[8 marks] Static hedging with options. Consider a parametrised family of European contingent claims with the payoff X(L) at time T given by the following expression X(L)=min(2KST+KST,L) where a real number K>0 is fixed and L is an arbitrary real number such that L0. (a) Sketch the profile of the payoff X(L) as a function of the stock price ST and find a decomposition of X(L) in terms of terminal payoffs of standard call and put options with expiration date T. Notice that the decomposition of the payoff X(L) may depend on values of K and L. (b) Assume that call and put options are traded at time 0 at finite prices. For each value of L0, find a representation of the arbitrage price 0(X(L)) of the claim X(L) at time t=0 in terms of prices of call and put options at time 0 using the decompositions from part (a). (c) Consider a complete arbitrage-free market model M=(B,S) defined on some finite state space . Show that the arbitrage price of X(L) at time t=0 is a monotone function of the variable L0 and find the limits limL3K0(X(L)), limL0(X(L)) and limL00(X(L)) using the representations from part (b). (d) For any L>0, examine the sign of an arbitrage price of the claim X(L) in any (not necessarily complete) arbitrage-free market model M=(B,S) defined on some finite state space . Justify your answer. [8 marks] Static hedging with options. Consider a parametrised family of European contingent claims with the payoff X(L) at time T given by the following expression X(L)=min(2KST+KST,L) where a real number K>0 is fixed and L is an arbitrary real number such that L0. (a) Sketch the profile of the payoff X(L) as a function of the stock price ST and find a decomposition of X(L) in terms of terminal payoffs of standard call and put options with expiration date T. Notice that the decomposition of the payoff X(L) may depend on values of K and L. (b) Assume that call and put options are traded at time 0 at finite prices. For each value of L0, find a representation of the arbitrage price 0(X(L)) of the claim X(L) at time t=0 in terms of prices of call and put options at time 0 using the decompositions from part (a). (c) Consider a complete arbitrage-free market model M=(B,S) defined on some finite state space . Show that the arbitrage price of X(L) at time t=0 is a monotone function of the variable L0 and find the limits limL3K0(X(L)), limL0(X(L)) and limL00(X(L)) using the representations from part (b). (d) For any L>0, examine the sign of an arbitrage price of the claim X(L) in any (not necessarily complete) arbitrage-free market model M=(B,S) defined on some finite state space . Justify your