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Risk and Asset Allocation

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Risk and Asset Allocation

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16 Nov 2005 (Updated )

Software for quantitative portfolio and risk management

S_LognormalCopulaSimul.m
% this script simulates the copula of a bi-variate lognormal distribution
% see "Risk and Asset Allocation"-Springer (2005), by A. Meucci


clc; clear;close all
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% input parameters
Mu=[1  -1]';     
r=-.9;            
sigma=[2 10]';    

NumSimul=100000;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Sigma=diag(sigma)*[1 r;r 1]*diag(sigma);
Y=mvnrnd(Mu,Sigma,NumSimul); % bi-variate normal simulation
X=exp(Y);                    % bi-variate lognormal simulation
U_1=logncdf(X(:,1),Mu(1),sigma(1)); % grade 1 simulation
U_2=logncdf(X(:,2),Mu(2),sigma(2)); % grade 2 simulation
Copula=[U_1 U_2];			    % copula

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% plots
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

figure 
% marginals
NumBins=round(10*log(NumSimul));

subplot('Position',[.05 .3 .2 .6]) 
[n,D]=hist(Copula(:,2),NumBins);
barh(D,n,1);
[y_lim]=get(gca,'ylim')
set(gca,'xtick',[])
grid on

subplot('Position',[.3 .05 .6 .2]) 
[n,D]=hist(Copula(:,1),NumBins);
bar(D,n,1);
[x_lim]=get(gca,'xlim')
set(gca,'ytick',[])
grid on

% scatter plot
subplot('Position',[.3 .3 .6 .6]) 
h=plot(Copula(:,1),Copula(:,2),'.');
set(gca,'xlim',x_lim,'ylim',y_lim)
grid on
xlabel('grade 1');
ylabel('grade 2');

% 3-d histogram (~rescaled pdf)
NumBins3d=round(sqrt(NumSimul)/5);
figure
hist3(Copula(:,[1 2]),[NumBins3d NumBins3d]);

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