Proposed by Cuchulainn. His suggestion also gives a few clues about why closed-form solutions may not be as useful as you might think:"What is the added value of closed [-form] solutions if 1) they are based on an ANSATZ and 2) they are incredibly difficult to solve numerically? Example: The Heston 1993 paper 1) he assumes solution in a special form and 2) brittle complex integral"P

While closed form solutions 'can' be difficult, one of their advantages is thatthey are usually 'easy' to implement.If you are going to first study stochastic volatility, where do you start?If you are going to add price jumps to that, where do you start?If you are going to add volatility jumps to that, where do you start?If you are going to test a numerical or approximation method, what do you test it against?If you are going to try a calibration against 100+ options, using models with a half dozen or more parameters, how do you get it working?The answer is usually to start with a model that has a so-called closed-form or exact solution. Need I remind that the first closed-form model that attracted alot of attention was the Black-Scholes model. The many advantages of exact solutions seem obvious to me. Having said all that, I would acknowledge that one has to be quite carefulwith the Heston '93 model, especially at relatively large 'volatility of volatility'.There are several reasons to be careful. First, there are branch-cut crossings associated with a logarithm.But these can be dealt with or avoided. Second, one has to avoid 'over-reliance'on closed-form models just because they can be solved. This is true forany model. After all, you expect the model tobe 'wrong' quantitatively, but it may also be wrong/misleading in some qualitative way.Again, large vol. of vol. is an issue with the particular model in the examplebecause it makes the volatility reflect off the origin. In my opinion, this is 'bad'.But that is another story.

- Cuchulainn
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Step I--------I am not against techniques that describe the qualiitative properties of the solutiion to the fnancial model as it were. Things like smoothness, growth, and stuff. Then we can appreciate the global solution.Step II--------Finding a closed solution nice if it is possible. But should not be an end in itself. Step III---------Numerical approximation of the problem in I and/or IIMathematiciains very seldom believe that II is feasible, unless it's serendipity. They, as they have done for centuries, approximate the mathematical problem in I by some fnite-dimensional pronlems and then let the degree of the dimension go to infinity.Don't get me wrong. I am just trying to understand something I do not understand.To take an example; the complex integral in Heston 1993 is difficult IMO. Personally I have never done a complex integral numercally, but the Heston PDE is fine and I can change all parameters and still get an answer.IMO a lot of clever PURE maths goes into finding closed solutions. That's a skill I have neglected for some time. I have computers to get a solution.My feeling is/question: closed solution is fine for qualitative reasoning but may be diffcult to COMPUTE?Maybe I am missing the point. I am open to suggestions.

- Cuchulainn
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> Need I remind that the first closed-form model that attracted a lot of attention was the Black-Scholes modelFine, only these days things are more complex e.g. non-linear etc. It's greate when you can get an exact solution.

This is not specific for quants and being none ... partially this is a kindof 'credo'. There are extremely provocating positions on that (Zeilberger)and little common agreement what is 'closed form'. My personal opinion isthat you will need various views and a 'rich' structure to develope thingsfurther.A simple thing like logarithm can be given through its Taylor series, butit will give you almost no insight (and will practically fail in the large).Writing down a recursion to compute the normal distribution to arbitraryexactness is fine, but you will not be able to compare it to the bivariatecase or apply it to slightly modified situations.So you answered your Q yourself ... while Step 3 needs closed forms around :-)For Heston (ok, i never tried to work within your prefered techniques):besides taking care (as Alan said) it is like often - if you get used to itis seems not to be incredibly difficult and as far as i remember Heston didconsider his solution to be of closed form ... strange, but may be he hadsomething like QUADPACK (a Fortran package) in mind to get numerical values.

- Cuchulainn
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QuoteOriginally posted by: AVtSo you answered your Q yourself ... while Step 3 needs closed forms around :-)For Heston (ok, i never tried to work within your prefered techniques):besides taking care (as Alan said) it is like often - if you get used to itis seems not to be incredibly difficult and as far as i remember Heston didconsider his solution to be of closed form ... strange, but may be he hadsomething like QUADPACK (a Fortran package) in mind to get numerical values.I would be interested in knowing how many hours of sweat and emptied waste-paper bins Heston has before he got the formula. A practical problem is that some people do not use Fortran (I find this a pity) but are using C++. So they have to find an integrator in C++, C# for example.It's great having a closed solution if it adds to one's qualitatve insights or can be used as a stepping stone to more specific algorithms. From my experience I have seldom come across a closed solution that maps all that easily an approximtae one.

- Cuchulainn
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Avt,The algorithm for QuadPACK documented anywhere? It's just can I have not come in contact with complex integrals. How does it work?

i see ... so it is going a little bit aside and beyond the topic ...lot of what might help you is coverd by GNU GSL, an open C lib: integration http://www.gnu.org/software/gsl/manual/ ... SEC248cplx numbers http://www.gnu.org/software/gsl/manual/ ... l#SEC38but i do not know a C++ interfacing toolthe others might be 'commercial' libs like NAG

- Cuchulainn
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> i see ... so it is going a little bit aside and beyond the topic ...Yes and no, If I see QUAD for a closed solution, I can understand better. Probably an aside.However, let's get back on track.

I think that it is important to have a very efficient (or fast) way to value vanillas, for calibration purposes. At this stage of the battle it seems that closed form (or semi-closed form) is the best way to go. Maybe some time in the future we will be able to have a finite difference scheme that runs in 0.00001 sec and gives 16 digits of accuracy, then we will be able to use FDM for calibration purposes and it will be our new 'closed form solution'.I am busy with the implementation of FDM`s for the evaluation of exotics in a stochastic vol environment. I find that the 'close form' solutions on AVt`s website is very useful as benchmark (using reasonable parameters obviously).R

- KackToodles
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Only closed form solutions lend lucid insight.

- Cuchulainn
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QuoteOriginally posted by: KackToodlesOnly closed form solutions lend lucid insight.Any examples?Would you consider an integral of a complex (in the sense of x + iy) function on an infinite domain to be lucid, for example.My feeling is that closed forms describe the qualitative properties and numerics the quantitative properties of the solution. You need both.

Last edited by Cuchulainn on September 13th, 2006, 10:00 pm, edited 1 time in total.

- Traden4Alpha
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Closed-form illustrates the tremendous strengths and weakness of math. On the one hand, analytic solutions let one prove universal properties about the system and its solutions. Proving that the system will always do a certain thing regardless of parameters, inputs etc. is extremely powerful. Numerical methods can't prove that a system will always behave a certain way as even an exhaustive Monte Carlo or systematic exploration of the parameter space can miss significant anomalous regimes of behavior.On the other hand, a closed-form solution is only as good as its weakest axiom. If the real system doesn't obey the axioms required for the math, then the solution can have some deep flaws. This can impact numerical methods too if they use the same mathematical underpinning. Where numerical methods can shine is when they simulate mathematically intractable behaviors such as non-normally-distributed price movements or nonlinear multi-body dynamics.

So what are the boundary conditions for your Cauchy problem?Either closed-form or numerical...

- Cuchulainn
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QuoteOnly closed form solutions lend lucid insight. This is certainly true in Physics. In mathematics, it may be different.A formula on paper is great, but it comes to life when it produces the numbers that you expect. You got to do something with it, yes?

Last edited by Cuchulainn on October 3rd, 2006, 10:00 pm, edited 1 time in total.

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