function dPi = deltaPi_vs_lambda(lambda, D,G,Dstar,Gstar,C,min_deltaS,max_deltaS)
%
% lambda = the amount of the hedge instrument to 
% D = total Delta of instruments depending on the underlying with price S
% G = total Gamma of instruments depending on the underlying with price S
% Dstar = Delta of the hedge instrument
% Gstar = Gamma of the hedge instrument
% C = cost to buy/sell (trade) one unit of the the hedge instrument 
% min_deltaS = smallest move in the underlying 
% max_deltaS = largest most in the underlying 
%   
% Written by:
% -- 
% John L. Weatherwax                2006-12-31
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

dp = D + lambda * Dstar;
gp = G + lambda * Gstar;

deltaS_worst = - dp/gp;
if( (deltaS_worst < min_deltaS) || (deltaS_worst > max_deltaS) ) % we are outside of the allowable range 
  v = min( min_deltaS * dp + 0.5 * ( min_deltaS^2 ) * gp, max_deltaS * dp + 0.5 * ( max_deltaS^2 ) * gp );
else % we are inside of the allowable range
  v = - ( dp^2 / ( 2 * gp ) );
end

dPi = v - abs(lambda) * C;