What is semi analytical solution?

Question

I am searching for the definition of a semi analytical solution.

I can't seem to find anything online, any help?

Answer 1

"Analytical" and "semi-analytical" are not really technical terms, I think. Roughly speaking, "analytical" means "closed-form", i.e. expressed as a finite expression using "well-known" functions. What exactly is a "well-known" function may be debated: $\sin$ and $\cos$ certainly qualify, $\text{LambertW}$ and generalized hypergeometrics maybe not. As stated in the second page of the Carr and March article cited, the expression in this case is only "semi-analytical" because it involves non-analytical Laplace transforms of analytical functions.

Answer 2

A few years later, my current understanding of these words are:

Closed form / analytical solution: you can plug in the numbers and get the result directly.

semi-closed form / analytical solution: there is a numerical step involved, which is usually an integral or another operator that is computable efficiently and accurately.

These terms are mostly used in financial papers where we aim for the former types of solutions (for example Black-Scholes equation is an instance of it) but we usually end up with the latter for more complex models (for example pricing under Heston).