A closed-form solution (or closed form expression) is any formula that can be evaluated in a finite number of standard operations. Consider the formula


It is of closed-form because it is expressed as a finite number of operations: a logarithm, an addition, a division and a square root. An obvious question is: what is a “standard operation?” There is no general agreement on this. For example, someone might consider a Fourier  transform to be a standard operation, and someone else might not. In this regard, the concept of a closed-form solution is informal.

The formula


is an infinite sum. Because it entails an infinite number of standard operations, it is not a closed-form solution. Suppose we need to evaluate [2]. We can’t possibly perform an infinite number of calculations! One solution is to use our knowledge of calculus to rearrange [2] as



Formulas [2] and [3] are equivalent. Formula [3] is a closed-form solution for [2]. In this case, we were fortunate that [2] had a particularly simple form, which allowed us to find a closed-form solution for it. Often, we will be confronted with more complicated formulas, and it will be difficult or impossible to discover a closed-form solution. In that case, we may seek a numerical solution instead. What is a numerical solution? Consider


It is a numerical solution for [2]. It approximates[2] by adding up the first 10,000 terms of that infinite sum.

A numerical solution is any approximation that can be evaluated in a finite number of standard operations. Closed form solutions and numerical solutions are similar in that they both can be evaluated with a finite number of standard operations. They differ in that a closed-form solution is exact whereas a numerical solution is only approximate.

Compare numerical solution [4] with closed form solution [3]. Obviously, [4] will be more difficult to value than [3]. Numerical solutions often involve extensive numerical calculations. Before the advent of computers, this could be prohibitive, and practitioners would strive to find closed-form solutions. Today, computers make it easy to implement numerical solutions that would have been unimaginable in years past.

Let’s consider a practical example. We want to solve the polynomial equation:

ax2 + bx + c = 0


where ab and c are known constants. As any high school student knows, there is a closed-form solution to this problem called the quadratic formula:


If she has a good teacher, the student will also know that there are generalizations of the quadratic formula that solve third order polynomial equations

ax3bx2 + cx + d = 0


and fourth order polynomial equations

a4bx3 + cx2 + dx + e = 0


Those solutions are complicated formulas, easily filling an entire page, but they exist and are exact, so they are closed-form solutions.

What about fifth order polynomial equations? Is there a closed-form solution for them? No. This is not a matter of mathematicians looking for but not yet finding a solution. They have proven that no closed-form solutions exist for general fifth or higher order polynomial equations. Higher-order polynomial equations are solved using numerical methods. A standard solution is Newton’s method. This numerical method starts with an initial “guess” for a solution. It then uses that value to come up with a second, better “guess” at a solution. It then performs the same operation on the second “guess” to come up with a third, even better “guess.” Continuing in this manner, Newton’s method rapidly converges to an excellent approximation for a solution.

If you are confronted with a math problem and can’t find a closed-form solution, how do you find a numerical solution? Fortunately, there is a vast literature on this topic that covers a number of standard approaches for constructing numerical solutions. Those general approaches are called numerical methods. For example, Newton’s method is a numerical method. When it is applied to a specific polynomial equation, it becomes a numerical solution for that particular equation.

The Monte Carlo method is another standard numerical method. In finance, it is widely used by risk managers to solve problems related to quantifying risk. It is also used by financial engineers to price derivatives. Other numerical methods used by financial engineers are binomial trees and finite differences.