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DoubleTrouble
Topic Author
Posts: 83
Joined: November 9th, 2010, 9:46 pm

Pricing Barrier option in BSM model

Hi!I'm in need of some guidance. I want to learn how to find the price of a barrier option of European type paying the amount at time T. Where and . This is how I reason:but to get further I need to know the distribution of M(T), which I don't. I know the distribution of maximum of Brownian motion so my first idea was to apply the logarithm on the inequality and then move the logarithm inside the maximum operator. Which gives meBut I can't get any further with this. Any suggestions, or is this approach totally wrong?Thank you in advance!
Last edited by DoubleTrouble on August 9th, 2011, 10:00 pm, edited 1 time in total.

Alan
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Joined: December 19th, 2001, 4:01 am
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Pricing Barrier option in BSM model

Because of the drift, that appraoch may be a dead end. Easier: solve the BS pde with the KO boundary condition.

DoubleTrouble
Topic Author
Posts: 83
Joined: November 9th, 2010, 9:46 pm

Pricing Barrier option in BSM model

Hi Alan,thank you for your reply. How do you suggest I do to solve the PDE? Should I do some kind of transformation into something I know has a certain solution? If so, I can't figure out what kind of transformation to do.Thank you in advance EDIT: Oh wait, I should use Feynman-Kac, right?
Last edited by DoubleTrouble on August 7th, 2011, 10:00 pm, edited 1 time in total.

rprat
Posts: 30
Joined: December 12th, 2009, 9:45 am

Pricing Barrier option in BSM model

May be you can solve it with finite differences method using algorithms such as Crank-Nicolson.

DoubleTrouble
Topic Author
Posts: 83
Joined: November 9th, 2010, 9:46 pm

Pricing Barrier option in BSM model

rprat: It's possible to find an analytic closed expression of the price at time 0Alan: I do seem to be able to figure out how to solve the BS PDE as you suggested. Any hints?Best regards

Alan
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Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

Pricing Barrier option in BSM model

Hint 1: Show that, if f(S,t) is a pde soln, then so is (S/H)^a f(H^2/S,t). Here H is an arbitrary constant and I will let you determine 'a'.Hint 2: How does f(S,t) - (S/H)^a f(H^2/S,t) behave at S = H? Hint 3: Suppose f(S,t) is the pde soln with the payoff 1(S<H), but no KO condition prior to maturity.
Last edited by Alan on August 9th, 2011, 10:00 pm, edited 1 time in total.

DoubleTrouble
Topic Author
Posts: 83
Joined: November 9th, 2010, 9:46 pm

Pricing Barrier option in BSM model

Thank you very much Alan. That was about the amount of help I needed to solve it!Best regards!

mj
Posts: 3449
Joined: December 20th, 2001, 12:32 pm

Pricing Barrier option in BSM model

The problem has been well studied. See eg chapter 8 of my book for derivation of the formulas using the martingale approach. (the concepts and practice of mathematical finance.)

DoubleTrouble
Topic Author
Posts: 83
Joined: November 9th, 2010, 9:46 pm

Pricing Barrier option in BSM model

Why didn't I think of that!? I have a copy like not even 2 meters to my right Thanks Joshi!
Last edited by DoubleTrouble on August 14th, 2011, 10:00 pm, edited 1 time in total.