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).

Scatter plot of the simulated random uniform variables

FIGURE 33.8 Scatter plot of the simulated random uniform variables

 
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