Financial Simulation

Mike Pena

Prof. Aparna Gupta

Financial Simulation

HW10

1 Exchange Option

Note: 99% CI taken into account in Width.

OUTPUT

Q1. Exchange Option

Price: 15.3804, Width: 0.1421

[pic 1]

2 Compound Option

Note: 99% CI taken into account in Width.

OUTPUT

Q2. Compound Option

Price: 1.2389, Width: 0.0105

3 Control Variates

Note: 99% CI taken into account in Width. Additionally, there is a significant improvement in the tightness of the confidence interval for both options after control variates were implemented. Strong positive correlation is visible in the scatter plot replications of discounted exchange option pay-off against repliations of European Call pay-off at option maturity. This correlation was utilized in the design of the control variate method. Method used 12.9 from Risk Management and Simulation (Gupta) as we know the theoretical mean of the control variate.

Q3. Control Variates

Exchange Option: Price (15.3804), Width (0.0334)

Compound Option: Price (1.2389), Width (0.0000)

[pic 2]

MATLAB CODE

% First clear the memory and close all open figures:

clear all; close all;

% Clear the screen:

clc;

% Change the working directory: This is the working directory

% where MATLAB looks for files (data inputs, .m files such as

% functions, etc.) and outputs results.

cd 'C:\Users\penam2\Desktop\RPI\Spring 2017\Financial Simulation\Homework\HW10';

% If you want to read files from another folder (maybe you

% have data stored in another location), you can use the addpath

% function:

addpath 'C:\Users\penam2\Desktop\RPI\Spring 2017\Financial Simulation\Homework\HW10';

% Quandl authentication token

Quandl.api_key('xokQnzdREAWG5absGuY3');

%% Question 1. Exchange Option

% Obtain Data:

stock{1}='GOOG/NYSE_GS';

n=length(stock);

for i=1:n

    data{i}=Quandl.get(stock{i},'collapse','daily', ...

        'start_date','2014-1-1','end_date','2017-1-1','type','data');

end

% Extract prices and compute returns:

prices_Dates=data{1}(:,1);

prices_GS=data{1}(:,5);

returns_Dates=data{1}(1:end-1,1);

returns_GS=log(prices_GS(1:end-1,:)./prices_GS(2:end,:));

% Compute Statistics:

P0_GS=prices_GS(1);

Mean_GS=mean(returns_GS(:));

Std_GS=std(returns_GS(:));

% Annualize Statistics:

Mean_GS=mean(returns_GS(:))*250;

Std_GS=std(returns_GS(:))*sqrt(250);

% Obtain Data:

stock{1}='GOOG/NYSE_JPM';

n=length(stock);

for i=1:n

    data{i}=Quandl.get(stock{i},'collapse','daily', ...

        'start_date','2014-1-1','end_date','2017-1-1','type','data');

end

% Extract prices and compute returns:

prices_Dates=data{1}(:,1);

prices_JPM=data{1}(:,5);

returns_Dates=data{1}(1:end-1,1);

returns_JPM=log(prices_JPM(1:end-1,:)./prices_JPM(2:end,:));

% Compute Statistics:

P0_JPM=prices_JPM(1);

Mean_JPM=mean(returns_JPM(:));

Std_JPM=std(returns_JPM(:));

% Annualize Statistics:

Mean_JPM=mean(returns_JPM(:))*250;

Std_JPM=std(returns_JPM(:))*sqrt(250);

% Compute Correlation:

rho=corr(returns_GS,returns_JPM);

% Set variable inputs:

T=1;step=500;dt=T/step;

N=1e3;alpha=0.5;r=0.02;

K_GS=P0_GS;

K_JPM=P0_JPM;

% Joint GBM Simulation:

[spath_GS,spath_JPM]=Joint_GBM(P0_GS,P0_JPM,Mean_GS,Mean_JPM,Std_GS,Std_JPM,rho,T,step,N);

% Compute Exchange Option:

q_GS=0.5;q_JPM=1.2;

payoff_Ex_Option=max(q_GS.*spath_GS(:,end)-q_JPM.*spath_JPM(:,end),0).*exp(-r*T);

[p_Ex_Option,~,ci_Ex_Option,~]=normfit(payoff_Ex_Option,0.01);

width=(ci_Ex_Option(2)-ci_Ex_Option(1))/p_Ex_Option;

fprintf('Q1. Exchange Option \n');

fprintf('Price: %3.4f, Width: %3.4f\n',p_Ex_Option,width);

fprintf('\n');

% Plot Paths:

figure

yyaxis left

plot(0:dt:T,spath_GS(end,:));

hold on

yyaxis right

plot(0:dt:T,spath_JPM(end,:));

title('Q1. Joing GBM Paths');

legend('GS','JPM');

%% Question 2. Compound Option

% Set Variable Inputs:

p1=prices_GS;

S0=p1(1);

T1=0.5;T2=1;

K1=0.5;K2=S0;

nstep=1;dt=(T2-T1)/nstep;

alpha=0.5;

rf=0.0112;

N=1e3;

step=500;

% Calibrate Model:

mu=mean((p1(1:end-1)-p1(2:end))./(p1(2:end).^alpha)./((p1(2:end).^(1-alpha)).*dt));

sigma= (std((p1(1:end-1)-p1(2:end)-mu*dt.*p1(2:end))./(p1(2:end).^alpha)))/(sqrt(dt));

% Simulate Prices:

stockpath_lp=CEVmodel(S0,sigma,mu,T2,step,N,alpha);