function [C,P] = blackscholes(E,tm,S,r,sigma)
%
% Implements the Black Scholes analytic formula.  
% See page 79 of "The Mathematics of Financial Derivatives" 
% by Paul Willmott.
% 
% Inputs:
%   E: exercise price 
%   tm: time to expiry
%   S: current asset price
%   r: interest rate
%   sigma: asset volatility
% 
% Note: Any input can be a vector and the output will be a 
% corresponding vector.
% 
% Written by:
% -- 
% John L. Weatherwax                2007-11-13
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

% to prevent warnings, in the case of zero inputs we offset E/S some
E    = E+eps; 
S    = S+eps; 
num1 = log(S./E) + ( r + 0.5*(sigma.^2) ).*tm;
d1   = num1./(sigma.*sqrt(tm+eps));
num2 = log(S./E) + ( r - 0.5*(sigma.^2) ).*tm;
d2   = num2./(sigma.*sqrt(tm+eps)); 
% the call price: 
C    = S.*normcdf(d1,0,1) - E.*exp(-r*tm).*normcdf(d2,0,1);
% the put price: 
P    = E.*exp(-r*tm).*normcdf(-d2,0,1) - S.*normcdf(-d1,0,1);