r=0.02;
sigma=0.3;
T=1;
x=100;
K=200;

function [res]=f(g)
   d=prod(size(g));
   S_T=zeros(1,d);
   S_T=x .* exp((r - sigma*sigma/2)*T*ones(1,d)+sigma*sqrt(T)*g);
   res=exp(-r*T)*max(S_T-K,0);
endfunction

function res=approx_1_derivee_variance(lambda)
   Y=f(G).^2 .* exp(-lambda*G+lambda^2/2) .* (lambda - G);
   res=mean(Y);
endfunction

function res=approx_2_derivee_variance(lambda)
   Y=- G .* f(G+lambda).^2 .* exp(-2*lambda*G-lambda^2);
   res=mean(Y);
endfunction

n=10000;

K=100;
lambda=[0.0:0.05:3.0];
z1=0;z2=0;
G=grand(1,n,"nor",0,1);
for i=[1:prod(size(lambda))] do
	z1(i)=approx_1_derivee_variance(lambda(i));
	z2(i)=approx_2_derivee_variance(lambda(i));
end;
plot2d(lambda,z1);
plot2d(lambda,z2);

K=200;
z1=0;z2=0;
lambda=[0:0.05:6];
G=grand(1,n,"nor",0,1);
for i=[1:prod(size(lambda))] do
	z1(i)=approx_1_derivee_variance(lambda(i));
	z2(i)=approx_2_derivee_variance(lambda(i));
end;
plot2d(lambda,z1);
plot2d(lambda,z2);