function [v,vn]=fn_b(nsim)
% for option on midterm, 2003
%u=vector of uniform (0,1) input to generate %normal variate by inverse
randn('state',0)
s0=10 ;K=10; sigma=.2 ; dt=1;  s=[]; v=[]; r=0.045
for i=1:nsim
    s1=s0+r*s0*dt+sigma*(s0^.5)*sqrt(dt)*randn;
    s2=s1+r*s1*dt+sigma*(s1^.5)*sqrt(dt)*randn;
s=[s s2]; % s is value of stock
v=[v (s2>=8).*(s2-10).*(s2<=12)]; % v is the value of the option
end
v=(s>=8).*(s-10).*(s<=12);
%discount option values to present
v=exp(-2*r)*v;

randn('state',0)
s0=10 ;K=10; sigma=.2 ; dt=1;  s=[]; vn=[]; r=0.055
for i=1:nsim
    s1=s0+r*s0*dt+sigma*(s0^.5)*sqrt(dt)*randn;
    s2=s1+r*s1*dt+sigma*(s1^.5)*sqrt(dt)*randn;
s=[s s2]; % s is value of stock
end
vn=(s>=8).*(s-10).*(s<=12); % v is the value of the option
vn=exp(-2*r)*vn;  % discounted value of options, new r. 
c=cov([v',vn']);  % covariance estimator