Closed-form solutions

From WilmottWiki

A closed-form solution is a formula resulting as the solution of a problem or a mathematical model. It contrasts with a numerical algorithm for finding a solution.

In quantitative finance the most famous closed-form solutions are the Black-Scholes formulae for the values of vanilla options.

Closed-form solutions are liked in quantitative finance for several reasons:

• They are fast to compute compared with a numerical algorithm, so option prices and the greeks can be found quickly allowing more contracts to be priced and risk managed.
• They sometimes permit the structure of a solution to be seen clearly.
• They can be used as a check on the accuracy of numerical algorithms. These algorithms can then be used in situations where there aren't any closed-form solutions.
• Even though finding such closed-form solutions may require simplifications of a model, the resulting simplified model may not be much worse than the original complicated model. And it may be far more transparent.

Some closed-form solutions are so complicated that the term may not be helpful. An example is the value of a vanilla option under the Heston stochastic volatility model which requires numerical integration in the complex plane.