OSP-3 Multivariate Distributions and Copulas

Multivariate Distributions in Finance

Multivariate distributions (in particular, Gaussian copula models) arise in risk management for a portfolio of defaultable bonds. If one “obligor” defaults, it will influence the likelihood of other obligors defaulting. We need to consider the dependency to evaluate the overall risk of the portfolio.

Another example is that some options’ payoffs depend on multiple assets, e.g.

Basket option:

Barrier option:

Spread option:

Multivariate Distributions

Box-Muller Transform

We have known ways to generate samples from using the accept-reject method. The inverse CDF method is not available because .

The Box-Muller transform, however, make it easier to generate a pair of i.i.d. rather than on . The transform goes as below,

If and are independent random variabels such that

then

that is

(dependency)

Given , we get , thus

Bivariate Correlated Normal Distribution

Because of the fact

we can generate arbitrary (correlated) normal random vectors as follows,

generate from the standard bivariate normal distribution using Box-Muller transform

return

Here is the square root matrix that satisfies

There are two common ways to find such square root matrix: Spectral decomposition and Cholesky decomposition.

Spectral decomposition:

decomposes the matrix based on the eigenvalues and eigenvectors

where ( are eigenvalues of ), and is an orthogonal matrix whose columns are the eigenvectors of .

we can choose
note that is possible because the covariance matrix is always semi-positive definite, and thus .

Cholesky decomposition:

finds a lower triangular matrix such that
we can choose
in Python, np.linalg.cholesky(Sigma) returns

Although the square root matrices produced by Spectral and Cholesky decomposition are different, they both satisfy , and hence both yield the same distribution.

Example, to generate a bivariate normal distribution , where

we can apply the Cholesky decomposition by solving

The solution is , and thus

To generate the , first generate using Box-Muller transform, and then

and Distributions from Normal Distributions

distribution is the same as distribution. It can also be generated by

where

A -distribution with degrees of freedom can be generated by

where

distribution has heavier tails than in the sense that for . For example, in , while in , .

Beside, the quantiles of are larger than . But as , .

Multivariate GBM

For the multivariate computational finance problems, e.g., those involving multiple stocks, we need an appropriate multivariate SDE. Multidimensional is a common starting place.

Find such that using Cholesky decomposition, and then the multivariate -dimensional GBM is defined as

We can generate a price path by first generating by first generating

and then

Multivariate -distribution

Multivariate -distribution is denoted by , where is the degrees of freedom and is the covariance structure.

If , then and

We can generate as follows:

Generate and independently ( is a scalar)

Return

All the marginal distributions are scaled -distributions.

Copula

Copulas are multivariate distributions having all marginal distributions as uniform distributions. Specifically, if is a random vector where , then the joint distribution function

is called a Copula.

Copulas are useful for generating observations with arbitrary dependency structures, and arbitrary marginal distributions.

The inverse CDF method indicates that . Therefore

is a multivariate distribution sharing the same (rank) dependency structure as and having marginal distributions .

Note that is non-decreasing, thus it will not change the rank dependency like Spearman’s rho and Kendall’s tau between s, i.e.

but

Conversely, if is some multivariate distribution with marginal CDFs , then has a distribution function which is a copula.

Copulas Example: dependency structures

There are numerous families of copulas that are developed to replicate several dependency structures.

One of the most general forms of copulas is given by (Archimedean copula)

where is a continuous strictly decreasing convex function called the generator function satisfying and .

The pseudo-inverse is given by

If , the generator is called a strict generator.

Some common Archimedean Copulas

Clayton Family:

Characteristics:

Models strong lower tail dependence (captures dependence in small values)
No upper tail dependence
Suitable for modeling extreme co-movements in financial downturns

Gumbel Family:

Characteristics:

Models strong upper tail dependence (captures dependence in large values)
Suitable for modeling extreme positive co-movements in risk or insurance
No lower tail dependence

Joe Family:

Characteristics:

Strong upper tail dependence: It captures strong dependence in the upper tail (large values)
Weak lower tail dependence: The lower tail dependence is weaker or non-existent
Used in risk management, insurance, and finance where extreme positive events are of interest

Gaussian Copula

The Gaussian Copula is a copula function that captures dependence using a multivariate normal distribution, regardless of the marginal distributions.

The Gaussian copula allows us to keep normal-like dependency while using arbitrary marginals. But it does not capture extreme co-movements (e.g. CDOs in 2008 financial crisis used Gaussian copula to model the joint probability of defaults but it failed when all assets drop together).

The formula for the Gaussian Copula with zero mean and covariance is given by

where represents the CDF of .

To generate samples from the Gaussian Copula given a covariance matrix

Generate (dependency)

Return , where ( is the CDF of standard normal distribution)

The generated in all have distribution but are dependent.

If we want to have random vectors with Gaussian Copula but marginals which are , then further return

-Copula

If we are modeling financial risk, tail dependence, systemic failures or other extreme events, the t-copula is a better choice. It assumes t-distribution dependence.

The t-copula uses a degree of freedom to control the tail weight. To generate from copula

Generate (dependency)

Return , where ( is the CDF of distribution)

If we want to have random vectors with copula but marginals which are , then further return