This is a short introduction to simulation with copulas.

Definition & Main Result

A $d$-dimensional copula $C:[0, 1]^{d}\to[0, 1]$ is a cumulative distribution function with uniform marginals. Consider a $d$-dimensional CDF, $F$, with marginals $F_{1}, \dots, F_{d}$. Then tehre exists a copula $C_{X}$ such that

\[F(x_{1}, \dots, x_{d}) = C_{X}(F_{1}(x_{1}), \dots, F_{d}(x_{d}))\]

for all $x_{i}\in[-\infty, +\infty]$ and $i=1, 2, .., d$. $C_{X}$ is called the copula of $X$.

Parametric Copulas

Parametric copulas are copulas that are fully specified by a finite number of parameters. The most common parametric copulas are the Gaussian copula and the Student-t copula. We will focus on the Gaussian copula in this post.

The Guassian Copula

Let $X \sim \mathcal{N}_{d}(0, \Sigma)$, and let $P$ be the correlation matrix. Then the corresponding Guassian copula is defined as

\[C_{P}^{\text{Guass}}(u) := \Phi_{P}(\Phi^{-1}(u_{1}), \dots, \Phi^{-1}(u_{d}))\]

where $\Phi$ is the standard univariate normal CDF and $\Phi_{P}$ denotes the joint CDF of $X$.

Remark: Guassian copula is fully specified by a correlation matrix $P$. i.e. for any $Y\sim\mathcal{N}_{d}(0, \Sigma’)$, as long as $\text{Corr}(Y)=P$, $Y$ has the same copula as $X$.

Simualting the Guassian Copula

Guassian Copula allows us to simulate data $Y \in \mathbb{R}^{d}$ with any given marginal distribution $F_{1}, \dots, F_{d}$ and correlation structure $P$ . The algorithm goes as follows:

1. For any given covariance matrix, $\Sigma$, let $P$ be its corresponding correlation matrix.
2. Compute the Cholesky decomposition $P = A^{T}A$
3. Generate $Z \sim \mathcal{N}{d} (0, I{d})$
4. Set $X = A^{T}Z$
5. Return $U =(\Phi(X_{1}), \dots, \Phi(X_{d}))$ Then we can define $Y_{i} = F_{i}^{-1}(U_{i})$, and we have generated $Y\in\mathbb{R}^{d}$ with the desired marginal distribution $F_{1}, \dots, F_{d}$ and correlation matrix $P$.

Example: CDO pricing

An investment bank is structuring and pricing a collateralized debt obligation (CDO) made up of $d$ corporate issuer. Each issuer has its own one-year default probability, but defaults tend to correlates with each other because of shared economic and industry factors. The goal is to capture both individual default risk and their dependence when valuing the different tranches of the CDO.

We treat the default event for the $i$ th issuer as a Bernoulli outcome $Y_{i} \sim \text{Bernoulli}(p_{i})$ and introduce a latent standard-normal default random variable $X \in \mathbb{R}^{d}$ representing how likely a issuer is going to default: $Y_{i} = \mathbf{1}{ X_{i}\leq \Phi^{-1}(p_{i}) }$. Now we want to simulate $Y$.

• Marginals: Estimate each firm’s one-year default probability $p_i$ from CDS spreads or ratings.
• Dependence: Build a correlation matrix $P$ for latent normals $X_i$, using historical equity returns, industry buckets, or market “base correlations.”
• Simulation:
  1. Cholesky-decompose $P=A A^T$.
  2. For each trial, draw $Z \sim \mathcal{N}(0, I)$, set $X=A Z$, then $U_i=\Phi\left(X_i\right)$.
  3. Declare default: $Y_{i} = \mathbf{1}{ X_{i}\leq \Phi^{-1}(p_{i}) }$.
• Tranche Valuation: Aggregate losses per path, assign them to equity/mezzanine/senior tranches, discount expected losses to get fair spreads.
• Stress \& Calibrate: Adjust $P$ (“implied correlation”) until model tranche prices match market quotes; re-simulate under higher correlations for tail-risk analysis.

Reference: Martin Haugh(2016), An Introduction to Copulas