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Circular barrier

Posted: August 1st, 2019, 6:22 am
by Seagulls
Does anyone know of an article which prices a GBM one-touch option hitting a barrier that is circular in shape? Thanks.

Re: Circular barrier

Posted: August 1st, 2019, 2:25 pm
by DavidJN
Try Kunitomo and Ikeda (1992), which is likely referenced in more recent papers.

Re: Circular barrier

Posted: August 1st, 2019, 2:48 pm
by Alan
To me, a circle requires two dimensions. What are the two dimensions? 

Re: Circular barrier

Posted: August 1st, 2019, 11:44 pm
by bearish
I was imagining something like two (possibly log) exchange rates (FX - given the "one-touch" terminology) X and Y, with the barrier being defined in terms of [$] X^2+Y^2=c [$].  This feels like it would lead you into Bessel processes and the like, but probably well covered in the probability theory literature.

Re: Circular barrier

Posted: August 2nd, 2019, 4:53 am
by Seagulls
Try Kunitomo and Ikeda (1992), which is likely referenced in more recent papers.
Thank you for the reference. I'll check it out.

Re: Circular barrier

Posted: August 2nd, 2019, 5:04 am
by Seagulls
To me, a circle requires two dimensions. What are the two dimensions? 
Alan, I've been through the index and it looks like a must-have book. Unfortunately I don't think it addresses my problem which I've covered with a diagram below. Thanks anyway.

Re: Circular barrier

Posted: August 2nd, 2019, 5:06 am
by Seagulls
I was imagining something like two (possibly log) exchange rates (FX - given the "one-touch" terminology) X and Y, with the barrier being defined in terms of [$] X^2+Y^2=c [$].  This feels like it would lead you into Bessel processes and the like, but probably well covered in the probability theory literature.
Just one exchange rate would be enough! I've expanded on the problem below. 

Re: Circular barrier

Posted: August 2nd, 2019, 6:02 am
by Seagulls
https://www.dropbox.com/s/96sbappmcaial ... D.png?dl=0

The problem is establishing the probability of the asset price (moving in a GBM) hits the circle, which in effect means it hits the arc CHD. The circle is static with centre A and radius AC and the current asset price is known. Therefore I can establish points C andD.
As I'm no mathematician and sometime (vaingloriously) call my self a financial engineer I would solve this in a tedious long form by breaking the arc into small sections (one degree increments) and then take the two points on the circle and create horizontal barrier through the average height of the two points. The width being the width between the two points. I would then approximate the probability of the asset price striking this narrow horizontal barrier by subtracting the one-touch call with expiry at the right most end of the barrier from the one-touch call with barrier at the left most point of the barrier. I would do this for each one degree segment and then aggregate the lot to get the probability of the asset price hitting the circle.
I'm damned sure it's not the sophisticated way you guys would do it but would this work?

Re: Circular barrier

Posted: August 2nd, 2019, 11:35 am
by bearish
Now I'm with Alan.

Re: Circular barrier

Posted: August 2nd, 2019, 7:46 pm
by Alan
I think GBM can hit every point on that circle! -- well, possibly excepting one, which doesn't matter.

DavidJN's cite may be spot-on, although I haven't looked at that paper in years. 

Off-hand, the horizontal line through A identifies two key times (t1,t2), and the GBM's location at t1 splits the problem into two exclusive possibilities. (Which are, hopefully, obvious).

Approximating the circle with purely horizontal pieces might be too crude. I vaguely recall parabolic segments (for BM)  are solvable -- that might be useful. 
Or maybe just a brute force lattice of some sort with a zillion nodes. Or a PDE. Many possibilities. 

But, again, whatever the numerics: solve two separate problems, starting from t1, and conditioning on the location of the motion at t1. Once you have those solutions, just integrate them over the transition density of reaching those starting points, which is just an elementary lognormal.

Anyway, that's how I'd do it.

Re: Circular barrier

Posted: August 3rd, 2019, 3:29 am
by Seagulls
Now I'm with Alan.
Thanks for your time, bearish. 

Re: Circular barrier

Posted: August 3rd, 2019, 3:32 am
by Seagulls
I think GBM can hit every point on that circle! -- well, possibly excepting one, which doesn't matter.

DavidJN's cite may be spot-on, although I haven't looked at that paper in years. 

Off-hand, the horizontal line through A identifies two key times (t1,t2), and the GBM's location at t1 splits the problem into two exclusive possibilities. (Which are, hopefully, obvious).

Approximating the circle with purely horizontal pieces might be too crude. I vaguely recall parabolic segments (for BM)  are solvable -- that might be useful. 
Or maybe just a brute force lattice of some sort with a zillion nodes. Or a PDE. Many possibilities. 

But, again, whatever the numerics: solve two separate problems, starting from t1, and conditioning on the location of the motion at t1. Once you have those solutions, just integrate them over the transition density of reaching those starting points, which is just an elementary lognormal.

Anyway, that's how I'd do it.
Thanks for your time, Alan. Much appreciated.

Re: Circular barrier

Posted: August 3rd, 2019, 8:26 am
by Seagulls
Alan, the simple solution (for me) would be the ability to determine the probability that a BM will hit a non-horizontal straight line. I would then be able to work my way round the arc and, in effect, be able to aggregate all the probabilities.

Re: Circular barrier

Posted: August 3rd, 2019, 7:15 pm
by Alan
With the exception of horizontal lines and maybe parabolas, I believe everything else requires numerics. 

Re: Circular barrier

Posted: August 4th, 2019, 2:46 am
by Seagulls
With the exception of horizontal lines and maybe parabolas, I believe everything else requires numerics. 
OK. Thank you, Alan.