1.5 Multivariate models
1.5.6 Constructing multivariate distributions with copulas
It is possible to construct a multivariate model by merging together one- dimensional distributional assumptions for the marginals and one special multivariate function responsible for the modeling of the dependence be- tween them. This special function is called a copula. The word copula
is coined to emphasize that this is a function which “joins togethe” one- dimensional distribution functions to form multivariate distribution func- tions. Technically a function is a copula if it is a cumulative distribution function on the unit cube [0,1]n with uniformly distributed marginals. In
this section, we briefly describe this technique without delving into the de- tails of copula properties or exhaust all available methods for copula con- struction. Throughout this section, for a functionf(·), we denote byDomf and Ranf the domain and the range of f(·), i.e. f :Domf →Ranf.
Definition and basic properties
We give the definition18 and some basic properties.
Definition 10. An n-dimensional copula is a functionC(·)with domain
[0,1]nsuch thatC(·)is a cumulative distribution function and the marginals
Ck(·),k= 1,2. . . , nsatisfy Ck(x) =x, x∈[0,1].
The most important result regarding copulas is the following theorem known as Sklar’s theorem.
Theorem 12. LetH(·)be an n-dimensional c.d.f. with marginalsF1(·), F2(·), . . . , Fn(·). Then there exists a copula C(·) such that for all x∈Rn,
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For a more detailed introduction into the properties of the copula functions and their application, see Embrechts et al. (2003) and Nelsen (1999).
H(x1, . . . , xn) =C(F1(x1), . . . , Fn(xn))
IfF1(·), F2(·), . . . , Fn(·)are all continuous, thenC(·)is unique; otherwise the
copula is uniquely determined onRanF1×. . .×RanFn. Conversely, ifC(·)is
a copula andF1(·), F2(·), . . . , Fn(·)are distribution functions, then the func-
tionH(·) is an n-dimensional c.d.f. with marginalsF1(·), F2(·), . . . , Fn(·).
Proof. For a proof, see Sklar (1996).
From Sklar’s theorem, we see that for continuous multivariate c.d.f., the univariate marginals and the multivariate dependence structure can be separated and the dependence structure can be represented by a copula.
Next we define the generalized inverse of a c.d.f.
Definition 11. Let F(·) be a univariate distribution function. We de- fine the generalized inverse ofF(·) as
F−1(t) = inf{x∈R|F(x)≥t}
for allt∈[0,1] using the convention that inf{∅}=−∞.
Corollary 2. LetH(·)be an n-dimensional c.d.f. with continuous marginals
F1(·), F2(·), . . . , Fn(·) and a copula C(·). Then for any u = (u1, . . . , un) ∈
[0,1]n,
C(u1, . . . , un) =H(F1−1(u1), . . . , Fn−1(un))
The corollary can be used to calculate the copula of a given multivariate c.d.f. In the following examples we illustrate that.
Example 8. (The Gaussian copula) Let Φ(·) denote the standard one-dimensional normal distribution function andΦΣ(·)denote the standard
multivariate normal distribution function with correlation matrix Σ. Then
C(u1, . . . , un) = ΦΣ Φ−1(u1), . . . ,Φ−1(un) = Z Φ−1 (u1) −∞ . . . Z Φ−1 (un) −∞ f(x1, . . . , xn)dx1. . . dxn (1.42)
wheref(x1, . . . , xn) is the multivariate Gaussian p.d.f. given in (1.29), rep-
resents the Gaussian copula because we know that all marginals of the stan- dard multivariate normal distribution are one-dimensional standard normal distributions. Now it is possible to construct a multivariate modelG(·) that has a Gaussian copula and arbitrary continuous marginals by considering
G(x1, . . . , xn) = ΦΣ Φ−1(F1(x1)), . . . ,Φ−1(Fn(xn))
Example 9. (The t-distribution copula) Let tν(·) denote the one-
dimensional t-distribution c.d.f. withν degrees of freedom and tν,Σ(·)denote
the multivariate t-distribution c.d.f. withν degrees of freedom and dispersion matrixΣ. The copula of the t-distribution19 is given by
C(u1, . . . , un) =tν,Σ tν−1(u1), . . . , t−ν1(un) = Z t−1 ν (u1) −∞ . . . Z t−1 ν (un) −∞ f(x1, . . . , xn)dx1. . . dxn (1.43)
wheref(x1, . . . , xn)is the multivariate t-distribution p.d.f. given in equation
(1.30). Again, as in the previous example, one can construct a multivariate model with the t-distribution copula and arbitrary continuous marginals.
Example 10. (The skewed t-distribution copula) The skewed Stu- dent’s t copula is defined as the copula of the multivariate distribution ofX. Therefore, the copula function is
C(u1, . . . , un) =FX(F1−1(u1), . . . , Fn−1(un))
where FX is the multivariate distribution function of X and Fk−1(uk), k=
1, n is the inverse c.d.f of the k-th marginal of X. That is, FX(x) has the
densityfX(x) defined above and the density functionfk(x) of each marginal
is fk(x) = aK(ν+1)/2rν+(x−µk)2 σkk γ2 k σkk exp(x−µk)σγkkk ν+(x−µk)2 σkk γ2 k σkk −ν+1 4 1 +(x−µk)2 νσkk ν+1 2 , (1.44)
where x∈R, σkk is thek-th diagonal element in the matrix Σ.
A copula can also be specified explicitly, without a reference to some known multivariate distribution.
Example 11. (The bivariate Gumbel copula) Consider the bivariate family of functions Cθ(u, v) = exp −h(−logu)θ+ (−logv)θi1/θ (1.45)
where (u, v) ∈ [0,1]2 and θ ≥ 1. It is simple to verify that C
θ(u, v) is a
bivariate c.d.f. in the unit cube. Equation (1.45) is known as the bivari- ate Gumbel family. It is a class of copulas, which is a sub-class of the Archimedean copulas, see Embrechts et al. (2003) and Nelsen (1999).
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Let X1, X2, . . . , Xn be random variables with csontinuous c.d.f. F1(·), F2(·), . . . , Fn(·) and joint distribution functionH(·). Then there is a unique
copula describing the dependence of the vector (X1, X2, . . . , Xn)T,
H(x1, . . . , xn) =P(X1 < x1, . . . , Xn< xn) =C(F1(x1), . . . , Fn(xn))
The random variables are independent if and only if H(x1, . . . , xn) =
n
Y
k=1 Fk(xk)
and we can arrive at a particular expression for the copula of a vector with independent components.
Theorem 13. Let (X1, X2, . . . , Xn)T be a vector of continuous random
variables with copula C(·). X1, X2, . . . , Xn are independent if and only if
C(u1, . . . , un) =u1u2. . . un.
One nice property of copulas is that for strictly monotone transforma- tions of the random variables, copulas are either invariant, or change in certain simple ways. Note that if a random variable is continuous, a strictly monotone transformation of it is also a continuous random variable.
Theorem 14. Let (X1, X2, . . . , Xn)T be a vector of continuous random
variables with copulaC(·). If for k= 1,2, . . . , n,gk(·) is strictly increasing,
then the vector(g1(X1), g2(X2), . . . , gn(Xn))T has the same copula C(·).
Proof. For a proof, see Embrechts et al. (2003).
The general case, in which gk(·) are strictly monotone, is discussed for ex-
ample in Embrechts et al. (2003).
The reasoning in examples 8 and 9 shows that in order to construct a multivariate model, it is sufficient to have one-dimensional assumptions for the marginals and a different assumption for the dependence structure through the copula. Thus choosing an appropriate copula is related to pre- suming a proper dependence model. As intuition suggests, some dependence concepts are indeed copula properties. We shall consider only the concept of tail dependence, for a more thorough description see Embrechts et al. (2003). The concept of tail dependence relates to the amount of dependence in the upper-right-quadrant tail or the lower-left-quadrant tail and is appropriate for studying the dependence between extreme events. Since it appears that the tail dependence between two random variables X1 and X2 is a copula property, it is invariant under strictly increasing transformation of X1 and X2.
Definition 12. Let (X1, X2)T be a vector of continuous random vari- ables with marginal distribution functionsF1(·)andF2(·), and a copulaC(·).
a) The coefficient of upper tail dependenceλU of (X1, X2)T is defined as λU = lim u→1−P X2 > F −1 2 (u)|X1 > F1−1(u) or equivalently as λU = lim u→1− 1−2u+C(u, u) 1−u
provided that the limitλU ∈[0,1]exists. IfλU ∈(0,1],X1 andX2 are
said to be asymptotically dependent in the upper tail, or the copula is said to have upper tail dependence. If λU = 0, X1 andX2 are said to
be asymptotically independent in the upper tail, or the copula is said to have upper tail independence.
b) The coefficient of lower tail dependenceλL of (X1, X2)T is defined as λU = lim u→0+P X2 < F −1 2 (u)|X1 < F1−1(u) or equivalently as λU = lim u→0+ C(u, u) u
provided that the limitλL∈[0,1]exists. IfλL∈(0,1],X1 and X2 are
said to be asymptotically dependent in the lower tail, or the copula is said to have lower tail dependence. If λL= 0, X1 and X2 are said to
be asymptotically independent in the lower tail, or the copula is said to have lower tail independence.
It turns out that the Gaussian copula is upper and lower tail independent on condition that the correlation coefficient is below one. The t-distribution copula is upper and lower tail dependent which is influenced by the degrees of freedom and the correlation coefficient, a table of values forλU is given in
Embrechts et al. (2003). The tail dependence is radially symmetric in these two examples because they are particular cases of the family of the elliptical distributions. In the general case, symmetry is not necessarily present.
1.6
The Monte Carlo method
Numerical methods that are known as Monte Carlo methods can be loosely described as statistical simulation techniques. Statistical simulation is de- fined, in quite general terms, to be any method that utilizes sequences of random numbers to perform simulation. This technique derives its name
from the casinos in Monte Carlo — a Monte Carlo simulation uses random numbers to model a process.
In the field of mathematical finance, Monte Carlo methods are used to value and analyze basic financial models as well as complex instruments, portfolios and investments by simulating various sources of uncertainty af- fecting their value. Afterwards, on the basis of the generated scenarios, we determine a number of statistics. The advantage of Monte Carlo methods over other techniques become more evident as the dimensions (sources of uncertainty) of the problem increase.
Traditionally, Monte Carlo techniques depend on a number generation method that mimics randomness as well as possible. The general principle of random number generation is as follows. Given the current value of one or more (usually internally stored) state variables, apply an iterative mathematical algorithm to obtain a new set of values for the state variables, and use a specific formula to obtain a new uniform (0, 1) variate from the current values of all the state variables. The generated numbers can never be truly random, only pseudo-random as they are generated according to some deterministic algorithm and after a (large) number of random number generations the sequence will start repeating itself. The number of iterations before replication starts is a measure of the quality of a random number generator and usually is called theperiodof the random number generator. The most commonly used pseudo-random number generation methods are known ascongruential generators first proposed by Lehmer (1951). The basic idea is to produce integer valuesmnon a given interval [0, M−1],and
to return a uniform (0,1) variateunby rescaling20. The next integer variate
is calculated by the mod21 operation
mn+1 = (amn+c) mod M (1.46)
whereM,a, andcare called the modulus, the multiplier, and the increment, respectively.
Note that equation (1.46) is piecewise affine with the same multiplier over all pieces. Thus, it preserves the volume of any given subinterval of [0, M −1], which is why it is called a congruential generator. Frequently, the constantcin equation (1.46) is chosen to be zero, whence we commonly encounter the namelinear congruential generator. There is a lot of literature on good choices foraandM,and not all of it is trustworthy. One reasonable choice, proposed by Park and Miller (1988) is: a= 16807 andM = 231−1.
20
The common method for rescaling is to setun=mn/M.Sincemncan, however, take
on the value 0, which we usually want to avoid, it is recommended to rescale according to
un= (mn+ 1)/(M+ 1).
21
mod operation finds the remainder of division of one number by another. Given two numbers, a (the dividend) and n (the divisor), a modulo n (abbreviated as, and
The period of this generator is equal to 231−2.These constants are normally set in the code and are never user-defined.
Another random number generator technique that has become increas- ingly popular recently is the Mersenne Twister. The name is to indicate that the period of the sequence is a Mersenne number, i.e. a prime num- ber that can be written in the form 2n−1 for some n ∈ N. The period
of the Mersenne twister as published by Matsumoto and Nishimura (1998) is 219937−1. Clearly, for all practical purposes, this number generator can be assumed to have infinite periodicity. The Cognity scenario generation engine uses the Mersenne twister generator.
Most simulations entail sampling random variables or random vectors from distributions other than the uniform. A typical simulation procedure uses methods for transforming samples from the uniform distribution to samples from other distributions. There is a large literature on both gen- eral purpose methods and specialized algorithms for specific cases. In the appendix to this chapter, we provide details about two of most widely used methods: theinverse transform methodand theacceptance-rejection method.