% GBM_Euro_call_lookback.m
% Extends GBM_paths.m to calculate price of European fixed strike lookback call option with
% strike price K.


clear

K=100

Nrealiz=10^5
S=100 % stock price at time 0
sig=0.2 % volatility
T=1 % end-time (in years)
ri=0.06 % risk-free interest rate
div=0.03 % dividend yield
N=500 % number of time steps

dt=T/N;
nudt=(ri-div-0.5*sig^2)*dt;
sigsdt=sig*sqrt(dt);

x0=log(S); % initial value for x=ln(S)

%figure
%hold on
tic

sum_CT=0;
sum_CT2=0;
for i=1:Nrealiz
    x=zeros(N,1);
    t=zeros(N,1);
    x(1)=x0; % initial condition
    t(1)=0;
    r=randn(N,1);
    maxSt=0;
    for n=1:(N-1)
        x(n+1)=x(n)+nudt+sigsdt*r(n);
        t(n+1)=t(n)+dt;
        St=exp(x(n+1));
        if St>maxSt
            maxSt=St;
        end
    end

    CT=max(0,maxSt-K);
    sum_CT=sum_CT+CT;
    sum_CT2=sum_CT2+CT^2;
end

call_value=sum_CT/Nrealiz*exp(-ri*T)
SD=sqrt( (sum_CT2-sum_CT*sum_CT/Nrealiz)*exp(-2*ri*T)/(Nrealiz-1))
SE=SD/sqrt(Nrealiz)
toc