Fin285a:Computer Simulations and Risk Assessment Section 7.2...

Fin285a:Computer Simulations andRisk Assessment

Section 7.2Copulas

Danı́elson, 1.8

Overview

Fall 2016: LeBaron Fin285a: 7.2-5 list – 2 / 27

Copula objectives

Copula lesson 1: Generate a distribution

Copula lesson 2: Generate two random variables

Copula lesson 3: Empirical distributions (data)

Copula details

Matlab functions and fitting copulas

Copula wrap-up

Copula objectives

Copula objectives

Copula lesson 1:Generate adistribution

Copula lesson 2:Generate tworandom variables

Copula lesson 3:Empiricaldistributions (data)

Copula details

Matlab functions andfitting copulas

Copula wrap-up


Objectives


⊲ Generate dependence between X and Y⊲ Fix distribution of X alone, and Y alone

(Marginal distributions)⊲ Example

• Fit returns to a student-t• Then adjust dependence between them

Copula lesson 1: Generate adistribution

Copula objectives




Copula details


Copula wrap-up


Generate a distribution


⊲ F (x) is cumulative distribution function for some RVX

⊲ Generate a uniform random variablezi ∼ Uniform[0, 1]

⊲ Generate yi = F−1(zi)⊲ yi’s will follow the distribution represented by F ()⊲ matlab: copulaex1.m

Copula example 1


% Simple motivating example for copula technology

% Convert a uniform random variable into a student-t

nsample = 10000;

z = rand(nsample,1);

y = tinv(z,3);

histfit(y);

Message


⊲ Can turn a uniform into any distribution you want⊲ This can be useful sometimes in monte-carlo’s

Copula lesson 2: Generatetwo random variables

Copula objectives




Copula details


Copula wrap-up


Convert two normals to student-t


⊲ Generate (w1, w2) from a multivariate Normal⊲ Get u = F (w1), v = F (w2) where F (x) is normal

CDF⊲ Generate y = G−1(u), z = G−1(v)

G(x) is the student-t CDF⊲ matlab: copulaex2.m

Copula example 2


% Basic bivariate copula

% Normal copula, student-t marginals, matlab scatter plot

nsample = 10000;

% build variance/covariance matrix for normals

rho = 0.5; % correlation

omega = [1 rho; rho 1];

mn = [ 0 0];

% draw normals to vector w

w = normal(nsample,mn,omega);

% map to uniforms using the cdf

u = normcdf(w(:,1),0,1);

v = normcdf(w(:,2),0,1);

% map back to student-t (4 degrees of freedom)

y = tinv(u,4);

z = tinv(v,4);

% nice matlab routine for plotting

scatterhist(y,z);

Copula lesson 3: Empiricaldistributions (data)

Copula objectives




Copula details


Copula wrap-up


Generate desired correlation with empirical marginals


⊲ Generate (w1, w2) from a multivariate Normal⊲ Get u = F (w1), v = F (w2) where F (x) is normal

CDF⊲ Generate usret1mc = G−1(u), emretmc = G−1(v)

• G(x) is empirical CDF from data• Like bootstrap

⊲ matlab: copulaex3.m

What is an empirical CDF?


⊲ Rt vector of returns⊲ p = G(x) = fraction(Rt ≤ x)⊲ R̃(p) = G−1(p) = quantile(Rt, p)

Copula details

Copula objectives




Copula details


Copula wrap-up


The steps to generate X, Y


⊲ Draw (U, V ) from a joint distribution C(U, V ) wherethe marginal distributions of both U and V areU [0, 1]

⊲ Then send U, V through an inverse CDF,

• X = G−1(U)• Y = F−1(V )

⊲ The distribution C(U, V ) is tricky, this is whatcontrols dependence at the uniform density stage

⊲ C(U, V ) is the copula function

What was the copula?


F (), G() are marginal CDF’s for X,Y

(U, V ) = (F (X), G(Y )) (7.2.1)

Pr(U ≤ u, V ≤ v) = C(u, v) (7.2.2)

Pr(X ≤ x, Y ≤ y) = C(F (x), G(y)) = H(x, y) (7.2.3)

C() is the copula function and controls the joint distribution.Sklar’s Thm: any multivariate CDF, H(x1, x2, . . . , xd) can be writtenas

H(x1, x2, . . . , xd) = C(F1(x1), F2(x2), . . . , Fd(xd)) (7.2.4)

Picture of the Gaussian Copula


Uniform World (u, v) (dependence)

Multivariate Normal (Σ)

(w1, w2) → Normal CDF(Φ), u = Φ(w1), v = Φ(w2)

Target Marginal Inverse CDF’s

y = G−1(u)z = F−1(v)

Copula varieties


⊲ GaussianC(u) = ΦΣ(Φ

−1(u1),Φ−1(u2))

ΦΣ(x) is multivariate Normal CDF, mean zero,,var=1, var/cvar= Σ, Φ(x) is univariate N(0, 1)

⊲ Student-t: Same as Gaussian but student-t CDF⊲ Clayton (Archemedian class)

C(u) = ψ(ψ−1(u1) + ψ−1(u2))ψ(x) = (1 + x)−1/θ

ψ−1(x) = x−θ − 1⊲ Joe⊲ Gumbel

Matlab functions and fittingcopulas

Copula objectives




Copula details


Copula wrap-up


copulafit and copularnd


% program = copulaexfit.m

% map returns to uniforms using empirical cdf

eqret = (wldeqp(2:end,2:end)-(wldeqp(1:end-1,2:end)))./...

wldeqp(1:end-1,2:end);

uret(:,1) = empcdf(eqret(:,1),eqret(:,1));

uret(:,2) = empcdf(eqret(:,2),eqret(:,2));

% Fit a gaussian copula

[paramhat] = copulafit(’Gaussian’,uret);

umc = copularnd(’Gaussian’,paramhat,nmc);

% Fit a student-t copula (nu = degrees of freedom)

[paramhat, nu] = copulafit(’t’,uret);

umc = copularnd(’t’,paramhat,nu,nmc);

% Fit a Clayton copula (fatter left tail)

[paramhat] = copulafit(’clayton’,uret);

umc = copularnd(’clayton’,paramhat,nmc);

Matlab implementation


⊲ matlab: copulaexfit.m⊲ Fit 3 copulas (Clayton, Student-t, Gaussian) to

uniform CDF⊲ Estimate extreme right and left tail probabilities⊲ Estimate VaR (extreme)⊲ Scatter plots

Copula wrap-up

Copula objectives




Copula details


Copula wrap-up


Advantages


⊲ Separates problems

• Fitting marginal distribution• Modeling dependence

⊲ May beat the bootstrap for small samples

• Generating joint (x, y) extreme observations• Using actual data as marginals distributions!!

⊲ Key problem: Which copula do you use?

Puzzles in multivariate returns


⊲ Do correlations change with volatility and risk?⊲ Some think yes

• When volatility is high, correlations are high?• Ability to diversify risk is low

⊲ This gets more complex⊲ Changing copulas⊲ ”Risk on/risk off” behavior

Risk on/risk off


⊲ Traders appear to correlate in buying and sellingrisky assets

⊲ Statistic: Days where more than 90 percent ofstocks are moving in the same direction

• Up to 2006 this is about 3-5 days per year• 2008-2011 this is above 30 (or about 1/7 (14

percent)trading days)

⊲ Matlab: rollcorr.m

Overview


Copula objectives

Copula lesson 1: Generate a distribution

Copula lesson 2: Generate two random variables

Copula lesson 3: Empirical distributions (data)

Copula details

Matlab functions and fitting copulas

Copula wrap-up

Fin285a:Computer Simulations and Risk Assessment Section 7.2...

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