%% Simulate Conditional Variance Model
% This example shows how to simulate a conditional variance model using simulate.
%% Step 1. Load the data and specify the model.
% Load the Deutschmark/British pound foreign exchange rate data included with the toolbox, and convert to returns. Specify and fit a GARCH(1,1) model.

load Data_MarkPound
r = price2ret(Data);
T = length(r);
Mdl = garch(1,1);
EstMdl = estimate(Mdl,r);
v0 = infer(EstMdl,r);

%
%% Step 2. Simulate foreign exchange rate returns.
% Use the fitted model to simulate 25 realizations of foreign exchange rate returns and conditional variances over a 1000-period forecast horizon. Use the observed returns and inferred conditional variances as presample innovations and variances, respectively.

rng default; % For reproducibility
[V,Y] = simulate(EstMdl,1000,'NumPaths',25,...
    'E0',r,'V0',v0);

figure
subplot(2,1,1)
plot(v0)
hold on
plot(T+1:T+1000,V)
xlim([0,T+1000])
title('Conditional Variances')
hold off

subplot(2,1,2)
plot(r)
hold on
plot(T+1:T+1000,Y)
xlim([0,T+1000])
title('Returns')
hold off

%
%% Step 3. Plot the returns distribution at a future time.
% Use simulations to generate a forecast distribution of foreign exchange returns 500 days into the future. Generate 1000 sample paths to estimate the distribution.

rng default; % For reproducibility
[V,Y] = simulate(EstMdl,500,'NumPaths',1000,...
    'E0',r-EstMdl.Offset,'V0',v0);

figure
histogram(Y(500,:),10)
title('Return Distribution in 500 Days')

% Copyright 2015 The MathWorks, Inc.