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TT

Closed form solution available?

September 19th, 2001, 12:39 am

I am trying to value a variant of an up-and-out option. This option is similar to a standard up-and-out barrier option without rebate, except that to be "out," the stock price must be at or above the barrier for 30 consecutive trading days. I'd be interested if you know of a paper or other source that has a closed form solution for valuing this option. Thanks in advance for your assistance.
 
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Collector
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Closed form solution available?

September 19th, 2001, 1:29 am

I don't know of any published paper on this barrier variant. This is a strong path dependent option. One way is naturally to use Monte Carlo simulation (at least if European style, American more difficult). I know this special case of a barrier option is quite popular in the equity market (as warrants). A closed form solution (or approximation) would be nice to have (also for a second edition of my book ) . Numerical methods are naturally also of interest. A closed form solution can at least be used as benchmark for other numerical models, and would give us some additional insight.So many papers are written on how to come up with a new and better? Asian option approximation. Instead of coming up with more closed form approximations for Asian options I hope researchers will spend some time on this interesting and exotic barrier option !I have also seen another variant of the option TT is mentioning, trading in the equity market. A variant where you for example have a 30 -day rolling average that need to hit the barrier to get knocked out (or in). There is actually a hole class of "rolling underlying" barrier options out there. unfotunatley I can not see anyone that have published formulas for these?. Monte Carlo naturally works, but it is quite slow even in C++.
 
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Paul
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Joined: July 20th, 2001, 3:28 pm

Closed form solution available?

September 19th, 2001, 6:00 am

This is what's known as a Parisian option, God knows why though! As C says, it's strongly path dependent. While this doesn't rule out closed-form solution it certainly makes it harder to find. The only published works I know of are1. Chesney et al Risk Magazine 1997 (not sure which issue). That's supposed to be about closed-form solution via Laplace transforms which is rather messy.2. Haber, Schonbucher and PW which is all about the numerical solution. I've uploaded my paper, and a piece of software for pricing these things. Aren't you impressed with the service!!!P
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parisian.zip
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reza
Posts: 6
Joined: August 30th, 2001, 3:40 pm

Closed form solution available?

September 19th, 2001, 3:39 pm

hi, interestingly I had to implement this at work and the first approach I used was the “closed form” you mentioned in RISK. By the way, the reason it is called Parisian Option is simply that it was derived by a bunch of academics from Paris: Monique Jeanblanc, M. Chesney, M. YorYou can take a look at the paper "Brownianexcursion and Parisian Barrier options" which can be found at www.univ-evry.fr/labos/lami/maths/pages ... nc.htmlThe problem is that the solution is a Laplace Transform that needs to be inverted numerically. There are many methods for this however this inversion is very very non-robust and unstable (I emailed Monique Jeanblac and she confirmed) so the method is NOT practically helpful.I then used an explicit version (Trinomial Tree) of the Haber-Schonbucher-Wilmott grid and it seems to be working well. (No flattery intended)A very similar approach was described in Avellaneda and Wu's paper "Pricing Parisian options with a trinomial lattice method" available on http://www.math.nyu.edu/research/lixin/ (but be careful, unlike Haber-Schonbucher-Wilmott, they suppose for the option to knock-out stock has to be STRICTLY above barrier and also they use a closed-form for the knock-in probability once the stock is above the barrier which unlike Haber-Schonbucher-Wilmott, would not allow you to price American options)
 
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Paul
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Joined: July 20th, 2001, 3:28 pm

Closed form solution available?

September 19th, 2001, 3:42 pm

reza, you've made my day!P
 
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moonh
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Joined: December 19th, 2001, 8:27 pm

Closed form solution available?

December 19th, 2001, 8:32 pm

I downloaded parisian option zip file, but pdf file is not good. Some of the characters are broken. Can you upload another one or can you email them to me? Thank You!!!
 
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Paul
Posts: 6604
Joined: July 20th, 2001, 3:28 pm

Closed form solution available?

December 19th, 2001, 9:18 pm

Here you go. No figures though.P
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paris newpdf.zip
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Oracolo

Closed form solution available?

December 22nd, 2001, 9:48 am

Interesting the Parisian options, but the hedging of this options.... is dangereus for gamma control!