# Chapter 3. Multivariate Volatility Models

### Matlab and Julia

Copyright 2011 - 2023 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: www.gnu.org/licenses.

##### Listing 3.1/3.2

##### % Download stock prices in MATLAB

```
p = csvread('stocks.csv',1,0);
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

```
using CSV, Statistics, DataFrames;
p = CSV.read("stocks.csv", DataFrame);
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

```
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

```
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

```
[par, Ht] = o_mvgarch(y,2, 1,1,1);
Ht = reshape(Ht,4,T)';
OOrho = Ht(:,3) ./ sqrt(Ht(:,1) .* Ht(:,4));
```

##### Listing 3.5/3.6

##### OGARCH in Julia

```
```

##### Listing 3.7/3.8

##### % DCC in MATLAB

```
[p, lik, Ht] = dcc(y,1,1,1,1);
Ht = reshape(Ht,4,T)';
DCCrho = Ht(:,3) ./ sqrt(Ht(:,1) .* Ht(:,4));
```

##### Listing 3.7/3.8

##### DCC in Julia

```
using ARCHModels, Plots;
dcc = fit(DCC{1, 1, GARCH{1, 1}}, y; meanspec = NoIntercept);
H = covariances(dcc);
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

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

##### Listing 3.9/3.10

##### Correlation comparison in Julia

```
```

##### Financial Risk Forecasting

Market risk forecasting with R, Julia, Python and Matlab. Code, lecture slides, implementation notes, seminar assignments and questions.© All rights reserved, Jon Danielsson,