please help to answer question (c):

1. Given the risk-neutral process of a non-tradable market index as, dst S , - = Y(1)dt + odz where y(t) is a time function and o is a constant. Assume also that risk-free interest rate r is constant and flat. (a) Use risk-neutral pricing to determine the futures price Kr of the index with maturity at T. Note : Maturity payoff of a futures contract is defined as Fr = Sr- Kr, where Kr is the futures delivery price defined at current time. The choice of Kr is defined in the way that current price of a futures contract is zero for which there is no cost on both sides in entering the agreement. (15 points) (b) Consider a cash-or-nothing digital option written on the market index with strike price L and maturity at time . The maturity payoff of this option is given by P, if ST > L (ST, D)= 10, if STSL Suppose the risk-neutral drift y() is not known. Use futures price defined in (a) to calibrate the market index at option's maturity under risk-neutral preference, and show that the current price of the digital option is given by fo = erTPN 108(7) -021 OVT Evaluate also the forward price of the digital option f(S, () conditional to a given market index S, at time t during the life of the option. (20 points) (c) Consider the compound options written on the market index. A digital-on-digital option is an option with maturity t and strike X written on the digital option in (b) with maturity T'> t. Payoff of the digital-on-digital option at maturity t is given by if f ( S . , T ) > x h ( S