fxoptions( S0, X, rd, rf, T, vol, style)

function fxoptions( S0, X, rd, rf, T, vol, style)
%   fxoptions - valuation of European and American call and put options 
%   on foreign exchange using Garman-Kohlhagen model.
%
%       fxoptions( S0, X, rd, rf, T, vol, style)
%
%*************************************************************************  
%
%INPUTS:
%
%   S0    	- Current spot exchange rate (1 unit of foreign currency per
%                 domestic currency)
%
%   X       - Strike (i.e., exercise) price of the option.
%
%   rd      - Annualized continuously compounded domestic risk-free rate of return
%                 over the life of the option, expressed as a positive decimal
%                 number.
%
%   rf      - Annualized continuously compounded foreign risk-free rate of return
%                 over the life of the option, expressed as a positive decimal
%                 number.
%
%   T       - Time to expiration of the option, expressed in years.
%
%   vol     - Annualized exchange rate volatility.
%
%   style   - 'E' or 'e' for a European style call or put option,
%             'A' or 'a' for an American style call or put option.
%
%**************************************************************************
%
%   Example: from John Hull's book, 15.25:
%      Suppose that the spot price of the Canadian dollar is US $0.85 and
%      that the CAD|USD exchange rate has a volatility of 4% per annum. The 
%      risk-free rates of interest in canada and the United States are 4% 
%      and 5% per annum, respectively.
%      Calculate the value of a European call option to buy one Canadian dollar
%      for US$0.85 in nine months.
%
%
%      fxoptions( .85, .85, 5/100, 4/100, 9/12, 4/100, 'e')
%
%       European Call: $0.014703
%       European Put:  $0.008540
% 
%**************************************************************************
% Rodolphe Sitter - MSFM Student - The University of Chicago
% April 17, 2009
%**************************************************************************


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% EUROPEAN OPTIONS - GARMAN-KOHLHAGEN MODEL

if strcmp(style, 'E')||strcmp(style, 'e')
    F=S0*exp((rd-rf).*T);
    d1=log(F./X)./(vol.*sqrt(T))+vol.*sqrt(T)/2;
    d2=log(F./X)./(vol.*sqrt(T))-vol.*sqrt(T)/2;

    European_call = exp(-rd.*T).*(F.*normcdf(d1)-X.*normcdf(d2));
    European_put = European_call+(X-F)*exp(-rd.*T);
    fprintf('European Call: $%0.5g  \n',European_call)
    fprintf('European Put:  $%0.5g  \n\n',European_put)
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% AMERICAN OPTIONS - GARMAN-KOHLHAGEN MODEL

if strcmp(style, 'A')||strcmp(style, 'a')
    
    %TRINOMIAL TREE

    N=10^3;
    dt=T/N;         
    dx=vol*sqrt(3*dt);
    S=S0*exp((N:-1:-N)*dx)';

    American_put=max(X-S,0);
    American_call=max(S-X,0);

    nu= rd-rf-vol^2/2;         
    Pu=.5*((vol^2*dt+nu^2*dt^2)/dx^2+nu*dt/dx);          
    Pd=.5*((vol^2*dt+nu^2*dt^2)/dx^2-nu*dt/dx);          
    Pm= 1-(vol^2*dt+nu^2*dt^2)/dx^2;
    D=exp(-rd*dt);

    for t=N-1:-1:0  
        American_put=max(X-S(N-t+1:N+t+1),D*(Pu*American_put(1:2*t+1)+Pm*American_put(2:2*t+2)+Pd*American_put(3:2*t+3)));
        American_call=max(S(N-t+1:N+t+1)-X,D*(Pu*American_call(1:2*t+1)+Pm*American_call(2:2*t+2)+Pd*American_call(3:2*t+3)));   
    end

    % BINOMIAL TREE

    u=exp(vol*sqrt(dt)); d=1/u;
    p=(exp((rd-rf)*dt)-d)/(u-d);

    Up=flipud(cumprod(ones(N+1,1)*u))/u;
    Down=cumprod(ones(N+1,1)*d)/d;
    ST=S0*(Up.*Down);

    American_put_b=max(X-ST,0);
    American_call_b=max(ST-X,0);

    for i=1:N
        Up=flipud(cumprod(ones(N-i+1,1)*u))/u;
        Down=cumprod(ones(N-i+1,1)*d)/d;
        ST=S0*(Up.*Down);
        American_put_b=max(X-ST,D*(p*American_put_b(1:end-1)+(1-p)*American_put_b(2:end)));
        American_call_b=max(ST-X,D*(p*American_call_b(1:end-1)+(1-p)*American_call_b(2:end)));
    end
    
    fprintf('AMERICAN CALL:\n')
    fprintf('$%0.5g   (Binomial Tree)\n',American_call_b)
    fprintf('$%0.5g   (Trinomial Tree)\n',American_call)
    
    fprintf('\nAMERICAN PUT:\n')
    fprintf('$%0.5g   (Binomial Tree)\n',American_put_b)
    fprintf('$%0.5g   (Trinomial Tree)\n\n',American_put)
end