# CHAPTER 5

# Monte-Carlo Pricing Issues

Monte-Carlo techniques may be fairly considered as the easiest way to deal with multi-asset or path-dependent payoffs. In the first section of this chapter, a method for sampling a set of correlated assets under the multivariate Gaussian model will be developed. In the second section, we will investigate some techniques to reduce the variability of simulation outputs, whatever the payoff.

## 5.1 MULTI-ASSET SIMULATION

### 5.1.1 The correlations issue

Higher dimensionality adds some complexity due to the co-dependency between assets. In general, the mutual interaction of two variables is measured by a **copula**, i.e., a bivariate cumulative probability defined as a function of the marginal cumulative probabilities of the two variables:

In this chapter, we will deal only with **multivariate Gaussian** distributions: it means that mutual dependencies among assets are measured, two at a time, by the coefficient of linear **correlation** or, more simply, the *correlation ρ*. This parameter is embedded in standard diffusion models such as Black–Scholes:

### 5.1.2 The Gaussian case

Mutual dependencies in a group of *n* assets can be summarized then by a correlation matrix, derived from the **Covariance** matrix

The multivariate Gaussian density function of a set of correlated variates ...