Statatistics and Data Analysis for Financial Engineering - Chapter 8 Copulas

A copula is a multivariate CDF whose univariate marginal distribution are all U(0,1).

By Sklar’s theorem:

$F_Y(y_1,..,y_d) = C_Y(F_{Y_1}(y_1),..,F_{Y_d}(y_d))$

data and code from the book can be found here: https://people.orie.cornell.edu/davidr/SDAFE2/ - which has numerous errors running on R version 4.1.2 (2021-11-01)

Solution to R-lab

Special copulas

d-dimensional independent copula of
co-monotonicity describes positive dependence and
counter-monotonicity copula which has negative dependence and if $d \leq 2$ . If d > 2 then a lower bound for copulas: $max(u_1+..+u_d+1-d, 0) \leq c(u_1,..,u_d)$

For Gaussian and t-copulas, let $\Omega$ be the correlation matrix:
1. $\Omega \rightarrow C_y$ is 1-1 to the Gauss copula
2. $\Omega = \textbf{I} \rightarrow$ Meta-Gaussian distribution, which is the independent copula (1.)
3. $\Omega = \textbf{1} \rightarrow C_+$
4. $\Omega = \textbf{-1} \rightarrow C_-$

Archimedean Copulas

Archimedean copula with generator function:

$C(u_1,..,u_d) = \varphi^{-1}(\varphi(u_1)+..+\varphi(u_d))$

Satisfies 3 conditions:

$\varphi$ is continuous, strictly increasing, and convex on $\varphi : [0,1] \rightarrow [0, \infty)$
and $\varphi(0) = \infty$
and $\varphi(1) = 1$

Frank copula

Generator function:

$C(u|\theta) = -ln(\frac{e^{-\theta u} -1)}{e^{-\theta} -1)}), -\infty < \theta < \infty$

$\theta \rightarrow 0 \implies c(u_1, u_2) \rightarrow C_0$
$\theta \rightarrow \infty \implies c(u_1, u_2) \rightarrow C_+$
$\theta \rightarrow -\infty \implies c(u_1, u_2) \rightarrow C_-$

Crayton copula

Generator function:

$C(u|\theta) = \frac{1}{\theta}(u^{-\theta}-1), \theta > 0$

$\theta \rightarrow 0 \implies c(u) \rightarrow C_0$
$\theta \rightarrow \infty \implies c(u_1, u_2) \rightarrow C_+$
$\theta \rightarrow -1 \implies c(u_1, u_2) \rightarrow C_-$

Gumbel Copula adsfads

Generator function:

$C(u|\theta) = -ln(u)^{\theta}, \theta \geq 1$

$\theta = 1 \implies c(u) \rightarrow C_0$
$\theta \rightarrow \infty \implies c(u_1, u_2) \rightarrow C_+$

Joe copula

Generator function:

$C(u|\theta) = -ln(1-(1-\theta)^{\theta}), \theta \geq 1$

$\theta =1 \implies c(u) \rightarrow C_0$
$\theta \rightarrow \infty \implies c(u_1, u_2) \rightarrow C_+$

Joe copula is similar to Gumbel. It cannot have negative dependence. It allows stronger upper tail dependence and is closer to being a reverse Clayton copula in the positive dependence case. is closer to a reverse Clayton copula.

Plot of generator function for Frank cupola

library(copula)
u= seq(0.000001, 1, length=500)
frank = iPsi(copula=archmCopula(family="frank", param=1), u)
plot(u, frank, type="l", lwd=3, ylab=expression(phi(u)))
abline(h=0)
abline(v=0)

Scatter plot of 9 bivariate Frank copulas

set.seed(5640)
theta = c(-100, -50, -10, -1, 0, 5, 20, 50, 500)
par(mfrow=c(3,3), cex.axis=1.2, cex.lab=1.2, cex.main=1.2)
for(i in 1:9){
  U= rCopula(n=200, copula=archmCopula(family="frank", param=theta[i]))
  plot(U, xlab=expression(u[1]), ylab=expression(u[2]), 
                          main=eval(substitute(expression(paste(theta, " = ", j)),
                         list(j = as.character(theta[i])))))
}

Figure 2: Bivariate random samples of size 200 from various Frank copulas.

Scatter plot of 9 bivariate Clayton copulas

Figure 3: Bivariate random samples of size 200 from various clayton copulas.

Scatter plot of 6 bivariate Gumbel copulas

Figure 4: Bivariate random samples of size 200 from various gumbel copulas.

Scatter plot of 6 bivariate Joe copulas

Figure 5: Bivariate random samples of size 200 from various joe copulas.

Rank correlation: 1. Kendall’s Tau, 2. Spearman’s rank correlation coefficient

Tail dependence

rho = seq(-1, 1, by=0.01)
df = c(1,4,25, 240)
x1 = -sqrt((df[1]+1)*(1-rho)/(1+rho))
lambda1 = 2*pt(x1, df[1]+1)
x4 = -sqrt((df[2]+1)*(1-rho)/(1+rho))
lambda4 = 2*pt(x4, df[2]+1)
x25 = -sqrt((df[3]+1)*(1-rho)/(1+rho))
lambda25 = 2*pt(x25, df[3]+1)
x250 = -sqrt((df[4]+1)*(1-rho)/(1+rho))
lambda250 = 2*pt(x250, df[4]+1)
par(mfrow=c(1,1), lwd=2, cex.axis=1.2, cex.lab=1.2)
plot(rho, lambda1, type="l", lty=1, xlab=expression(rho), ylab=expression(lambda[l]==lambda[u]))
lines(rho, lambda4, lty=2)
lines(rho, lambda25, lty=2)
lines(rho, lambda250, lty=2)
legend("topleft", c( expression(nu==1), expression(nu=4), expression(nu==25), expression(nu=250) ), lty=1:4)

Example: flows in pipeline

library(copula)
library(sn)
dat = read.csv("datasets/FlowData.csv")
dat = dat/10000
n = nrow(dat)
x1 = dat$Flow1
fit1 = st.mple(matrix(1,n,1), y=x1, dp=c(mean(x1), sd(x1),0,10))
est1 = fit1$dp              #vector or list of estimated DP parameters
u1 = pst(x1, dp=est1)       # vector of probabililities - skew-t
x2 = dat$Flow2
fit2 = st.mple(matrix(1,n,1), y=x2, dp=c(mean(x2), sd(x2),0,10))
est2 = fit2$dp
u2 = pst(x2, dp=est2)
U.hat = cbind(u1,u2)
z1 = qnorm(u1)
z2 = qnorm(u2)
Z.hat = cbind(z1,z2)

library(ks) 
fhatU = kde(x=U.hat, H=Hscv(x=U.hat))
par(mfrow=c(2,2), cex.axis=1.2, cex.lab=1.2) #, cex.max=1.2
hist(u1, main="(a)", xlab=expression(hat(U)[1]), freq=FALSE)
hist(u1, main="(b)", xlab=expression(hat(U)[2]), freq=FALSE)
plot(u1, u2, main="(c)", xlab = expression(hat(U)[1]), ylab = expression(hat(U)[2]), mgp = c(2.5, 1, 0))
plot(fhatU, drawpoints=FALSE, drawlabels=FALSE, cont=seq(10, 80, 10), 
     main="(d)", xlab=expression(hat(U)[1]), ylab=expression(hat(U)[2]), mgp = c(2.5, 1, 0))

Figure 6: Pipeline data. Density histograms (a), and (b) and a scatterplot (c) of the uniform-transformed flows. The empirical copula C.hat, is the empirical CDF of the data in (c). Contours (d) from an estimated copula density c.hat via a two-dimensional KDE of (c)

fhatZ = kde(x=Z.hat, H=Hscv(x=Z.hat))

#pdf("norm_flows_hist_plot.pdf", width=7, height=7)
#
par(mfrow=c(2,2), cex.axis=1.2, cex.lab=1.2, cex.main=1.2)
qqnorm(z1, datax=T, main="(a)") ; qqline(z1)
qqnorm(z2, datax=T, main="(b)") ; qqline(z2)
plot(z1, z2, main="(c)", xlab = expression(hat(Z)[1]), ylab = expression(hat(Z)[2]), mgp = c(2.5, 1, 0))
plot(fhatZ, drawpoints=FALSE, drawlabels=FALSE, cont=seq(10, 90, 10), 
     main="(d)", xlab=expression(hat(Z)[1]), ylab=expression(hat(Z)[2]), mgp = c(2.5, 1, 0))

Figure 7: Pipeline date. Normal quantile plots (a) and (b), a scatterplot (c) and KDE density countours from the normal-transformed flows.

options(digits=3)
cor.test(u1, u2, method="spearman")

## Warning in cor.test.default(u1, u2, method = "spearman"): Cannot compute exact
## p-value with ties

## 
##  Spearman's rank correlation rho
## 
## data:  u1 and u2
## S = 9e+06, p-value = 1e-11
## alternative hypothesis: true rho is not equal to 0
## sample estimates:
##    rho 
## -0.357

cor.test(u1, u2, method="kendall")

## 
##  Kendall's rank correlation tau
## 
## data:  u1 and u2
## z = -7, p-value = 3e-11
## alternative hypothesis: true tau is not equal to 0
## sample estimates:
##    tau 
## -0.242

sin(-0.242*pi/2)

## [1] -0.371

cor.test(u1, u2, method="pearson")

## 
##  Pearson's product-moment correlation
## 
## data:  u1 and u2
## t = -7, df = 340, p-value = 8e-12
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.448 -0.263
## sample estimates:
##    cor 
## -0.359

cor.test(z1, z2, method="pearson")

## 
##  Pearson's product-moment correlation
## 
## data:  z1 and z2
## t = -7, df = 340, p-value = 2e-10
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.426 -0.238
## sample estimates:
##    cor 
## -0.335

Table 1: Estimates of copula parameters, maximized log-likelihood, and AIC using the uniform-transfomred pipline flow data.

Next: fitting the paramteric pseudo-maximum likelihood: Btw how lazy is this code in the text-book! .Last.value is such a hack.

library(knitr)
library(kableExtra)
omega = -0.371

options(digits=4)

Ct = fitCopula(copula=tCopula(dim = 2), data=U.hat, method="ml", start=c(omega, 10)) 
#Ct@estimate
mle_t = loglikCopula(param=Ct@estimate, U.hat, copula=tCopula(dim = 2))
t_AIC = -2*mle_t + 2*length(Ct@estimate)
#
Cgauss = fitCopula(copula=normalCopula(dim = 2), data=U.hat, method="ml", start=c(omega)) 
#Cgauss@estimate
mle_g = loglikCopula(param=Cgauss@estimate, U.hat, copula=normalCopula(dim = 2))
g_AIC = -2*mle_g + 2*length(Cgauss@estimate)
# Not run
Cgu = fitCopula(copula=gumbelCopula(2, dim=2), data=U.hat, method="ml")

## Warning in .local(copula, tau, ...): For the Gumbel copula, tau must be >= 0.
## Replacing negative values by 0.

## Warning in fitCopula.ml(copula, u = data, method = method, start = start, :
## optim(*, hessian=TRUE) failed: non-finite finite-difference value [1]

# Not run
Cjoe = fitCopula(copula=joeCopula(2, dim=2), data=U.hat, method="ml")

## Warning in .local(copula, tau, ...): For the Joe copula, tau must be >= 0.
## Replacing negative values by 0.

## Warning in .local(copula, tau, ...): optim(*, hessian=TRUE) failed: non-finite
## finite-difference value [1]

#
Cfr = fitCopula(copula=frankCopula(1, dim=2), data=U.hat, method="ml")
#Cfr@estimate
mle_f = loglikCopula(param=Cfr@estimate, U.hat, copula=frankCopula(dim = 2))
f_AIC = -2*mle_f + 2*length(Cfr@estimate)
#
Ccl = fitCopula(copula=claytonCopula(1, dim=2), data=U.hat, method="ml")
#Ccl@estimate
mle_c = loglikCopula(param=Ccl@estimate, U.hat, copula=claytonCopula(dim = 2))
c_AIC = -2*mle_c + 2*length(Ccl@estimate)

# Put the data in a dataframe:
df <- data.frame(CopulaFamily=c('t', '', 'Gaussian', 'Frank', 'Clayton'),
        variable=c("t.hat", "v.hat", "p.hat", "theta.hat","theta.hat"),
        estimates=c(Ct@estimate[1], Ct@estimate[2], Cgauss@estimate,  Cfr@estimate,Ccl@estimate[1]),
         maximized_ll=c(mle_t, "", mle_g, mle_f, mle_c),
         AIC=c(t_AIC, "", g_AIC, f_AIC, c_AIC))

df$maximized_ll <- as.numeric(df$maximized_ll)
df$AIC <- as.numeric(df$AIC)
kbl(df, digits=2)

CopulaFamily	variable	estimates	maximized_ll	AIC
t	t.hat	-0.34	20.98	-37.97
	v.hat	22.44	NA	NA
Gaussian	p.hat	-0.33	20.36	-38.72
Frank	theta.hat	-2.25	23.07	-44.14
Clayton	theta.hat	-0.17	9.87	-17.74

Introduction to copulas (Part 1) https://medium.com/@financialnoob/introduction-to-copulas-part-1-2ccbc5373d2e

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README.md		README.md
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Statatistics and Data Analysis for Financial Engineering - Chapter 8 Copulas

Special copulas

Archimedean Copulas

Frank copula

Crayton copula

Gumbel Copula adsfads

Joe copula

Plot of generator function for Frank cupola

Scatter plot of 9 bivariate Frank copulas

Scatter plot of 9 bivariate Clayton copulas

Scatter plot of 6 bivariate Gumbel copulas

Scatter plot of 6 bivariate Joe copulas

Rank correlation: 1. Kendall’s Tau, 2. Spearman’s rank correlation coefficient

Tail dependence

Example: flows in pipeline

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Statatistics and Data Analysis for Financial Engineering - Chapter 8 Copulas

Special copulas

Archimedean Copulas

Frank copula

Crayton copula

Gumbel Copula adsfads

Joe copula

Plot of generator function for Frank cupola

Scatter plot of 9 bivariate Frank copulas

Scatter plot of 9 bivariate Clayton copulas

Scatter plot of 6 bivariate Gumbel copulas

Scatter plot of 6 bivariate Joe copulas

Rank correlation: 1. Kendall’s Tau, 2. Spearman’s rank correlation coefficient

Tail dependence

Example: flows in pipeline

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