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);

 
    GARCH(1,1) Conditional Variance Model (Gaussian Distribution):
 
                  Value       StandardError    TStatistic      PValue  
                __________    _____________    __________    __________

    Constant    1.0538e-06     3.5053e-07        3.0063       0.0026443
    GARCH{1}       0.80653       0.012913        62.458               0
    ARCH{1}        0.15439       0.011578        13.335      1.4463e-40

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')