# Chapter 3. Multivariate Volatility Models (in MATLAB/Julia)

Copyright 2011 - 2020 Jon Danielsson. This code is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This code is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. The GNU General Public License is available at: https://www.gnu.org/licenses/.

##### Listing 3.1/3.2: Download stock prices in MATLAB Last updated August 2016

p = p(:,[1,2]);      % consider first two stocks
y = diff(log(p))*100;          % convert prices to returns
y(:,1)=y(:,1)-mean(y(:,1)); % subtract mean
y(:,2)=y(:,2)-mean(y(:,2));
T = length(y);

##### Listing 3.1/3.2: Download stock prices in Julia Last updated July 2020

using CSV, Statistics;
y1 = diff(log.(p[:,1])).*100; # consider first two stocks
y2 = diff(log.(p[:,2])).*100; # convert prices to returns
y1 = y1 .- mean(y1);           # subtract mean
y2 = y2 .- mean(y2);
y = hcat(y1,y2);     # combine both series horizontally
T = size(y,1);       # get the length of time series


##### Listing 3.3/3.4: EWMA in MATLAB Last updated June 2018

%% create a matrix to hold covariance matrix for each t
EWMA = nan(T,3);
lambda = 0.94;
S = cov(y);          % initial (t=1) covar matrix
EWMA(1,:) = S([1,4,2]);        % extract var and covar
for i = 2:T          % loop though the sample
S = lambda*S+(1-lambda)* y(i-1,:)'*y(i-1,:);
EWMA(i,:) = S([1,4,2]);    % convert matrix to vector
end
EWMArho = EWMA(:,3)./sqrt(EWMA(:,1).*EWMA(:,2)); % calculate correlations

##### Listing 3.3/3.4: EWMA in Julia Last updated July 2020

## create a matrix to hold covariance matrix for each t
EWMA = fill(NaN, (T,3))
lambda = 0.94
S = cov(y)           # initial (t=1) covar matrix
EWMA[1,:] = [S[1], S[4], S[2]]           # extract var and covar
for i in 2:T         # loop though the sample
S = lambda*S + (1-lambda)*y[i-1,:]*(y[i-1,:])'
EWMA[i,:] = [S[1], S[4], S[2]]       # convert matrix to vector
end
EWMArho = EWMA[:,3]./sqrt.(EWMA[:,1].*EWMA[:,2]); # calculate correlations


##### Listing 3.5/3.6: OGARCH in MATLAB Last updated August 2016

[par, Ht] = o_mvgarch(y,2, 1,1,1);
Ht = reshape(Ht,4,T)';
%% Ht comes from o_mvgarch as a 3D matrix, this transforms it into a 2D matrix
OOrho = Ht(:,3) ./ sqrt(Ht(:,1) .* Ht(:,4));
%% OOrho is a vector of correlations

##### Listing 3.5/3.6: OGARCH in Julia Last updated July 2020

## No OGARCH code available in Julia at present


##### Listing 3.7/3.8: DCC in MATLAB Last updated August 2016

[p, lik, Ht] = dcc(y,1,1,1,1);
Ht = reshape(Ht,4,T)';
DCCrho = Ht(:,3) ./ sqrt(Ht(:,1) .* Ht(:,4));
%% DCCrho is a vector of correlations

##### Listing 3.7/3.8: DCC in Julia Last updated July 2020

using ARCHModels, Plots;
## Multivariate models in ARCHModel package
dcc = fit(DCC{1, 1, GARCH{1, 1}}, y; meanspec = NoIntercept);
## Access covariances
H = covariances(dcc);
## Getting correlations
DCCrho = [correlations(dcc)[i][1,2] for i = 1:T];
plot(DCCrho, title = "Correlations", legend = false)


##### Listing 3.9/3.10: Correlation comparison in MATLAB Last updated June 2018

plot([EWMArho,OOrho,DCCrho])
legend('EWMA','DCC','OGARCH','Location','SouthWest')

##### Listing 3.9/3.10: Correlation comparison in Julia Last updated July 2020

## No OGARCH code available in Julia at present