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Cuchulainn
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Who understands Black Scholes Boundary Conditions, really

May 27th, 2005, 9:47 am

I am really confused!
Last edited by Cuchulainn on May 26th, 2005, 10:00 pm, edited 1 time in total.
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chiron
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Who understands Black Scholes Boundary Conditions, really

May 27th, 2005, 11:24 am

what you mean... for call V(st,t)=max[st-k,0]???
 
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TraderJoe
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Who understands Black Scholes Boundary Conditions, really

May 27th, 2005, 11:41 am

What, I thought you wrote the book on pde's??? This can't be so!
 
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ppauper
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Who understands Black Scholes Boundary Conditions, really

May 27th, 2005, 12:24 pm

For americansThe value of the portfolio is continuous, plus smooth pasting.I guess this has to do with extending conditions on the free boundary to 2 or more dimensions ?If you can post the specific problem (or message me with it), I may be able to help
 
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blade
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Who understands Black Scholes Boundary Conditions, really

May 27th, 2005, 12:32 pm

Hi Cuchulainn, I seem to be having a fairly quiet afternoon, so I'll take a bite. What seems to be the misunderstanding with the B.C's ? Are you talking about spatial boundary conditions, ie S = 0, S = infinity, or temporal boundary conditions ? Blade
 
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DavidJN
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Who understands Black Scholes Boundary Conditions, really

May 27th, 2005, 2:32 pm

A really good place to read about this stuff is the seminal article by Robert Merton in the Bell Journal of Economics, 1973(!). I think it's called "The Theory of Rational Option Pricing".
 
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JamesH83
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Who understands Black Scholes Boundary Conditions, really

May 27th, 2005, 3:12 pm

You only need the payoff function at expiry (ie max[s-x,0]) to solve the Black-Scholes PDE for a european call right?
Last edited by JamesH83 on May 26th, 2005, 10:00 pm, edited 1 time in total.
 
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Cuchulainn
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Who understands Black Scholes Boundary Conditions, really

May 27th, 2005, 3:32 pm

QuoteOriginally posted by: TraderJoeWhat, I thought you wrote the book on pde's??? This can't be so! Of course, but I am looking for a complete rounded mathemtically sound theory (BTW I am getting pretty close to it) not a hand-waving string theory analogy with boxes of oscillators that you rant on about, bloody jerk.Stick to your Off topics, nobody likes you there either.Get lost JERK.
Last edited by Cuchulainn on May 26th, 2005, 10:00 pm, edited 1 time in total.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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Cuchulainn
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Who understands Black Scholes Boundary Conditions, really

May 27th, 2005, 3:35 pm

QuoteOriginally posted by: JamesH83You only need the payoff function at expiry (ie max[s-x,0]) to solve the Black-Scholes PDE for a european call right?That's no problem when the exact solution is known. Problem is when we truncate the semi-infinite interval and we have to use boundary conditions. So when to use: Dirichlet Neumann Convexity (this is the difficult one that has NOT been addressed sufficiently)
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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Cuchulainn
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Who understands Black Scholes Boundary Conditions, really

May 27th, 2005, 3:42 pm

QuoteOriginally posted by: bladeHi Cuchulainn, I seem to be having a fairly quiet afternoon, so I'll take a bite. What seems to be the misunderstanding with the B.C's ? Are you talking about spatial boundary conditions, ie S = 0, S = infinity, or temporal boundary conditions ? BladeWell, on the CFD thread there are lots of boundary conditions to choose from but the challenges are: A. Financial motivation B. Mathematical analysis C. Numerical approximation and stabilityThe CFD groups have a lot of experience in C (Alan, Sammus, Arnheim, Blade, Yomi), point B is probably discussed by some Russians somewhere and for A I got two diffrerent answers for a basket option.A, B and C are the challenges. I have not seem them addressed.Newcomer TJ (Pedro 666) is hanging around here. Blade, if you don't knoew him you can ignoire him.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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Cuchulainn
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Who understands Black Scholes Boundary Conditions, really

May 27th, 2005, 3:45 pm

QuoteOriginally posted by: chironwhat you mean... for call V(st,t)=max[st-k,0]???No, for this this is the Terminal condition (t = T). boundary conditions are when S = Smin and S = Smax.In the financial literature people speak of boundary C. but they mean bdy. t = T only.The payoff is OK even in n factors.
"Compatibility means deliberately repeating other people's mistakes."
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Cuchulainn
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Who understands Black Scholes Boundary Conditions, really

May 27th, 2005, 3:48 pm

Ppauper Here is PROBLEM I: mathematically and numerically it is fine, but a poster here says it is not Financially OK. What's wrong with Bhansali's approach?Yomiin Bhansali's book, p. 209 he looks at 2 factor baskets and he uses CN.PayoffP = max (K - a1S1 - a2S2, 0)BCS1 = 0: max (K - a2S2, 0)S1 = S1Max: 0.0 (Dirichlet)S2 = 0: max (K - a2S2, 0)S2 = S2Max: 0.0 (Dirichlet)Is this reasonable?For some problems one needs to solve the FDM scheme in a triangle?? Any ideas?
Last edited by Cuchulainn on May 26th, 2005, 10:00 pm, edited 1 time in total.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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Cuchulainn
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Who understands Black Scholes Boundary Conditions, really

May 27th, 2005, 3:50 pm

PROBLEM 2 : one-factor bond modelAlan states That is, d/dr B(r, t) = 0 at r = r_max is the natural truncation.The ORGINAL FORMUATION WAS working on the 1-factor bond model with CIR (Tavella page 126):Bt + 0.5 * sig^2 * Urr + (a-br)Ur - rB = 0BC are:r = 0Bt + ABr = 0 (PDE degenerates at origin)r = infinityUrr = 0Payoff Usual
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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http://www.datasim.nl
 
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Cuchulainn
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Who understands Black Scholes Boundary Conditions, really

May 27th, 2005, 3:54 pm

Problem 3 : boundary conditions Uss = 0It is well known that the Black Scholes PDE is defined in S direction on a semi-infinite interval. Since we cannot put infinity into the computer we must truncate so that a new PDE is defined on an interval [0, SMax]. Since this problem is now bounded we must define some condition at SMax, for example the linearity (convexity) BC:(1) Vss = 0 (second derivative of option V)A number of us (Sammus, Arnheim) are working on problems in which we must find some finite difference approximation to (1). I am working on ADE (explicit) methods and the use of FDM and ghost points for (1) are causing big problems. Some initial gut feeling remarks are:a) I have never seen BC like (1) anywhere in physics but option pricing PDE is the first time. What is common are Dirichlet (V = 0) or Neumann (Vs = 0).b) Have not been able to find financial motivation for (1) that is reasonably exact in treatmentc) I have difficulty incorporating (1) into ADE schemes. The schemes become very ‘unnatural’On a more detailed level, if we integrate (1) twice we get(2) V = A + BS at S = SMaxwhere A and B are arbitrary constants. Of course we may need to know values for A and B. One possibility is to use the compatibility relations between the payoff function (let’s take a call):(3) Payoff (S) = max (S – K, 0)If we let S become very big in (3) we se that V is essentially equal to S when S is big (fo example SMax). So, if we demand compatibilityPayoff(SMax) = V at SMax from (1)We should get A = 0 and B = 1. So at SMax we have (4) V(S = SMax) = SMaxor maybe (5) Vs = 1 at S = SMax (as is the case with Heston 1993)So, my questions are:A) Can we replace linearity BC by an equivalent (or more exact) BC?B) Is the above logic OK or am I missing some vital piece of information?C) Who invented the linearity condition?Any help welcome. I am sure some quant gurus out there can cast some light on this. I am in the dark here. ThanksDanielP.S. Personally, like (5) because it says that V increases linearly in S. This probably figures from the exact BS formula as well. This is also the same kind as in Heston 1993. It me
"Compatibility means deliberately repeating other people's mistakes."
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Cuchulainn
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Who understands Black Scholes Boundary Conditions, really

May 27th, 2005, 3:57 pm

QuoteOriginally posted by: DavidJNA really good place to read about this stuff is the seminal article by Robert Merton in the Bell Journal of Economics, 1973(!). I think it's called "The Theory of Rational Option Pricing".But I'm not sure if he handles the boundary conditions that I need.In QF, when people say BC they usually mean Payoff function.
Last edited by Cuchulainn on May 26th, 2005, 10:00 pm, edited 1 time in total.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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http://www.datasim.nl