The Simulation Algorithm for Standardized Normal Variables using Single-factor Models
The simulations of these standard normal variables are normal standard inverse of uniform variables following U(0, 1). With standardized variables, the models take more familiar forms, with F being the standardized stock returns, Z being the standardized index return, and the new residuals X being both the same independent standard normal variable.
In these equations, all variables are now normal standard. The issue is to generate random correlated standard normal variables using the standardized factor model. We know that the factor and the residual are normal independent. For generating random returns for a single stock, we simply generate random standard normal variable for the factor and the residual.
The algorithm for generating correlated normal standardized variables Y and F2 dependent on a common factor Z collapses to the generation of two normal standard variables with correlation coefficient p. The algorithm is the following:
• generate three uniform independent variables following U(0, 1), uv uy u3
• take the normal inverse of u for generating a normal standard variable Z
• take the normal inverse of w2 for generating a normal standard variable X
• take the normal inverse of w3 for generating a normal standard variable X2
For each set of three values of Z, Xv Xy calculate Yl and Yy using the same standardized equation. The values of Y and F2 have correlation p. Table 34.1 shows the first 10 simulations.
TABLE 34.1 Sample simulations of two correlated variables depending on a single factor
FIGURE 34.2 Simulation of two dependent standardized returns
The first simulations in the case of a single factor are shown in Table 34.1.
Figure 34.2 plots the results of 400 simulations for the pair of standardized returns.
For moving back to non-standard returns, we need to revert from standard to non-standard variables. Since all variables are normal standard, they can be expressed directly as a function of the independent uniform U(0, 1) variables.
These formulas bridge the gap between the forthcoming copula approach and the classical approach.
The procedure would be similar with several factors since the residual is always independent from the factors. It is even simpler with independent factors because we can proceed exactly as above by generating as many independent normal factors as needed. If the factors are not independent, we need to use the Cholesky technique for generating random values of factors complying with their variance-covariance matrices. The procedure is described below.