Please answer with all steps and equations. Preferably don't use excel, but if you do, please show equations.

Consider a European Call in the Black-Scholes GBM model. Prove that its Gamma is always strictly positive. You should provide a full mathematical derivation that Gamma_Call > 0. See the formula sheet for the relevant formulas you may use. Formulas that may be potentially useful: F = S_0e^(r - delta)T F = S_0e^rT - FV_T(Div) C - P = PV(F - K) R = Sigma_iP(0, T_i)F(T_i)/Sigma_iP(0, T_i) S_nh = S_0 d^n(u/d)^x, X ~ Bin(n, q) u = e^(r - delta)h + sigma Squareroot h, d = e^(r - delta)h - sigma Squareroot h q = e^r - delta)h - d/u - d C = e^-rh(qC_u + (1 - q)C_d) Delta = e^-delta h C_u - C_d/S_u - S_d B = e^rh (uC_d - dC_u)/(u - d) C = Delta S + B C_0 = e^delta T S_0N(d_1) - Ke^-rT N(d_2) P_0 = Ke^-rT N(-d_2) - e ^deltaT S_0 N(-d_1) d_1, 2 = log(S_0/K) + (r - delta)T plusminus sigam^2 T/2/sigma Squareroot T Delta_Call = e^-deltaT N(d_1) Delta_Put = -e^-delta T N(d_1) S_t = S_0e^(r - delta - sigma^2/2)t + sigma Squareroot t epsilon, epsilon ~ N (0, 1) E[S_t] = S_0e^(r - delta)t