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calibration_FinitePoint4.m
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calibration_FinitePoint4.m
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function [Leverage_surf, impv_surf, x_L]=calibration_FinitePoint4(x, z, Maturity, L_initial, market_impv, r,q,kappa,theta,xi,rho,alpha,y0, epsilon,epsilon2, step_size, NN, Market_strikes, Market_maturities)
x_L=x;
dx=(max(x)-min(x))/(length(x)-1);
dz=(max(z)-min(z))/(length(z)-1);
dense_para_x=20;
x_dense=repelem(x', dense_para_x)';
x_increased=x(1):(x(2)-x(1))/dense_para_x:x(2);
x_increased(end)=[];
x_increased=x_increased-x(1);
x_increased=repmat(x_increased,1,length(x));
x_sparse=x_dense;
x_dense=x_dense+x_increased';
x_dense(end-dense_para_x+2:end)=[];
x_sparse(end-dense_para_x+2:end)=[];
dx_dense=dx/dense_para_x;
dense_para_z=1;
z_dense=repelem(z', dense_para_z)';
z_increased=z(1):(z(2)-z(1))/dense_para_z:z(2);
z_increased(end)=[];
z_increased=z_increased-z(1);
z_increased=repmat(z_increased,1,length(z));
z_dense=z_dense+z_increased';
dz_dense=dz/dense_para_z;
Maturity=[0; Maturity];
[initial_p,dt]= initial_by_bisection(x_dense,z_dense,1,r,q, kappa,theta,xi,rho,y0);
p=initial_p;
nT=length(Maturity);
%boundary condition for L
lb=-1; rb=1;
V_initial=max(1-exp(x_dense),0);
V_initial_put=max(exp(x_dense)-1,0);
L=L_initial(:,1);
Leverage_surf=zeros(length(x),length(Maturity)-1);
impv_surf=zeros(length(x),length(Maturity)-1);
for i=1:nT-1
t_start=Maturity(i);
t_end=Maturity(i+1);
v=zeros(length(x),NN+1);
s=zeros(length(x),NN+1);
beta_1=0.9;
beta_2=0.9;
for n=1:NN
impv=zeros(length(x),1);
vega=zeros(length(x),1);
L_dense=interp1(x,L,x_dense,'spline');
L_old=L;
G=log(L);
if t_end<dt
[initial_p,dt]= initial_by_bisection(x_dense,z_dense,1,r,q, kappa,theta,xi,rho,y0);
p=initial_p;
V=OptionPrice2(x_dense,z_dense,t_start,t_end,5,L_dense, initial_p, r, q, V_initial, y0);
V_put=OptionPrice2_put(x_dense,z_dense,t_start,t_end,5,L_dense, initial_p, r, q, V_initial_put, y0);
elseif t_start<dt && t_end>dt
[initial_p,dt]= initial_by_bisection(x_dense,z_dense,1,r,q, kappa,theta,xi,rho,y0);
V=OptionPrice2(x_dense,z_dense,t_start,dt,5,L_dense, initial_p, r, q, V_initial, y0);
V_put=OptionPrice2_put(x_dense,z_dense,t_start,dt,5,L_dense, initial_p, r, q, V_initial_put, y0);
V_middle=V;
V_middle_put=V_put;
p=adi_p_b(x_dense, z_dense, L_dense, initial_p, dt, t_end, (t_end-dt)/1, r,q,kappa,theta,xi,rho,alpha,y0 );
V=OptionPrice2(x_dense,z_dense,dt,t_end,5,L_dense, p, r, q, V_middle, y0);
V_put=OptionPrice2_put(x_dense,z_dense,dt,t_end,5,L_dense, p, r, q, V_middle_put, y0);
else
p=adi_p_b(x_dense, z_dense, L_dense, initial_p, t_start, t_end, (t_end-t_start)/1, r,q,kappa,theta,xi,rho,alpha,y0 );
V=OptionPrice2(x_dense,z_dense,t_start,t_end,5,L_dense, p, r, q, V_initial, y0);
V_put=OptionPrice2(x_dense,z_dense,t_start,t_end,5,L_dense, p, r, q, V_initial_put, y0);
end
%
psi=p*exp(-r*(Maturity(i+1)));
if i>2
G_T=(G-log(Leverage_surf(:,i-1)))/(t_end-t_start);
else
G_T=zeros(length(x),1);
end
% visulization
subplot(1,2,1)
plot(x_L, G, 'r*')
subplot(1,2,2)
plot(x_L, L, 'b*')
pause(0.000001);
sign1=zeros(length(x),1);
for j=1:length(x)
% using out of the money option to compute impv
if x(j)>0
if V(x_dense==x(j))>=exp(-q*t_end)
sign1(j)=1;
impv(j)=nan;
else
impv(j)=blsimpv(1,exp(x(j)),r,t_end,V(x_dense==x(j)),'Yield',q,'Class', {'Call'});
end
else
if V_put(x_dense==x(j))>=exp(-r*t_end)*exp(x(j))
sign1(j)=1;
impv(j)=nan;
else
impv(j)=blsimpv(1,exp(x(j)),r,t_end,V_put(x_dense==x(j)),'Yield',q,'Class', {'Put'});
end
end
if isnan(impv(j))
continue
else
vega(j)=blsvega(1,exp(x(j)),r,t_end, impv(j),q);
end
if vega(j)==0
vega(j)=1e-6;
end
end
gradient=zeros(length(x),1);
for t=1:length(Market_strikes)
for h=1:length(Market_maturities)
if market_impv(t,h)==0
continue
else
int_psi=y0*(psi*exp(z_dense))*dz_dense;
int_psi=int_psi(x_dense==x_sparse);
gradient=gradient+exp(x+2*G).*(impv-market_impv(t,h))./vega.*int_psi./(1e-6+(exp(x)-Market_strikes(t)).^2+(t_end-Market_maturities(h)).^2).^4;
end
end
end
gradient(2:end-1)=gradient(2:end-1)-epsilon*exp(-2*x(2:end-1)).*((G(3:end)+G(1:end-2)-2*G(2:end-1))/(dx^2)-(G(3:end)-G(1:end-2))/(2*dx))+epsilon2*G_T(2:end-1);
gradient(isinf(gradient))=sign(gradient(isinf(gradient)));
gradient(isnan(gradient))=0;
gradient=gradient+1e-2*sign1;
gradient=inpaint_nans(gradient,3);
v(:,n+1)=beta_1*v(:,n)+(1-beta_1)*gradient;
s(:,n+1)=beta_2*s(:,n)+(1-beta_2)*gradient.^2;
v_modefied=v(:,n+1)/(1-beta_1^(n+1));
s_modefied=s(:,n+1)/(1-beta_2^(n+1));
g_modefied=(step_size*v_modefied)./(sqrt(s_modefied)+10^(-3));
G=G-g_modefied;
G(1)=-lb*dx+G(2);
G(end)=rb*dx+G(end-1);
% limit the maximum and minimum of the leverage function
G(G<-10)=-10;
G(G>3)=3;
L=exp(G);
err_L=norm(L_old-L);
err_G=abs(L_old-L);
% constrain the error within a small interval containing the market points
err_G=max(err_G(exp(x)>=Market_strikes(1)-0.5 & exp(x)<=Market_strikes(end)+0.5));
if err_G<1e-1
fprintf('Maturity:%d, inter:%d, error_L:%d \n', t_end,n,err_L)
Leverage_surf(:,i)=L_old;
impv_surf(:,i)=impv;
V_initial=V;
V_initial_put=V_put;
initial_p=p;
break
end
if n==NN
fprintf('Not converge')
Leverage_surf(:,i)=L_old;
impv_surf(:,i)=impv;
V_initial=V;
initial_p=p;
V_initial_put=V_put;
end
end
end