# 10
Advanced Monte Carlo Techniques

In this chapter, we present two classes of methods to improve the speed and efficiency of Monte Carlo simulations. First, we discuss tools that reduce the variance of the estimators while preserving other qualities, such as unbiasedness, at the same time. An alternative approach is the so-called Quasi Monte Carlo (QMC) method, which uses low-discrepancy sequences in place of sequences of pseudo-random numbers.

## 10.1 VARIANCE REDUCTION TECHNIQUES

Variance reduction techniques try to obtain statistically efficient estimators by reducing their variance. The common underlying principle is the utilization of additional information about the problem in order to reduce the effect of random sampling on the variance of the observations. The accomplishable efficiency gain often depends on the contract parameters of the instrument to be valuated, for instance, on the strike price. For practical implementations, the most important consideration is thus the efficiency trade-off: Simple and easy to implement techniques, such as the control variate technique, often provide already significant variance reduction. More sophisticated methods usually also entail a higher computational cost, and it depends on the specific problem whether this additional cost can be compensated by still higher variance reduction.

### 10.1.1 Antithetic Variates

Suppose we would like to estimate *μ = E*[*f*(*Z*)] = *E*[*Y*], where *f*(*Z*) is, for instance, a function used for the valuation of an instrument ...