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terance
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Boundary conditions for Black-Scholes-Merton SDE

March 4th, 2013, 7:17 pm

Hello!The familiar PDE:under the assumption, that boundary conditions:Would you be so kind to explain me the following:1) Are there any arguments behind the number of these conditions?2) What this boundary condition (basically, it says that after time of expiration the slope of the price of european option should be equal to 1)Thank for your answers! PS V=C, made a mistake.
Last edited by terance on March 3rd, 2013, 11:00 pm, edited 1 time in total.
 
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LocalVolatility
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Boundary conditions for Black-Scholes-Merton SDE

March 4th, 2013, 11:54 pm

1.a) In any arbitrage free model, the level zero has to be absorbing for the stock price. Thus, if the stock price is ever zero it has to stay there forever with probability one. Consequently, all call options on it expire worthless with probability one.1.b) You might want to replace this with . As the stock price becomes large, the probability of exercising the option at maturity goes to one. Thus the proportion of the time value that comes from the optionality component (and not the financing) goes to zero. The call option is then like a forward contract where the forward price is paid immediately and this is exactly the above value.1.c) This is just the payoff function.2) This makes no sense to me. Once the option is expired, the delta is not defined any more.
 
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terance
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Boundary conditions for Black-Scholes-Merton SDE

March 5th, 2013, 11:42 am

QuoteOriginally posted by: LocalVolatility1.a) In any arbitrage free model, the level zero has to be absorbing for the stock price. Thus, if the stock price is ever zero it has to stay there forever with probability one. Consequently, all call options on it expire worthless with probability one.1.b) You might want to replace this with . As the stock price becomes large, the probability of exercising the option at maturity goes to one. Thus the proportion of the time value that comes from the optionality component (and not the financing) goes to zero. The call option is then like a forward contract where the forward price is paid immediately and this is exactly the above value.1.c) This is just the payoff function.2) This makes no sense to me. Once the option is expired, the delta is not defined any more.Thank you very much for your answer!Btw, is it possible for you to clarify why is it enough to solve PDE with three boundary conditions (for call option price)? Mb, it is enough 2 boundary conditions?
 
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Cuchulainn
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Boundary conditions for Black-Scholes-Merton SDE

March 5th, 2013, 11:59 am

Quote1.a) In any arbitrage free model, the level zero has to be absorbing for the stock price. Thus, if the stock price is ever zero it has to stay there forever with probability one. Consequently, all call options on it expire worthless with probability one.1.b) You might want to replace this with . As the stock price becomes large, the probability of exercising the option at maturity goes to one. Thus the proportion of the time value that comes from the optionality component (and not the financing) goes to zero. The call option is then like a forward contract where the forward price is paid immediately and this is exactly the above value.These BCs can also be 'deduced' mathematically from Fichera theory and that the solution is continuous at S = 0 and at S = SMax.Better still, transform S to (0,1) and apply Fichera again. You could use Fichera for puts as well. If financial motivation breaks down, use Fichera and vice versa.BTW S = SMax is a numerical trick and not part of PDE formulation. QuoteBtw, is it possible for you to clarify why is it enough to solve PDE with three boundary conditions (for call option price)? Mb, it is enough 2 boundary conditions? This jargon is wrong. There are 2 BC and 1 initial (or terminal) condition, although QF literature does use this incorrect terminology.t = T is NOT a boundary.//Integate once in t to give an unknown constantIntegate twice in S in t to give 2 unknown constants3 unknowns ! Quote2) What this boundary condition (basically, it says that after time of expiration the slope of the price of european option should be equal to 1)This BC cannot be motivated by Fichera AFAIK, maybe it is pathological case?(?)
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LocalVolatility
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Boundary conditions for Black-Scholes-Merton SDE

March 6th, 2013, 3:38 am

QuoteThese BCs can also be 'deduced' mathematically from Fichera theory and that the solution is continuous at S = 0 and at S = SMax.Thank you for pointing that out - very interesting. I just had a look at your paper http://papers.ssrn.com/sol3/papers.cfm? ... id=1552926. I noticed three little typos and thought you'd appreciate to be notified about them:i) Equation 31: The l.h.s. should be .ii) Equation 47: The two last terms on the r.h.s. should be .iii) Equation 48: Beta should be defined as .
 
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Cuchulainn
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Boundary conditions for Black-Scholes-Merton SDE

March 6th, 2013, 6:28 am

Thanks, LocalVolatility for the heads up. You are correct(I have an extended and corrected chapter in my recent C# book on ADE).The Fichera theory has been discussed quite a bit on Wilmott. Domain transformation(orthogonal to ADE) is also very useful.In the next few weeks I plan to post a note on how find BCs from 1st principles without having to invoke Fichera nor finance motivation. The example is CIR and I need an inner product/variatonal approach like in FEM but due to the unbounded reaction term at infinity I need to use a weighted Sobolev space in which to prove the energy inequalities. Then we can 'reproduce' the Feller condition (and more) in this way. The more I think of it the more I am beginning to believe that FEM is a candidate for these PDEs.BTW in the Wilmott qfcl project the prototype FD2 implements a 2-factor ADE. Might be useful.
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ppauper
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Boundary conditions for Black-Scholes-Merton SDE

March 6th, 2013, 8:46 am

QuoteOriginally posted by: teranceHello!2) What this boundary condition (basically, it says that after time of expiration the slope of the price of european option should be equal to 1)i) for American options, you have that condition (dV/dS=1) at the free boundaryii) for Europeans, you are talking about after expiration so you have replaced the option with max(S-X,0) which has a slope of 1 if S>X and 0 if X<S.
 
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Cuchulainn
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Boundary conditions for Black-Scholes-Merton SDE

March 6th, 2013, 1:59 pm

For early exercise, 3 BCs are needed- at far field- 2 conditions at the (unknown) moving boundary.
 
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ppauper
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Boundary conditions for Black-Scholes-Merton SDE

March 6th, 2013, 2:22 pm

QuoteOriginally posted by: CuchulainnFor early exercise, 3 BCs are needed- at far field- 2 conditions at the (unknown) moving boundary.indeed, and for american calls, those 2 conditions are V=S-X and the (dV/dS)=1 that he had
 
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terance
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Boundary conditions for Black-Scholes-Merton SDE

March 9th, 2013, 7:35 pm

Big thanks to everybody for your reply in this thread!