There is something I still don't understand. The BS PDE is usually defined on a semi-infinite interval and for the far-field condition we truncate to a bounded interval. Then at S = 0 we let the PDE be satisfied identically. The questions are:1. What is the fincancial motivation for the case S = 0 (of course, negative S is financially impossible but ...)2. Is there some 'deep' reason why we cannot let s be defined in the whole real line?It's probably a dumb question with a very simple answer. The advantage of the whole real line is that it is a nice Cauchy problem.

QuoteOriginally posted by: CuchulainnThere is something I still don't understand. The BS PDE is usually defined on a semi-infinite interval and for the far-field condition we truncate to a bounded interval. Then at S = 0 we let the PDE be satisfied identically. The questions are:1. What is the fincancial motivation for the case S = 0 (of course, negative S is financially impossible but ...)2. Is there some 'deep' reason why we cannot let s be defined in the whole real line?It's probably a dumb question with a very simple answer. The advantage of the whole real line is that it is a nice Cauchy problem.It is just positivity. Many IR models allow for negative rates, and depending on the model you sometimes need to stretch the axis to -infty. SABR is an interpolation between normal and lognormal - for beta=0 it is normal, beta=1 lognormal; for beta<1/2 the axis goes to -infty, for beta>1/2 it stops at 0.If you prefer the whole real line, use log price or log moneyness instead of price.

QuoteOriginally posted by: figaroQuoteOriginally posted by: CuchulainnThere is something I still don't understand. The BS PDE is usually defined on a semi-infinite interval and for the far-field condition we truncate to a bounded interval. Then at S = 0 we let the PDE be satisfied identically. The questions are:1. What is the fincancial motivation for the case S = 0 (of course, negative S is financially impossible but ...)2. Is there some 'deep' reason why we cannot let s be defined in the whole real line?It's probably a dumb question with a very simple answer. The advantage of the whole real line is that it is a nice Cauchy problem.It is just positivity. Many IR models allow for negative rates, and depending on the model you sometimes need to stretch the axis to -infty. SABR is an interpolation between normal and lognormal - for beta=0 it is normal, beta=1 lognormal; for beta<1/2 the axis goes to -infty, for beta>1/2 it stops at 0.If you prefer the whole real line, use log price or log moneyness instead of price.That would be a good one x = log(S). But this transformation may not change the other parameters in the PDE too adversely. Some remarks are:1. The BS PDE is always (nearly) wriitten with variable S (I am lazy , don't want to do the transformation and may not be possible in general). So just using the original *would* be optimal in my case.2. But negative S, is that bad, wrong from any perspsective?

Cuch (or anyone) on a related topic what are the implications for doing a solitonic transform like

QuoteOriginally posted by: CuchulainnThere is something I still don't understand. The BS PDE is usually defined on a semi-infinite interval and for the far-field condition we truncate to a bounded interval. Then at S = 0 we let the PDE be satisfied identically. The questions are:1. What is the fincancial motivation for the case S = 0 (of course, negative S is financially impossible but ...)2. Is there some 'deep' reason why we cannot let s be defined in the whole real line?It's probably a dumb question with a very simple answer. The advantage of the whole real line is that it is a nice Cauchy problem.Let say that you define S on the whole real axis. You solve your Cauchy problem, and then study your solution on the positive real S axis only. Do you think you might find something new (it would be an interesting problem)?

QuoteLet say that you define S on the whole real axis. You solve your Cauchy problem, and then study your solution on the positive real S axis only. Do you think you might find something new (it would be an interesting problem)? HiWell, my rationale is :1. I find both far field and S = 0 boundary conditiions to be arbitrary to a certain extent. I can solve them in n factors but the main question is whether the REAL mathematical problem is a Cauchy problem (someone in the Annals of QF must have written the original article).I suppose each solution is OK but with Cauchy we do not have boundaries and that is nice. Of course, negative S is left out of the reporting. 2. Another option is to use a Moebius transformation to transform the problem to a circle, but I am not sure if it is generally applicable.So, having an infinite interval leads to two possible solutions above.

I guess when you boil down my question, are there any fundamental objections to combining time and space aswith solitons, besides creating messy BC's? I appreciate what you said elsewhere cuch cf messy Bc's/easier PDE, easy BC/messy PDE!

QuoteOriginally posted by: zetaI guess when you boil down my question, are there any fundamental objections to combining time and space aswith solitons, besides creating messy BC's? I appreciate what you said elsewhere cuch cf messy Bc's/easier PDE, easy BC/messy PDE!I want to get rid of BCS now!

you mean deal with a messy PDE?

QuoteYou solve your Cauchy problem,Karatzas and Shreve page 378, equation (8.35) define a Cauchy problem for BS PDE on THE SEMI_INFINITE REGION (0, Sinfnity). I think this is incorrect and is not a Cauchy problem.Their formulation does not lead fom their Theorem 7.6 which IS correctly posed as a Cauchy problem. I'll check it but the leap from infinite to semi-infinite interval is not clear.So, the question is: how did we arrive at a semi-infinite region while all the Feynman_Kac stuff talks about the whole real line? What piece of information am I missing, or is it a leap of faith? DefinitionCauchy problem means infinite region is all directions.Think it's going to be some warm milk tonite before bedtime!

zeta:solitons are not the natural symmetry for BS. You would use solitons if you had travelling wave solutions, whereas log S is drift-diffusive, with drift rt + convexity adjustment, and diffusivity vol^2.The FX convention is to work with BS call delta, D = N(d1) where d1 = (log F/K + 1/2 vol^2 t ) / vol sqrt(t) - that rescales the domain to [0, 1] and the S <-> D inversion is straightforward given that you know the BS vols. Advantage, already scaled to optionality so the domain is fixed. Disadvantage, structure in strike space manifests itself through thinning boundary layers in the delta space so you need adaptive discretization.Cuchulainn:You need to ensure that the implied pdf does not give you nonzero probability over negative strikes, otherwise your prices are off. This does happen with interest rates - even though nobody has ever written an option struck at negative rate, the fact we give them nonzero probability for hedging imposes a lot of structure on implied vol and pdf for positive strikes.Also, rather than truncating, maybe you can extrapolate? Something like domain rescaling with moving end points carrying the extrapolation information, where the extrapolation is BS mod your extrapolation of implied volatilities? Just a thought.

This is all in x=log S though. The leap consists of moving from price to log price. With nonuniform implied vol there is some extra work calculating the convexity adjustment to get the right forwards, but that is it.QuoteOriginally posted by: CuchulainnQuoteYou solve your Cauchy problem,Karatzas and Shreve page 378, equation (8.35) define a Cauchy problem for BS PDE on THE SEMI_INFINITE REGION (0, Sinfnity). I think this is incorrect and is not a Cauchy problem.Their formulation does not lead fom their Theorem 7.6 which IS correctly posed as a Cauchy problem. I'll check it but the leap from infinite to semi-infinite interval is not clear.So, the question is: how did we arrive at a semi-infinite region while all the Feynman_Kac stuff talks about the whole real line? What piece of information am I missing, or is it a leap of faith? DefinitionCauchy problem means infinite region is all directions.Think it's going to be some warm milk tonite before bedtime!

figaro,The issue is now a generic PDE that has a Cauchy problem becoming a PDE in a semi-infinite region. It is now a pure maths PDE problem, separate from any specific application.Compare to a heat equation in 1) finite rod 2) infinite rod, but 3)? formulation of heat equation in semi-infinite rod ??? (is there a source?)I have reached an impasse. What's confusing me is that BS PDE with S does not specifically say that S MUST be positive. Of course, x = log S does give us what we want, then back to BC that we must specify.

Actually that is not true. It is impossible to get propagation into negative S because of the S^2 zero of the diffusion coefficient at S=0; you can do some analysis generally - if the diffusion coeff is O(S^k) at S=0, propagation stops at S=0 for k>=1 and continues into negative S for k<1. That is what happens in SABR - SABR has Fokker Planck eqn u_t = S^(2 beta) u_SS. beta>=1/2 restricts S to be positive, beta<1/2 allows it to go negative. BCs don't come into it.Generally speaking, you need to check the diffusion coefficient at zero; if it is fundamental (k>=1) you can do the log transform to make the domain infinite, otherwise ignore the BC and let it go negative if it wants, hence the domain already is infinite.

QuoteIt is impossible to get propagation into negative S because of the S^2 zero of the diffusion coefficient at S=0; gotcha, I'm back on track again at S = 0. So for a basket put I just use the exact BS solution or solve a 1 dim PDE.