Editing Monte Carlo methods in finance (section)

==Applicability==

===Level of complexity===
Many problems in [[mathematical finance]] entail the computation of a particular [[integral]] (for instance the problem of finding the arbitrage-free value of a particular [[derivative (finance)|derivative]]). In many cases these integrals can be valued [[analytic solution|analytically]], and in still more cases they can be valued using [[numerical integration]], or computed using a [[partial differential equation]] (PDE).  However, when the number of dimensions (or degrees of freedom) in the problem is large, PDEs and numerical integrals become intractable, and in these cases [[Monte Carlo method]]s often give better results.

For more than three or four state variables, formulae such as [[Black–Scholes model|Black–Scholes]] (i.e. [[analytic solution]]s) do not exist, while other [[numerical method]]s such as the [[Binomial options pricing model]]  and [[finite difference method]]s  face several difficulties and are not practical.  In these cases, Monte Carlo methods converge to the solution more quickly than numerical methods, require less memory and are easier to program. For simpler situations, however, simulation is not the better solution because it is very time-consuming and computationally intensive.

Monte Carlo methods can deal with derivatives which have path dependent payoffs in a fairly straightforward manner. On the other hand, Finite Difference (PDE) solvers struggle with path dependence.

=== American options ===
Monte-Carlo methods are harder to use with [[Option style|American option]]s. This is because, in contrast to a [[partial differential equation]], the Monte Carlo method really only estimates the option value assuming a given starting point and time.

However, for early exercise, we would also need to know the option value at the intermediate times between the simulation start time and the option expiry time. In the [[Black–Scholes model|Black–Scholes]] PDE approach these prices are easily obtained, because the simulation runs backwards from the expiry date. In Monte-Carlo this information is harder to obtain, but it can be done for example using the [[least squares]] algorithm of Carriere (see link to original paper){{Citation needed|date=May 2021}} which was made popular a few years later by Longstaff and Schwartz (see link to original paper){{Citation needed|date=May 2021}}.