% 
% Solve the transcendental equation associated with the local solution 
% near the free boundary of an American option.  This corresponds to equation
% 7.24 in Wilmott's book.
% 
% Written by:
% -- 
% John L. Weatherwax                2008-06-11
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

clc; 
close all; 

fn = inline( '2*sqrt(pi)*(x^3)*exp(0.25*x^2)*normcdf(x,0,sqrt(2))-2*(2-x^2)', 'x' ); 

figure; 
x = linspace(0.7,1,100); 
y = zeros(size(x)); 
for ii=1:length(x)
  y(ii) = fn(x(ii)); 
end
plot( x, y, '-x' ); grid on; axis tight; 

rt = fsolve( fn, 0.9 ); 
fprintf('root found is %10.6f\n',rt);