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.
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3 Answers
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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".
answered Nov 16, 2017 at 14:04
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$\begingroup$ I subscribe, for basket products Monte Carlo is the way to go. $\endgroup$ Nov 16, 2017 at 14:37
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$\begingroup$ To vouch for Monte Carlo's application for high dimensionality problems and to put in a completely different perspective, I did undergrad research work for my physics department which contributed to LIGO's ability to "reverse engineer" gravitational waves to learn about the system that produced it. These systems were fully described with 15 parameters. We used a system that ran several Markov chain Monte Carlo of varying granularity simultaneously. This is called parallel tempering $\endgroup$ Nov 16, 2017 at 18:48
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$\begingroup$ I'm not very familiar with FDM, but the big benefit that the parallel tempering Markov chain Monte Carlo (PTMCMC) provided is that it can relatively quickly search a high dimension parameter space. The more finely grained chains do well at exploring local extrema, while the less finely grained ones can discover new regions of extrema and then help the finely grained ones jump there and explore. This is sort of like having people with telescopes and microscopes running a search party. The telescopes can find interesting regions (say, a cave), and then the microscopes can scour it in detail $\endgroup$ Nov 16, 2017 at 18:56
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$\begingroup$ @BenSandeen actually this is a problem from my interview. And the interviewer mentioned the
convergence speed, is there any difference between MC and FDM under the high dimension? I know quasi-MC is always higher than MC. $\endgroup$– A.OreoNov 17, 2017 at 4:55 -
$\begingroup$ @A.Oreo I'm sorry, but I don't know enough about FDM to answer your question :/ $\endgroup$ Jan 24, 2018 at 23:44
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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.
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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.
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$\begingroup$ ADE is second order in space and time, but once twice the calc time as explicit Euler. It's also trivial to code for 1 dimension and pretty easy to apply to 2 or even 3. $\endgroup$ Mar 21, 2019 at 6:00
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$\begingroup$ The overall computational work is of the same order with ADE or ADI. I don't think the original question is about ADE vs ADI or vs another finite difference technique. In my experience, enforcing proper boundary conditions with ADE is not trivial. And regarding efficiency, Coats, Ê. H. and Μ. H. Terhune, "Comparison of Alternating Direction Explicit and Implicit Procedures in Two-Dimensional Flow Calculations" seem to favor ADI. $\endgroup$– jherekMar 21, 2019 at 11:55
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$\begingroup$ Some problems have trivial BCs, others not. I didn't find ADI's BCs any harder or easier than ADE's. But the tridiagonal solvers for ADI was much more finicky than the dual sweeps. ADE supports uncertain volatility very naturally, not so ADI. $\endgroup$ Mar 21, 2019 at 20:44