function Value = Binomial(u,d,R,n,S0)
%    Value  = Binomial(u,d,R,n,S0)
%   
%   Returns the value of an European simple option with payoff Phi(S_T)
%   when the stock price follows a Binomial model on t=0,1,...,n, and 
%   the money-market has interest rate R per period.
%   The program uses the Binomial algorithm (or Backward Induction)/
%   Input Parameters:
%       u = Upward factor
%       d = Downward factor
%       R = Risk-free interest rate per unit time
%       n = Number of time steps
%       S0 = Initial option price
%       Payoff = function that returns the payoff at maturity (which is
%       assumed to be a function of S_n). This function should be coded
%       below. Now is coded to return the value of a European Call Option.
%   Output:
%       The value of the Option at time t=0.
%
%   Additional information:
%       1. You need to save this code with the name Binomial.m in some
%       folder
%       2. Open the MATLAB and using the command cd foldername, move to the
%       folder where the program is saved
%       3. from the command line, juts type Binomial(u,d,R,n,S0).
%       4. You can check if the code is working by typing 
%       Binomial(1.5,.5,0,3,80), which should return 27.5 (Example in
%       class).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Vectors of option value
    Vt = zeros(1,n+1);
    Vt_1 = zeros(1,n+1);
    
    
%Computation of the risk-neutral probabilities
    qu = (1+R-d)/(u-d); 
    qd = (u-(1+R))/(u-d);

% Initialization 
for k=1:n+1  % k-1 = # of upward movements.
    Vt(k)=Payoff(S0*u^(k-1)*d^(n-(k-1)));
end

% Backward induction
for j=1:n
    for k=1:n-j+1
        Vt_1(k) = (qu*Vt(k+1)+qd*Vt(k))/(1+R);
    end
    Vt= Vt_1;
end

Value = Vt(1);

end
    
    
function P = Payoff(s)
    % This function returns the value of the payoff.
    % Right now is code to return the payoff of a European call option.
    
    K = 80; % Strike 
  
    if s>K
        P = s-K;
    else
        P=0;
    end
end