$ and plugging in $b$ it's equal to zero, so the PDF should be something like this

$(e^{-((x-u)/a)^2/2}+e^{-((x+u-2b)/a)^2/2)})/(2\pi)^{1/2}$

and with the help of wolfram alpha I got this:

I don't know how to get this part: $(\frac{p}{b})^{a}$ Every barrier option pricing formula has this distinctive feature, but i don't know how you get that from $e^{a^2t+2b}$

Here's the integral

https://upload.wikimedia.org/math/9/d/8/9d80d384d06e1e2068c1463e08fe8a61.png

## Answer by q.t.f. (score 4)

https://quant.stackexchange.com/a/17920

Try this paper by Rolf Poulsen : http://colloquium.mathfinance.de/abstracts/poulsen.pdf. He derives barrier option prices in the Black-Scholes model using only reflection and Girsanov's Theorem, and then discusses extensions.

## Answer by Thomas Baert (score 1)

https://quant.stackexchange.com/a/18214

an up and out call involves an absorbing barrier. There is no pricing formula with a reflective barrier.

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