%  skewness_preference.m 
%  Messing around with skewness preference using Arrow-Pratt-like derivatives of u
%  Backus, Chernov, and Martin, "Equity index options"
%  Written:  April 2009 and after 
format compact 
clear all
close all

disp(' ')
disp('Skewness preference') 
disp('--------------------------------------------')
disp(' ')

%  ---------------------------------------------------------------------------------------
%  1. Cumulants 
syms alpha alpha0 alpha1 c 

alpha0 = 3
alpha1 = 0.01
logc = log(c)
alphac = alpha0 - alpha1*logc
u = exp(-alphac*logc)

u1 = diff(u,c,1)
u2 = diff(u,c,2)
u3 = diff(u,c,3)

simplify(u3)

return