1. Given the risk-neutral process of a non-tradable market index as, ds, = y(t)dt + odzi Si 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 Ks of the index with maturity at T. Note : Maturity payoff of a futures contract is defined as Fr= St Kt, where Ky is the futures delivery price defined at current time. The choice of Ky 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 T. The maturity payoff of this option is given by f(St, 7) = = { P, if S > 0 if ST SL Suppose the risk-neutral drift y(t) 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=e+IPN Evaluate also the forward price of the digital option f(Si, t) conditional to a given market index S; at time t during the life of the option. (20 points) 1. Given the risk-neutral process of a non-tradable market index as, ds, = y(t)dt + odzi Si 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 Ks of the index with maturity at T. Note : Maturity payoff of a futures contract is defined as Fr= St Kt, where Ky is the futures delivery price defined at current time. The choice of Ky 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 T. The maturity payoff of this option is given by f(St, 7) = = { P, if S > 0 if ST SL Suppose the risk-neutral drift y(t) 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=e+IPN Evaluate also the forward price of the digital option f(Si, t) conditional to a given market index S; at time t during the life of the option. (20 points)