Consider a one-period arbitrage-free model (B, S) of market composed of a bank account B with interest rate r, and one underlying S. Answer the following questions.

(a) [5 points] Write down a formula for the function f s.t. f(S1, m) is the payoff of the derivative which, at maturity T = 1, gives its buyer a call option with strike K on the underlying S, plus the amount m R if the call with strike K is in the money (i.e. if ST K).

(b) [5 points] From now on let (B, S) be given by the binomial model with

r = 0, S0 = 12, S1(H) = 20, S1(T) = 4, K = 12. Compute the initial value X0(m) of the derivative with payoff X1(m) := f(S1, m).

(c) [5 points] In the above binomial model (B, S), compute the price M0 of the money-back call option, which at maturity T = 1 gives its buyer a call option on S with strike K = 12, plus it repays the initial cost M0 if the call with strike K finished in the money.