- ZmeiGorynych
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A closed form solution can be useful in the early stages of exploring a problem. The complete closed form solution (as a function of parameters, if any; and complete with proof that there are no other solutions) is great for understanding what the solutions can look like.If, in addition, it can be evaluated numerically more efficiently than solving the original equation, it's clearly useful. Otherwise, it's just one more representation of the equation, eg in integral vs. differential form, and sometimes one representation can However, for me integrals (especially complex and over unbounded areas) begin to stretch the definition of 'closed form'. When people say closed form is great I'd say they usually think of closed form as 'combination of known functions', such as BS.

Last edited by ZmeiGorynych on December 9th, 2006, 11:00 pm, edited 1 time in total.

Lets reinterpret this question a little bit.How do you really understand a phenomenon? Closed form does not always necessarily lead to greater understanding. In finance, it is crucial to understand the dynamics - what essence is captured, what is missed.I would be with Feynman on this. Models are only rough appoximations that help to build and clarify intuition - the phenomenon is much more. To really know something: to live, breathe and understand it, is something totally different. I thinking asking the right questions before beginning the modelling is rather more important. When you change the question a little bit you often get a new model, so whether there is a closed form solution to one particular set of questions (i.e., model) is hardly as interesting as trying to formulate the more relevant question.

- exneratunrisk
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in axiomatic mathematics, closed form solution means, "all quantifiers are eliminated"In other words, you can use it for infinitive many "cases" (and do not need to test finite many of them to trust...)"For all n; does there exist a k; so that sum(i, i=1,..n)=k?" yes, k=n(n+1)/2 (for all and existence quantifiers eliminated)In the sense of axiomatic matematics "sum i" and "n(n+1)/2" are "identical" because they have identical i/o relations.In algoritmic math, the difference makes sense. Remark. Closed form solutions are often bound to small, constrained "worlds". ("for all" quantifier is too mighty?).

Last edited by exneratunrisk on May 15th, 2007, 10:00 pm, edited 1 time in total.

- Cuchulainn
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Closed form solutions give us an idea of the qualitative properties of an equation. This may or may not be necessary; for example, to prove the existence and uniqueness of the equation. The solution may or may not be computable.On the other hand, constructive mathematics allows us to prove existence and uniqueness as well as find the solution as in the Brouwer's (btw a Ducth guy who was @zeta's uncle I believe) fixed point theorem. FTP is used a lot in numerical analysis, Leonardo of Pisa was the first to use it in 1225 to find the roots of equation///////x^3 + 2x^2 + 10x - 20 = 0write as x = g(x) == 20/(x^2 + 2x + 10)Now do iteration on the computerx(n+1) = g(x(n)) with x(0) = 1///////The same idea holds for more complex applications, because g() is a contraction mapping. So, even in the above simple case we have both analytics and numerics.But this is an old discussion between mathematicians. FPT I suppose the choice depends on what is needed and whether the solution fits the requirements.

Last edited by Cuchulainn on September 6th, 2007, 10:00 pm, edited 1 time in total.

Maybe a newbie point of view may help.For a quant, I guess closed-form refer to analytical expressions which are more or less difficult to implement in a computer but are faster to evaluate than the underlying experiment they refer to. Like ln2 for instance.Another advantage comes when one tries to modify the hypotheses related to the closed form and cover more general cases or slightly different ones. Then, one way to go is to try to find another closed form, which is a 'perturbation' of the first one, and have criteria for the validity of this expansion.Also, as in many fields, closed form allows for determining which parameters are relevant, and how they drive the closed-form.About the meaning of any closed form, I think in most cases a closed-form brings very little to our understanding. Most often, it is its derivation that contains most of what is useful, and of course, the hypotheses made.Let's give few examples of my point of view :1 - Assume you would like to compute an integral of a very complicated functions over a stochastic process. But because you are clever, you notice that both the process and the integrand have very peculiar symmetries which, once combined, will give you the trivial result : 0. I guess this could be called the ultimate closed-form. It is clearly very useful to anyone to know that form : no one knowing it would run a simulation of corresponding paths of the process and compute the values of the function along them. But if someone wants to modify a little the function, he should check first whether the new function also has the very peculiar symmetries that simplifies the all stuff.2 - Your problem is to find a way to bike over a road with a random profile carrying a given load of stone, but most of all, to find the work that you'll do doing this. Someone else tells you that if the road profile, even random, follows some general rules, then he's able to find a fancy formula for the work. Again, this is very helpful, but your only knowledge will be what comes out of the formula. On the other hand, if you try to derive it yourself, you could come with the observation that the more the road goes up, the greater the chance it will go down after that. At each time step, you'll find also that you'll have to stay on your bike, which will involve some dynamical corrections of your position on it, to balance your weight with the load, the slope of the road that has suddenly change, and your speed. Again, the closed-form formula is useful but not as is the complete understanding to obtain it. [I'm sure the analogy with what I'm thinking to is obvious.]3 - You would like to know how does a ball oscillate in a quartic well. In fact, your well is not completely quartic, it is quadratic, but with a rather small quartic contribution to its curvature. Again, if someone gives you a fancy formula, you're happy with it, provided of course that the guy also told you when you can apply it, e.g when the ratio of quartic over quadratic is small enough. So, if you need that because you spend all your day long at the desk waiting for phone calls like : hey, I have a well of this kind and a ball of that one, what will be the oscillation period? You don't really care about the derivation of your expression. But if you sit in an obscure office quite far from that desk, and ask yourself what you could do to improve this closed-form with, say, cubic terms in the well profile, you must understand how the expression was derived and that, as soon as correction to quadratic are small, the new period will be modified by adding multiple of the first one : T + 2T + 3T ... Therefore, if by chance, you bring some coffee to the very nice guy sitting at the desk the day he receives an unusual phone call about an unusual well profile, you can just tell him : if corrections to quadratic are small, don't take them into account, it will just add harmonics, but the main oscillation will prevail.

Quoteall the haphazard traits eraseyou'll see then that the world is fairclosed forms r beautiful and simple. scientists want simplicity, they want to remove all unnecessary details, and look at the essence, which is supposed to be simple.ok, it has nothing to do with quant fin, sorry

Below is a link to the best explanation I have been able to find on this topic.Closed-form

- Cuchulainn
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Having done a CFS(either doing it by hand or in Mathematica, for example), one can hopefully get new insight, such as:1. Qualitative behaviour of the dependent variables (bounds, positivity, asymptotics etc.)2. Numerics3. Maybe other insights?In many examples, it is difficult to realise 1 because of the complexity of the expression and numerical schemes are not well-posed with respect to the parameters. CFS based on series and asymptotic expansions seem to be sensitive.In this case CFS are not so useful. This might be subjective... But that is a part of the discussion/equation here.

Last edited by Cuchulainn on July 8th, 2008, 10:00 pm, edited 1 time in total.

- Cuchulainn
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If you take multi-asset options, then each type leads to a new closed formula (e.g. Margrabe), so each time there is a new payoff then we have to do some maths for some time to get a solution.This is very elegant and very time-consuming.With FDM/FEM you just model the PDE and algo once and change the payoff and off you go.

Last edited by Cuchulainn on September 1st, 2011, 10:00 pm, edited 1 time in total.

- SierpinskyJanitor
**Posts:**1069**Joined:**

Exactly! This is one of the few aspects wherein the boundary between contemplative ( blue-sky ) academic theoretical research and practical real-market usefulness is clearly defined. Despite the apparent "soundness" of closed-form approaches to price basket options ( example ) i strongly suspect that not even for benchmarking and/or validation purposes these techniques are used in capital markets P&L driven banks/hfs/assetmgrs*. As Dr. Cuch clearly explained, it´s either PDE or full MC, that is, in the real world, now as far as the various Fin.Eng Ivory Towers are concerned, then by all means, let's lose some time finding the semi-obscure closed-formula for Himalayans. It will definitely baffle any academic jury ( ie, where usually no prior market experience is found ).*variants of these formulae are usually applicable in Credit Risk batches for CVA/exposure estimation and model-callibration ( ie, extracting "implied" risk factors ) and even there, usually simplified for performance reasons. ( 1M$ for someone who has seen anything like the formula above being used in any FO/MO ) - So as long as these are subject to simplification in order to have real practical usage, then why bother complexifying the problem by coming up with a closed-formula in the first place?

Last edited by SierpinskyJanitor on September 13th, 2011, 10:00 pm, edited 1 time in total.

- Cuchulainn
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An interesting follow-on is that having found an elegant closed solution you may want to solve it numerically, for example by using function from a library. This could be expensiveAnother example is the logistic function that has an exact solution but maybe computationally more efficient to solve it as an ODE?

Last edited by Cuchulainn on April 24th, 2015, 10:00 pm, edited 1 time in total.

- Traden4Alpha
**Posts:**23951**Joined:**

QuoteOriginally posted by: CuchulainnAn interesting follow-on is that having found an elegant closed solution you may want to solve it numerically, for example by using function from a library. This could be expensiveAnother example is the logistic function that has an exact solution but maybe computationally more efficient to solve it as an ODE?Quite true. And one form or the other might be easier to compute in parallel (either in terms of using parallel resources to evaluate the function at a single point or in evaluating multiple values in SIMD architectures in which one wants to avoid iterative algorithms.)P.S. How does the KISS principle affect things. If the user thinks that the logistic function library call uses the exact solution math, might a different implementation violate the user's expectations for how the library call behaves (e.g, computation time, precision, handling of extreme or corner case values, etc.)

Last edited by Traden4Alpha on April 24th, 2015, 10:00 pm, edited 1 time in total.

- Cuchulainn
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QuoteOriginally posted by: Traden4AlphaQuoteOriginally posted by: CuchulainnAn interesting follow-on is that having found an elegant closed solution you may want to solve it numerically, for example by using function from a library. This could be expensiveAnother example is the logistic function that has an exact solution but maybe computationally more efficient to solve it as an ODE?Quite true. And one form or the other might be easier to compute in parallel (either in terms of using parallel resources to evaluate the function at a single point or in evaluating multiple values in SIMD architectures in which one wants to avoid iterative algorithms.)P.S. How does the KISS principle affect things. If the user thinks that the logistic function library call uses the exact solution math, might a different implementation violate the user's expectations for how the library call behaves (e.g, computation time, precision, handling of extreme or corner case values, etc.)Some formulae in the past were not used because they were computationally too intensive. But new hardware/software technologies may cause them to be interesting again. People tend to use things they are used to and forget to look back.

- Traden4Alpha
**Posts:**23951**Joined:**

QuoteOriginally posted by: CuchulainnQuoteOriginally posted by: Traden4AlphaQuoteOriginally posted by: CuchulainnAn interesting follow-on is that having found an elegant closed solution you may want to solve it numerically, for example by using function from a library. This could be expensiveAnother example is the logistic function that has an exact solution but maybe computationally more efficient to solve it as an ODE?Quite true. And one form or the other might be easier to compute in parallel (either in terms of using parallel resources to evaluate the function at a single point or in evaluating multiple values in SIMD architectures in which one wants to avoid iterative algorithms.)P.S. How does the KISS principle affect things. If the user thinks that the logistic function library call uses the exact solution math, might a different implementation violate the user's expectations for how the library call behaves (e.g, computation time, precision, handling of extreme or corner case values, etc.)Some formulae in the past were not used because they were computationally too intensive. But new hardware/software technologies may cause them to be interesting again. People tend to use things they are used to and forget to look back.Very true! What was scarce yesterday (and drove the selection of solution approaches) is not scarce today or tomorrow. Big O analysis ignores both the Moore's law gains in computational power and the even larger improvements algorithm efficiency.By the same token, symbol manipulation technologies (e.g., Mathematica, Maple, etc.) let one compute exact solutions that would have been intractable in the days of pencil and paper.

- Cuchulainn
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QuoteOriginally posted by: Traden4AlphaQuoteOriginally posted by: CuchulainnQuoteOriginally posted by: Traden4AlphaQuoteOriginally posted by: CuchulainnAn interesting follow-on is that having found an elegant closed solution you may want to solve it numerically, for example by using function from a library. This could be expensiveAnother example is the logistic function that has an exact solution but maybe computationally more efficient to solve it as an ODE?Quite true. And one form or the other might be easier to compute in parallel (either in terms of using parallel resources to evaluate the function at a single point or in evaluating multiple values in SIMD architectures in which one wants to avoid iterative algorithms.)P.S. How does the KISS principle affect things. If the user thinks that the logistic function library call uses the exact solution math, might a different implementation violate the user's expectations for how the library call behaves (e.g, computation time, precision, handling of extreme or corner case values, etc.)Some formulae in the past were not used because they were computationally too intensive. But new hardware/software technologies may cause them to be interesting again. People tend to use things they are used to and forget to look back.Very true! What was scarce yesterday (and drove the selection of solution approaches) is not scarce today or tomorrow. Big O analysis ignores both the Moore's law gains in computational power and the even larger improvements algorithm efficiency.By the same token, symbol manipulation technologies (e.g., Mathematica, Maple, etc.) let one compute exact solutions that would have been intractable in the days of pencil and paper.Good remark. The Collector in his book(page 319 for a 2-factor) has a series solution for non-path-dependent options using Bernoulli paths. Looking at the formula you would think it would be slower than a binomial method but maybe not because the loops can be parallelized and you can use precomputed lookup tables for factorials and a clever way to do pow().

Last edited by Cuchulainn on April 28th, 2015, 10:00 pm, edited 1 time in total.

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