// exact DP - infinite - VI

clc
clear

////// Variables //////
// n - number of states
// Niter - number of iterations
// J - n x Niter matrix, J(i,t) is the cost function starting from i after t steps of VI
// Q - vector of dimension n, Q(i) is short for Q(HOLD, i) 
// price_list - 1 x n vector, price_list(i) is the stock price associated with state i

////// Parameters //////

K = 1;        // Strike Price
S = 1;        // Initial Price
p = 0.4;      // Probability of Moving Up
u = 1+1e-3;   // Growth Rate
d = 2-u;      // Diminish Rate
Niter = 1000;     // Number of iterations
n = 10;       // Number of States
alpha = 0.95; // Discount Factor


//// Initialization ////
// Initialize All Possible States
price_list = S*[d.^((n/2-1):-1:0), u.^(1:(n/2))]; 
// Initialize J and Q
J = zeros(n,Niter);
Q = zeros(n,1);

//// VI ////

for t = 2 : Niter
 
    for i = 1 : n
      
      //////// Find Q-values by using J(:,t-1)
      
      //////// Write here:

      
      ////////// Update J(i,t) by using new Q values
      
     //////// Write here:

    end
   
end

figure(1);
plot(price_list, J);
plot(price_list, J(:,Niter),'*');
title('Convergence of Value Functions (Dotted line: J^*)');
xlabel('Stock Price');
ylabel('Option Price (Value Fucntion)');

figure(2);
plot(price_list,Q,'-*');
plot(price_list,price_list-K);
title('Q Values and Optimal Strategy)');
xlabel('Stock Price');
ylabel('Q Value');
legend('Q Values of the control HOLD','Q Values of the control EXERCISE')