Application: Simulating Two Uniform Standard Variables with Conditional Copula

Before proceeding, we need to be reminded that simulating two uniform standard variables that are dependent allows the generation of dependent variables Y and Z following any distribution, normal or not, by looking at those two dependent uniform variables as percentiles of the distribution function F of the two "targets" Y and Z. The process for simulating non-normal dependent variables starts with the above building block and proceeds by deriving the simulated values of Z and F, writing that F(Z) — U and F(Y) — V. The two "target" dependent variables are Z = F-U) and Y=F~V).

The core building block consists of simulating the dependent uniform standard variables, [/and V. The process can be decomposed as follows. We start by simulating U= Uv a random uniform standard number within (0, 1) using a random number generator (such as the one in Excel™). Say that u — u — 0.525. Next, we simulate a second random uniform standard number, which represents the percentile a, using a random number generator. We find U2 — u2 = 0.810. This simulation provides only independent variables Ul and U2. U2 is an intermediate variable. The third step calculates the argument of the cumulative standard normal function of which percentile is v since:

In this equation, u = 0.525 and u2 = 0.810. We need to input the correlation of the Gaussian copula. Let us use p — 70%. The numerical calculation becomes:

Replacing:

Finally, the second dependent uniform variable Kis:

The algorithm is summarized below.

By running simulations several times, for example 200 times, it is possible to check that the random uniform variables t/and K are dependent. The scatter plot of U and Kis elongated along the first diagonal, illustrating graphically the positive dependence (Figure 33.8).

FIGURE 33.8 Scatter plot of the simulated random uniform variables