Choosing Monte Carlo or Finite Differences for Basket Option Pricing

# use Monte Carlo or FDM to price Basket option


# use Monte Carlo or FDM to price Basket option












In the real practice, do we use `Monte Carlo` or `finite difference method` of PDE to price the `Basket option`(say 20 underlyings)?

And could you show some reasons in detail.

## Answer by Antoine Conze (score 6, accepted)

https://quant.stackexchange.com/a/36929

Multidimensional finite differences (such as ADI schemes) are only practical up to 3 dimensions, higher dimension are too demanding in terms of computer memory and computing time.

For higher order problems Monte Carlo is usually the method of choice. Using low discrepancy quasi random suites (e.g. Sobol) along with the Brownian bridge technique leads to reasonable computing times. See for instance Jaeckel's book "monte carlo methods in finance".

## Answer by Brian B (score 1)

https://quant.stackexchange.com/a/36930

Quasi Monte Carlo (QMC) as suggested by Antoine's answer will work fine if you're not planning on having a portfolio of these things to deal with. If you're on the buy side or just playing around, go with QMC.

For more serious applications, the answers to this question: Basket option pricing: step by step tutorial for beginners include the most common industry practices of

- Moment matching (link to a paper), and

- Proxy volatilities

along with a suggestion by Choi (2018), which I have not reviewed, that a quadrature schemes works best (according to his paper of 2018 [Arxiv]).

Moment matching and proxies achieve the necessary level of accuracy in a (very) small fraction of the computation time a QMC scheme would require.

## Answer by jherek (score 0)

https://quant.stackexchange.com/a/44688

In finite difference methods, assuming the Basket is composed of $p$ assets, the solution of systems of size $N^p$ is going to be involved where $N$ is the grid discretization size per dimension. With the ADI technique, you can solve those in linear time, that is in $O(N^p)$. Furthermore the typical finite difference method is order-2. Higher order methods usually require more time to solve the system. Very roughly, the overall computational work is thus $O(N^{-2/p})$.

In the Monte-Carlo method, the convergence order is $O(N^{-1/2})$, regardless of the basket size $p$. With quasi-Monte-Carlo methods, it can be pushed closer to $O(N^{-1})$. Comparing the two, you arrive at the conclusion that finite difference methods are advantageous for $p=1,2$, and possibly $p=3$.

The framework for the computational effort order has been described in Broadie & Glassermann A stochastic mesh method for pricing high-dimensional American options and more explicitely with regards to FDM vs MC vs Quadratures in the encyclopedia of quantitative finance Quadrature methods article.