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Good morning to all,

I wanted to post this question here hoping to have more details. The concern, in my opinion, comes from the fact that the concept of "closed-form" is not clear. Because, on the one hand, we have the algebraists who say that the gaussian function (error function) cannot have an analytical expression and, on the other hand, we have the financiers who accept that any formula for the price of a product is considered a closed-form if it is written with the gaussian function (as in the case of the Black-Scholes formula) or any other function whose values are known in the tables.

My question is thus the following one: In option pricing theory, what exactly does a closed-form formula mean ?

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It typically means one can price the option in terms of a "simple" formula as opposed to having to resort to numerical methods such as Monte-Carlo, numerical PDEs, numerical integration, trees, etc.

Here "simple" would refer to elementary functions and possibly power series expansions (this should cover the Error function)

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  • $\begingroup$ So if I understood correctly, the normal distribution function is considered to be a "simple" function as you say, and this despite the fact that it has only an integral form whose resolution is generally done with numerical methods. $\endgroup$ – MAKacef Jan 2 at 17:59
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    $\begingroup$ If it would require one to actually use numerical integration, then maybe people's perspective would change, but I think the normal cdf can be calculated efficiently with a series expansion. $\endgroup$ – d_797 Jan 2 at 18:37
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In fact, the frontier between closed formulas and "opened" ones is a litle bit fuzzy. In fact, as soon as you use special functions like log, exp, erf, erfc and so on, you are relying on expansions methods to evaluate their values. So for algebraists, even these formulas shouldn't be considered "closed". However their numeric calculus are based on expansions providing near or exact machine double precision so that we may consider them to be "closed".

Now, in quantitative finance, i would adopt the following more flexible and pragmatic definition to say what is closed or not:

1/ If your option pricing formula is based on a characteristic function (or others) you know explicitly given special functions for which we have machine double precision, provided you have tested integration scheme with a precision under the basis point, you may consider this one to be a candidate "closed" formula. Of course, to assess this, you will need to run a lot of stress-tests to ensure the stability of your formula while its parameter are shocked in all possible ways.

2/ This candidate is then ok if your overall computation time is similar to the Back Scholes formula at least in terms of order of magnitude. I tell you that because i can obtain a precision under the bp for almost any numeric method like PDE or Monte-Carlo except that it will need a much more important computation time.

I hope this will help your thinking.

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  • $\begingroup$ Thank you very much for your response. If I understand correctly, a pricing formula is said to be "closed" if it is written with simple functions (exp, erf, log,...) and if the study of the sensitivity of the input parameters has been carried out and has given good results compared to numerical methods (Monte-Carlo simulations). $\endgroup$ – MAKacef Jan 3 at 18:21
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There are many types of closed-form expression as you can find in the link here below https://en.wikipedia.org/wiki/Closed-form_expression#Comparison_of_different_classes_of_expressions

In quantitative finance, IMO, closed-form expression rather corresponds to analytic expression.

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