Monte Carlo and Quasi-Monte Carlo Methods

Chapter 12 discusses Monte Carlo and quasi-Monte Carlo methods and demonstrates how these techniques can be used to compute functionals of multidimensional diffusions. Monte Carlo methods feature prominently in this book, in particular we discuss how to use Lie Symmetry methods to construct unbiased Monte Carlo estimators in Chap. 6, and we discuss how to construct unbiased Monte Carlo estimators for Wishart processes in Chap. 11. In Chap. 12, we focus on two novel themes which have recently emerged in the context of Monte Carlo methods, namely the exact simulation of general stochastic differential equations and multilevel methods. In the second part of the chapter we discuss quasi-Monte Carlo methods. The focus of this part is on scrambled nets, and we show how they can produce faster convergence rates than standard Monte Carlo methods. The chapter concludes by illustrating how to apply quasi-Monte Carlo methods under the benchmark approach introduced in Chap. 1. We recall the Minimal Market Model from Chap. 3 and price financial derivatives on realized variance in this model.

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