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From:
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Date: 08 Oct 1999
Time: 07:52:18
A function has a closed form if it can be written exactly with a finite number of standard functions (powers, exponentials, sins, etc). Many functions are well defined, but cannot be expressed in this manner. If we want to evaluate such functions, we can only do so approximately with numerical methods (numerical integration, Monte Carlo simulation, series expansion). A common example is from probability. The probability density function for the standard normal distribution has a closed form. You can look it up in any probability text. Its cumulative distribution function (the integral of the PDF) does not have a closed form. It can only be approximated using numerical methods. That is why people build and use standard normal tables.
The solution to an option-pricing problem is a function. Sometimes it has a closed form, as with the Black-Scholes formula for an equity option paying no dividends. Other times it has no closed form solution and must be evaluated with Monte Carlo simulation or some other means.
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