S_OptionReplication.m

% This script illustrates the option-based portfolio insurance (OBPI) strategy
% see Meucci, A. (2010) "Review of Dynamic Allocation Strategies - Convex versus Concave Management"
% available at http://ssrn.com/abstract=1635982

clear; clc; close all;
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% input parameters

Initial_Investment = 1000;
Time_Horizon = 6/12; % in years
Time_Step = 1/252; % in years

m=0.2;  % yearly expected return on the underlying
s=0.40; % yearly expected percentage volatility on the stock index
r=0.04; % risk-free (money market) interest rate

NumSimul=30000;

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% level of aggressivness bw 0 and 1
Aggressivness = .2;


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% initialize values
Underlying_Index=Initial_Investment;  % value of the underlyting at starting time, normalzed to equal investment
Start=Underlying_Index;
Elapsed_Time=0;
Portfolio_Value=Initial_Investment;

options=[];
Strike = fsolve('Solve4Strike',Start,options,Time_Horizon,Underlying_Index,s,r,Initial_Investment,Aggressivness*Initial_Investment/3);
Underlying_in_Portfolio_Percent = Delta(Time_Horizon-Elapsed_Time,Underlying_Index,s,Strike,r);


Underlyings_in_Portfolio=Portfolio_Value*Underlying_in_Portfolio_Percent;
Cash_in_Portfolio=Portfolio_Value-Underlyings_in_Portfolio;

% initialize parameters for the plot (no theory in this)
close all;
Portfolio_Series=Portfolio_Value;
Market_Series=Underlying_Index;
Percentage_Series=Underlying_in_Portfolio_Percent;

% asset evolution and portfolio rebalancing
while Elapsed_Time < Time_Horizon -10^(-5)  % add this term to avoid errors
    % time elapses...
    Elapsed_Time=Elapsed_Time+Time_Step;
    
    % ...asset prices evolve and portfolio takes on new value...
    Multiplicator = exp((m-s^2/2)*Time_Step+s*sqrt(Time_Step)*randn(NumSimul,1));
    Underlying_Index = Underlying_Index.*Multiplicator;
    Underlyings_in_Portfolio = Underlyings_in_Portfolio.*Multiplicator;
    Cash_in_Portfolio = Cash_in_Portfolio*exp(r*Time_Step);
    Portfolio_Value = Underlyings_in_Portfolio+Cash_in_Portfolio;
    
    % ...and we rebalance our portfolio
    Underlying_in_Portfolio_Percent = Delta(Time_Horizon-Elapsed_Time,Underlying_Index,s,Strike,r);
    Underlyings_in_Portfolio=Portfolio_Value.*Underlying_in_Portfolio_Percent;
    Cash_in_Portfolio=Portfolio_Value-Underlyings_in_Portfolio;
    
    
    % store one path for the movie (no theory in this)
    Portfolio_Series = [Portfolio_Series Portfolio_Value(1)];
    Market_Series=[Market_Series Underlying_Index(1)];
    Percentage_Series=[Percentage_Series Underlying_in_Portfolio_Percent(1)];
end

% play the movie for one path
Time=[0:Time_Step:Time_Horizon];
y_max=max([Portfolio_Series Market_Series])*1.2;
for i=1:length(Time)
    subplot(2,1,1);
    plot(Time(1:i),Portfolio_Series(1:i),'linewidth',2.5,'color','b');
    hold on;
    plot(Time(1:i),Market_Series(1:i),'linewidth',2,'color','r');
    axis([0 Time_Horizon 0 y_max]);
    grid on;
    ylabel('value','fontsize',12,'fontweight','bold');
    title('investment (blue) vs underlying (red) value','fontsize',12,'fontweight','bold')
    
    subplot(2,1,2);
    bar(Time(1:i),Percentage_Series(1:i),'stack','r');
    axis([0 Time_Horizon 0 1]);
    grid on;
    xlabel('time','fontsize',12,'fontweight','bold');
    ylabel('%','fontsize',12,'fontweight','bold');
    title('percentage of underlying in portfolio','fontsize',12,'fontweight','bold')
    
    
    set(gcf,'Units','normalized','Position', [0.1, 0.1, .8, .8]);
    F(i) = getframe;
end
%break
% plot the scatterplot
figure

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

subplot('Position',[.05 .3 .18 .6])
[n,D]=hist(Portfolio_Value,NumBins);
barh(D,n,1);
[y_lim]=get(gca,'ylim');
set(gca,'xtick',[])
grid on

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

% joint scatter plot
subplot('Position',[.3 .3 .6 .6])
plot(Underlying_Index,Portfolio_Value,'LineStyle','none','marker','.');
hold on;
so=sort(Underlying_Index);
plot(so,so,'r');
set(gca,'xlim',x_lim,'ylim',y_lim)
grid on;
xlabel('underlying at horizon (~ buy & hold )','fontweight','bold');
ylabel('investment at horizon','fontweight','bold');

set(gcf,'Units','normalized','Position', [0.1, 0.1, .8, .8]);