// exact DP - VI
// finite time binomial call option

clc
clear

////// Varaibles //////

// n - number of states
// T  - horizon
// J  - n x T matrix
// policy - n x T matrix

// J(i,t)      - the value function at time t and state i
// policy(i,t) - the policy at time t and state i, 
              // 1: Exercise, 2: Hold


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

K = 1;      // Strike Price
S = 1;      // Initial Price
p = 0.495;  // Probability of Moving Up
u = 1.01;   // Growth Rate
d = 0.9;    // Diminish Rate
T = 1000;   // Horizon, or Duration
n = 100;    // Number of States
alpha = 1;  // Discount Factor
 
r = u;

price_list = S*[d.^((n/2-1):-1:0), u.^(1:(n/2))]; // List of All Possible States

////// Initialization //////
J = zeros(n,T);
policy = ones(n,T); // 1 - Exercise, 2 - Hold

// Cost vector at time T
J(:,T) = max(price_list-K, zeros(1,n))'; 

//// DP Algorithm
for t = (T-1):-1:1 
    for i = 1:n
        //// Write the one-step-backward DP algorithm here:
        ///////////////////////////
        
        
        //////////////////////////
end

//// Plot the Option Prices and Exercising Strategies

figure(1)
surf(1:100,price_list,policy(:,T/100*(1:100)));ylabel('Stock Price');xlabel('Time');
title('Optimal Exercising Policy (blue: exercise, red: hold)');
//print(1,'exact-finite-policy-2','-depsc')

figure(2)
surf(1:100,price_list,J(:,T/100*(1:100)));ylabel('Stock Price');xlabel('Time');
title('Optimal Value Fucntion (Option Price at given time and stock price)');
//print(2,'exact-finite-J-2','-depsc')