maxima_abformula.wxm

assume(alpha > 0);
assume(nu>0);
z(y) := nu/log((sqrt(1+2*rho*nu/alpha*y+nu^2*y^2/alpha^2)-rho-nu*y/alpha)/(1-rho));
y(K) := (K^(1-beta) - f^(1-beta))/(1-beta);
M0(K) := log(f/K)*z(y(K));
taylor(M0(K),K,f,1);
psi(K) := sqrt(alpha^2+2*alpha*rho*nu*y(K)+nu^2*y(K)^2)*K^beta;
M1(K) := -M0(K)*z(y(K))^2*log(M0(K)*sqrt(K*f/(psi(K)*psi(f))));
taylor(M1(K),K,f,1);
gamma(K) := (K^beta - f^beta)/(K-f)
taylor(M0(K)*(1+alpha*rho*nu*gamma(K)*T)+M1(K)*T,K,f,2);

M0(K) := (f-K)*z(y(K));
taylor(M0(K),K,f,1);
M1(K) := -M0(K)*z(y(K))^2*log(M0(K)*sqrt(1/(psi(K)*psi(f))));
taylor(M1(K),K,f,1);


assume(alpha > 0);
assume(nu>0);
xsi(y) := nu/log((sqrt(1+2*rho*nu/alpha*y+nu^2*y^2/alpha^2)-rho-nu*y/alpha)/(1-rho));
y(z) := ((f*exp(z))^(1-beta) - f^(1-beta))/(1-beta);
M0(z) := -z*xsi(y(z));
taylor(M0(z),z,0,2);
psi(z) := sqrt(alpha^2+2*alpha*rho*nu*y(z)+nu^2*y(z)^2)*(f*exp(z))^beta;
M1(z) := -M0(z)*xsi(y(z))^2*log(M0(z)*sqrt(f*exp(z)*f/(psi(z)*psi(0))));
taylor(M1(z),z,0,2);
g(z) := ((f*exp(z))^beta - f^beta)/(f*exp(z)-f);
taylor(g(z),z,0,2);
taylor(M0(z)*(1+alpha*rho*nu*g(z)*T)+M1(z)*T,z,0,2);