Serving the Quantitative Finance Community

 
User avatar
Cuchulainn
Topic Author
Posts: 64439
Joined: July 16th, 2004, 7:38 am
Location: Drosophila melanogaster
Contact:

Who understands Black Scholes Boundary Conditions, really

November 4th, 2008, 3:41 pm

Quote2. I have tried mapping x in (0,infinity) to (0,1) in related 2D problems (Heston model) in which this process plays a role, withoutsuccess. Also, have yet to see anybody succeed in that (in a finite difference solver) for that model or similar. Have to say, I only tried one schemeWhich mappings (2, yes?) did you use?
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl
 
User avatar
Cuchulainn
Topic Author
Posts: 64439
Joined: July 16th, 2004, 7:38 am
Location: Drosophila melanogaster
Contact:

Who understands Black Scholes Boundary Conditions, really

November 4th, 2008, 3:50 pm

Alan,did you just change a > 1/2 to a > 1? Or maybe I just imagined it. I understand because you have v = 1 and no factor 1/2 in CIR PDE?For x = 0 and a >= 1 ==> no BC allowed in my view.a < 1 ==> need BC.
Last edited by Cuchulainn on November 3rd, 2008, 11:00 pm, edited 1 time in total.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl
 
User avatar
Alan
Posts: 10612
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

Who understands Black Scholes Boundary Conditions, really

November 4th, 2008, 3:57 pm

Just tried mapping the volatility coordinate to (0,1) using Tanh Yes, made a whole bunch of edits exactly because of that factor of 1/2 -- my earlier post soln isnow correct, pretty sure .
Last edited by Alan on November 3rd, 2008, 11:00 pm, edited 1 time in total.
 
User avatar
Cuchulainn
Topic Author
Posts: 64439
Joined: July 16th, 2004, 7:38 am
Location: Drosophila melanogaster
Contact:

Who understands Black Scholes Boundary Conditions, really

November 4th, 2008, 4:55 pm

QuoteOriginally posted by: AlanJust tried mapping the volatility coordinate to (0,1) using Tanh .Is there anything known about asymptotic behaviour of the solution? A kind of "asymptotic TBC"??????edit:QuoteThe maps are easy to cook up -- just try getting them to work in a 2D solver. No problem, I just define a new PDE function.Tanh() is a pain (differentiation, limits) etc. What about trying exp(-S) for CIR PDE? I get Feller condition and BC but have not done the numerics yet. And CIR is a bit like Heston..
Last edited by Cuchulainn on November 3rd, 2008, 11:00 pm, edited 1 time in total.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl
 
User avatar
Alan
Posts: 10612
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

Who understands Black Scholes Boundary Conditions, really

November 4th, 2008, 5:20 pm

See Ch. 10 of my (existing) book for a way to derive it.
 
User avatar
JohnLeM
Posts: 515
Joined: September 16th, 2008, 7:15 pm

Who understands Black Scholes Boundary Conditions, really

November 4th, 2008, 5:23 pm

I was having a look to the existing web literature for Heston model and the numerical schemes devoted to them.I found this link: http://eprints.maths.ox.ac.uk/718/01/Se ... thesis.pdf, that seemed quite complete at a very quick first glance.Would you recommend it ?
 
User avatar
Alan
Posts: 10612
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

Who understands Black Scholes Boundary Conditions, really

November 4th, 2008, 5:45 pm

I glanced briefly. I commend the author for showing the error at the truncated domain corner (Smax,Vmax).Most authors don't.But, as you can see, those errors are huge, and due precisely to the problem I have raised in this thread:the lack of a TBC along the line V=Vmax < infinity. (Mostly, the whole literature on this model has the same problem --hence, the Fame and Glory stuff ...)
Last edited by Alan on November 3rd, 2008, 11:00 pm, edited 1 time in total.
 
User avatar
Cuchulainn
Topic Author
Posts: 64439
Joined: July 16th, 2004, 7:38 am
Location: Drosophila melanogaster
Contact:

Who understands Black Scholes Boundary Conditions, really

November 4th, 2008, 7:14 pm

QuoteOriginally posted by: AlanI glanced briefly. I commend the author for showing the error at the truncated domain corner (Smax,Vmax).Most authors don't.Was the author aware of the problem? It must be pointed out that one of the boundaries in author's BC (based on Heston 1993) is degenerate, i.e. v= 0 and no BC is allowed but only if Feller condition kappa*theta >= sig^2/2. Otherwise, it's a new discussion. Any other BC are unfounded imo.Imposing BC along S = Smax and on V = Vmax will introduce a discontinuity because of the way they are chosen, yes? Or is there another reason? Heston 1993 has dC/dv = 0 on Vmax and dC/dS = 0 on Smax, so how can we ever get a smooth solution!!!! Are these BC wrong?//I see no stess testing of rho, another issue (see Sheppard's thesis that uses Soviet splitting)
Last edited by Cuchulainn on November 3rd, 2008, 11:00 pm, edited 1 time in total.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl
 
User avatar
Cuchulainn
Topic Author
Posts: 64439
Joined: July 16th, 2004, 7:38 am
Location: Drosophila melanogaster
Contact:

Who understands Black Scholes Boundary Conditions, really

November 4th, 2008, 7:23 pm

QuoteOriginally posted by: CuchulainnQuoteOriginally posted by: AlanJust tried mapping the volatility coordinate to (0,1) using Tanh .Is there anything known about asymptotic behaviour of the solution? A kind of "asymptotic TBC"??????edit:QuoteThe maps are easy to cook up -- just try getting them to work in a 2D solver. Tanh() is a pain (differentiation, limits) etc. What about trying exp(-S) for CIR PDE? I get Feller condition and BC but have not done the numerics yet. And CIR is a bit like Heston..update
Last edited by Cuchulainn on November 3rd, 2008, 11:00 pm, edited 1 time in total.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl
 
User avatar
Alan
Posts: 10612
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

Who understands Black Scholes Boundary Conditions, really

November 4th, 2008, 8:16 pm

I will be interested to hear if some map works for the 2D Heston problem with correlation.Other cases (esp. 1D) seem much less interesting, unless there is some reason why theyshould also work in 2D.
Last edited by Alan on November 3rd, 2008, 11:00 pm, edited 1 time in total.
 
User avatar
Cuchulainn
Topic Author
Posts: 64439
Joined: July 16th, 2004, 7:38 am
Location: Drosophila melanogaster
Contact:

Who understands Black Scholes Boundary Conditions, really

November 4th, 2008, 8:18 pm

QuoteOriginally posted by: AlanI will be interested to hear if some map works for the 2D Heston problem only with correlation, asother cases seem much less interesting.you mean justU_t = rho * x * y * U_xy? edit: NoSo, the full model without taking Call or Put BC into consideration?
Last edited by Cuchulainn on November 3rd, 2008, 11:00 pm, edited 1 time in total.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl
 
User avatar
Alan
Posts: 10612
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

Who understands Black Scholes Boundary Conditions, really

November 4th, 2008, 8:25 pm

Better grammar: I will be interested to hear if some map works for the full 2D Heston problem forputs or calls, including a cross term for correlation. Other cases (esp. 1D) seem much less interesting. The BC are whatever is appropriate for the (unknown at this point) numerical method.
Last edited by Alan on November 3rd, 2008, 11:00 pm, edited 1 time in total.
 
User avatar
Cuchulainn
Topic Author
Posts: 64439
Joined: July 16th, 2004, 7:38 am
Location: Drosophila melanogaster
Contact:

Who understands Black Scholes Boundary Conditions, really

November 4th, 2008, 8:29 pm

QuoteOriginally posted by: AlanBetter grammar: I will be interested to hear if some map works for the full 2D Heston problem forputs or calls, including a cross term for correlation. Other cases (esp. 1D) seem much less interesting. The BC are whatever is appropriate for the (unknown at this point) numerical method.gotcha! That's what I was on.BTW, my view is that the original Heston BCs are WRONG!
Last edited by Cuchulainn on November 3rd, 2008, 11:00 pm, edited 1 time in total.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl
 
User avatar
Alan
Posts: 10612
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

Who understands Black Scholes Boundary Conditions, really

November 4th, 2008, 8:38 pm

Well, I am looking at eqn (9), Heston (1993). Which BC don't you like?
 
User avatar
JohnLeM
Posts: 515
Joined: September 16th, 2008, 7:15 pm

Who understands Black Scholes Boundary Conditions, really

November 5th, 2008, 10:54 am

Alan : I am looking to a pdf available paper that describes precisely how to price an European instrument written over a CIR process using explicit Monte Carlo simulations with a Bessel Square root process (or altenatively Ornstein Uhlenbeck process). It surely exists, but can't find it by google.Have you any suggestion ?