Given the risk-neutral process of a non-tradable market index as, ds, S = y(t)dt + adz,...

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Given the risk-neutral process of a non-tradable market index as, ds, S₁ = y(t)dt + adz, where y(t) is a time function and o is a constant. Assume also that risk-free interest rater is constant and flat. (a) Use risk-neutral pricing to determine the futures price Kr of the index with maturity at 7. 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 T. The maturity payoff of this option is given by JST, 1)= {0, if ST SL if ST > L 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 Jo=e+TPN (ing()-10²7) 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 OVT h(S₁, t) = { +) = {(S₂, 1)! if f( S₂, t) > X if f(S,, t) ≤X Show that the current price of this compound option is given by ho=e" PN: (B, ²7,-√), where ß solves e(T-PN (+2)-²7+ ovip) =X Note: The bivariate normal probability function is defined as N₂(x, b, p) = f du ₁(u) N pu (10 points) Given the risk-neutral process of a non-tradable market index as, ds, S₁ = y(t)dt + adz, where y(t) is a time function and o is a constant. Assume also that risk-free interest rater is constant and flat. (a) Use risk-neutral pricing to determine the futures price Kr of the index with maturity at 7. 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 T. The maturity payoff of this option is given by JST, 1)= {0, if ST SL if ST > L 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 Jo=e+TPN (ing()-10²7) 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 OVT h(S₁, t) = { +) = {(S₂, 1)! if f( S₂, t) > X if f(S,, t) ≤X Show that the current price of this compound option is given by ho=e" PN: (B, ²7,-√), where ß solves e(T-PN (+2)-²7+ ovip) =X Note: The bivariate normal probability function is defined as N₂(x, b, p) = f du ₁(u) N pu (10 points)

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