What Closed-Form Means in Practical Option Pricing
Summary
The document explores what practitioners mean by a closed-form option-pricing formula, given that familiar formulas may rely on special functions such as the error function. It contrasts a strict mathematical interpretation, where the boundary of closed form can be narrow, with a practical finance usage that distinguishes formulas from methods requiring numerical procedures such as simulation, trees, or differential-equation solvers.
One answer proposes treating explicit special-function formulas as candidates when their numerical evaluation is sufficiently precise and stable across stressed parameters. It also suggests comparing computation time with a standard analytical benchmark. Another answer notes that even special functions are evaluated numerically in practice, so the distinction is partly conventional. These are pragmatic criteria, not a universal definition: acceptable precision, stability, and speed depend on the use case. The discussion supplies conceptual guidance but no worked pricing example or empirical comparison.
Key ideas
- Closed form in finance commonly means a relatively direct formula rather than a larger numerical pricing procedure.
- Special functions can be included under practical definitions even if their values rely on numerical evaluation.
- Candidate formulas should be checked for accuracy and stability under parameter shocks.
- Computational cost is another practical way to compare a formula with numerical alternatives.
- The boundary between closed and numerical methods is not universally agreed.
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# Are there really closed-form pricing formulas? # Are there really closed-form pricing formulas? 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 ? ## Answer by d_797 (score 5) https://quant.stackexchange.com/a/60295 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) ## Answer by BenG73 (score 5) https://quant.stackexchange.com/a/60302 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. ## Answer by NN2 (score 2) https://quant.stackexchange.com/a/60304 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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