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

Copyright 2011 - 2019 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/.

The GARCH functionality in the econometric toolbox in Matlab cannot do univariate GARCH.

Kevin Sheppard's MFE toolbox is much better, while not as user friendly, it is much better written and is certainly more comprehensive. It can be downloaded here and the documentation here is quite detailed.

##### 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 Python Last updated June 2018

import numpy as np
p = p[:,[0,1]]                                        # consider first two stocks
y = np.diff(np.log(p), n=1, axis=0)*100               # calculate returns
y[:,0] = y[:,0]-np.mean(y[:,0])                       # subtract mean
y[:,1] = y[:,1]-np.mean(y[:,1])
T = len(y[:,0])


##### 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 Python Last updated June 2018

EWMA = np.full([T,3], np.nan)
lmbda = 0.94
S = np.cov(y, rowvar = False)
EWMA[0,] = S.flatten()[[0,3,1]]
for i in range(1,T):
S = lmbda * S + (1-lmbda) * np.transpose(np.asmatrix(y[i-1]))* np.asmatrix(y[i-1])
EWMA[i,] = [S[0,0], S[1,1], S[0,1]]
EWMArho = np.divide(EWMA[:,2], np.sqrt(np.multiply(EWMA[:,0],EWMA[:,1])))
print(EWMArho)


##### 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 Python Last updated June 2018

## Python does not have a proper OGARCH package 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 Python Last updated June 2018

## Python does not have a proper DCC package at present


##### 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 Python Last updated June 2018

## Python does not have a proper OGARCH/DCC package at present