An introduction to multilevel Monte Carlo for option valuation

Abstract

Monte Carlo is a simple and flexible tool that is widely used in computational finance. In this context, it is common for the quantity of interest to be the expected value of a random variable defined via a stochastic differential equation. In 2008, Giles proposed a remarkable improvement to the approach of discretizing with a numerical method and applying standard Monte Carlo. His multilevel Monte Carlo method offers a speed up of $O\left({ϵ}^{-1}\right)$, where ε is the required accuracy. So computations can run 100 times more quickly when two digits of accuracy are required. The ‘multilevel philosophy’ has since been adopted by a range of researchers and a wealth of practically significant results has arisen, most of which have yet to make their way into the expository literature. In this work, we give a brief, accessible, introduction to multilevel Monte Carlo and summarize recent results applicable to the task of option evaluation.

Keywords:

• computational complexity
• control variate
• Euler–Maruyama
• Monte Carlo
• option value
• stochastic differential equation
• variance reduction

2010 AMS Subject Classifications:

• 91G60
• 65C05
• 65C30

Acknowledgments

The author is grateful to Mike Giles for creating and placing in the public domain the code that was used as the basis for Figure 3.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

The author is funded by a Royal Society/Wolfson Research Merit Award and an EPSRC Digital Economy Fellowship.