Using Hurst Exponent for VaR Scaling

In this script we will demonstrated how Hurst exponent can be used for VaR scaling. The P&L data can be found in the P&L.xls spreadsheet. This data should be imported first.

Contents

Import P&L data

The data can be imported using xlsread command. Pls, change directory accordingly.

[~, ~, raw] = xlsread('\\Wil-homedrive03\c66144$\Desktop\P&L.xls','Sheet1');
raw = raw(2:end,:);

Calculating R/S Statistic

pnl = cell2mat(raw);
clearvars raw;
N=length(pnl);
for n=2:N
    % Calculate R statistic
    Deviation=cumsum(pnl(1:n))-cumsum(ones(n,1))*sum(pnl(1:n))/n;
    R(n-1)=max(Deviation)-min(Deviation);
    % Calculate S statistic
    S(n-1)=sqrt(sum(pnl(1:n).^2)/n-(sum(pnl(1:n))/n)^2);
    % Calculate R/S statistic
    Q(n-1)=R(n-1)/S(n-1);
    time(n-1)=n;
end

Estimating Hurst Exponent

The Hurst exponent is derived by plotting $Q(n)$ as a function of $\log(n)$ and fitting a straight line. The slope of the line gives $H$.

plot(log(time),log(Q),'bo')
hold on
fit=polyfit(log(time),log(Q),1);
H=fit(1)
plot(log(time),polyval(fit,log(time)),'r-',log(time),.5*log(time),'g--','LineWidth',2)
legend('R/S','Hurst Line', 'Normal Line','Location','North')

H =

    0.5853

Calculating 10-day P&L (Non-Overlapping)

We will calculate sliding 10-day P&L first and then pick P&L corresponding to non-overlapping intervals.

pnl_ovlp_10=movsum(pnl,10);
pnl_non_ovlp_10=zeros(1,120)';
for i=1:(N/10)
    pnl_non_ovlp_10(i)=pnl_ovlp_10(6+10*(i-1));
end

Comparing Hurst and Normal Scaling

Sample quantile corresponding to $\alpha=.95$ based on 1-day horizon can be scaled up using standard $\sqrt{10}$ assumption to a 10-day VaR. Alternatively, one can use Hurst exponent. The Hurst exponent results appear to be more accurate. This is just one realization of the P&L process and is used for demonstrational purposes only. Eventhough the Hust exponent appear to be close to 0.5, the scalors differ by more than 20% ($10^{H}/\sqrt{10}$).

m=100;
VaR10DayNO=zeros(1,m); %Unscaled non-overlapping 10-day VaR
VaR10DayO=zeros(1,m);  %Unscaled overlapping 10-day VaR
NSVaR10Day=zeros(1,m); %Normal-scaled 10-day VaR
HSVaR10Day=zeros(1,m); %Hurst-scaled 10-day VaR
quantiles=zeros(1,m);
normalerror=zeros(1,m);
hursterror=zeros(1,m);
overlaperror=zeros(1,m);
for i=1:m
    VaR10DayNO(i)=quantile(pnl_non_ovlp_10,(i+10)/(10*m));
    VaR10DayO(i)=quantile(pnl_ovlp_10,(i+10)/(10*m));
    NSVaR10Day(i)=sqrt(10)*quantile(pnl,(i+10)/(10*m));
    HSVaR10Day(i)=10^H*quantile(pnl,(i+10)/(10*m));
    quantiles(i)=1-(i+10)/(10*m);
    normalerror(i)=(NSVaR10Day(i)-VaR10DayNO(i))*100/VaR10DayNO(i);
    hursterror(i)=(HSVaR10Day(i)-VaR10DayNO(i))*100/VaR10DayNO(i);
    overlaperror(i)=(VaR10DayO(i)-VaR10DayNO(i))*100/VaR10DayNO(i);
end
figure
subplot(2,2,1)
% Comparing Hurst-scaled 10-day VaR to unscaled non-overlapping 10-day VaR
hold on
plot(quantiles,VaR10DayNO,'or',quantiles,HSVaR10Day,'*y')
legend('VaR','HuVaR','Location','South')
ylim([min([min(VaR10DayNO),min(HSVaR10Day),min(NSVaR10Day)]),max([max(VaR10DayNO),max(HSVaR10Day),max(NSVaR10Day)])])
subplot(2,2,2)
% Comparing Normal-scaled 10-day VaR to unscaled non-overlapping 10-day VaR
hold on
plot(quantiles,VaR10DayNO,'or',quantiles,NSVaR10Day,'*g')
legend('VaR','NoVaR','Location','South')
ylim([min([min(VaR10DayNO),min(HSVaR10Day),min(NSVaR10Day)]),max([max(VaR10DayNO),max(HSVaR10Day),max(NSVaR10Day)])])
subplot(2,2,3)
% Comparing the errors for the two scaling approaches
hold on
plot(quantiles,hursterror,'oy',quantiles,normalerror,'og')
legend('Hurst error, %','Normal error, %','Location','North')
subplot(2,2,4)
% Comparing the VaRs for the two scaling approaches
hold on
plot(quantiles,HSVaR10Day,'*y',quantiles,NSVaR10Day,'*g')
legend('HuVaR','NoVaR','Location','South')
ylim([min([min(VaR10DayNO),min(HSVaR10Day),min(NSVaR10Day)]),max([max(VaR10DayNO),max(HSVaR10Day),max(NSVaR10Day)])])

Comparing The Overlapping and Non-Overlapping VaRs

figure
subplot(1,2,1)
hold on
plot(quantiles,VaR10DayNO,'or',quantiles,VaR10DayO,'ob')
legend('Non-Overlapping VaR','Overlapping VaR','Location','South')
subplot(1,2,2)
hold on
plot(quantiles,overlaperror,'or')
legend('VaR Error due to Overlap, %','Location','North')


Published with MATLAB® R2016a