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ZmeiGorynych
Posts: 876
Joined: July 10th, 2005, 11:46 am

### Why do quants like closed-form solutions?

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.

flairplay
Posts: 130
Joined: September 26th, 2006, 1:34 pm

### Why do quants like closed-form solutions?

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
Posts: 3559
Joined: April 20th, 2004, 12:25 pm

### Why do quants like closed-form solutions?

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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### Why do quants like closed-form solutions?

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.
"Compatibility means deliberately repeating other people's mistakes."
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blondie
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Joined: June 11th, 2007, 1:34 pm

### Why do quants like closed-form solutions?

quantmeh
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Joined: April 6th, 2007, 1:39 pm

### Why do quants like closed-form solutions?

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

bilbo1408
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Joined: August 3rd, 2007, 12:50 pm

### Why do quants like closed-form solutions?

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

Cuchulainn
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### Why do quants like closed-form solutions?

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.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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Cuchulainn
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### Why do quants like closed-form solutions?

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.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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SierpinskyJanitor
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Joined: March 29th, 2005, 12:55 pm

### Why do quants like closed-form solutions?

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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### Why do quants like closed-form solutions?

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.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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### Why do quants like closed-form solutions?

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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### Why do quants like closed-form solutions?

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.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl

Posts: 23951
Joined: September 20th, 2002, 8:30 pm

### Why do quants like closed-form solutions?

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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### Why do quants like closed-form solutions?

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.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl