# 21

# Simulation Tools for Interest Rates and Other Financial Indexes

*Il y a plus d'outils que d'ouvriers. (La Bruyère)*

Market prices simulation is the basis of ALM and is required in:

- economic value computations;
- stress testing computations;
- risk measure computations (VaR, economic capital, etc.).

In risk sections, we have already developed some simulation tools for credit risk, stock market risk, etc.

Nowadays the most common simulations are stochastic simulations and for this reason, we start with the presentation of some basic stochastic calculation principles.

## 21.1 STOCHASTIC CALCULATION

### 21.1.1 Brownian motion definition

#### 21.1.1.1 Wiener process

The *Brownian motion* is defined in mathematics by the use of a continuous-time stochastic process called the *Wiener process*. The term Brownian comes from Robert Brown and the term Wiener process from Norbert Wiener. This process *W*_{t} is characterized by three facts:

*W*_{0} = 0;
*W*_{t} is almost surely continuous;
*W*_{t} has independent increments with normal distributions.

This means that for 0 < s < t < t_{2} < t_{3}:

(*W*_{t} − *W*_{s}) follows *N*(0, *t* − *s*)

(*W*_{t} − *W*_{s}) and (*W*_{t2} − *W*_{t3}) are independant variables

In addition, for an infinitesimal increment, the relationship continues:

To simulate a Wiener process, the A/L manager will use this relationship introducing normal distributions.

#### 21.1.1.2 Normal distribution

The normal distribution is also called a Gaussian ...