1. Given the risk-neutral process of a non-tradable market index as, ds, = y(t)dt + odz S, 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 Kt of the index with maturity at T. Note : Maturity payoff of a futures contract is defined as Fr=S7- 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 Sr>L RS5,1)={C, if , se 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 (1947), ) for e-TPN Evaluate also the forward price of the digital option f(s, t) 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 h(S., T) = = {ics, o) , if f(Sq, t)>X T), if f(S, T) SX Show that the current price of this compound option is given by ho = e 'T PNB, log() - T+OVEB where solves e(T-1) PN OVT-T = X oyT Note : The bivariate normal probability function is defined as Nax, b, p) = L, du ,(u) (-) (10 points) 1. Given the risk-neutral process of a non-tradable market index as, ds, = y(t)dt + odz S, 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 Kt of the index with maturity at T. Note : Maturity payoff of a futures contract is defined as Fr=S7- 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 Sr>L RS5,1)={C, if , se 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 (1947), ) for e-TPN Evaluate also the forward price of the digital option f(s, t) 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 h(S., T) = = {ics, o) , if f(Sq, t)>X T), if f(S, T) SX Show that the current price of this compound option is given by ho = e 'T PNB, log() - T+OVEB where solves e(T-1) PN OVT-T = X oyT Note : The bivariate normal probability function is defined as Nax, b, p) = L, du ,(u) (-) (10 points)
