Static hedging with options

2. [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 (2|K ST] + K ST, L) where a real number K > 0 is fixed and L is an arbitrary real number such that L>0. = (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 X(L) may depend on the value of the variable L. (b) Assume that call and put options are traded at time 0 at finite prices. For each value of L > 0, find a representation of the arbitrage price To(X(L)) of the claim X(L) at time t 0 in terms of prices of call and put options at time O using the decompositions from part (a). (c) Consider a complete arbitrage-free market model M= (B, S) defined on some finite state space 2. Show that the arbitrage price of X(L) at time t O is a monotone function of the variable L > 0 and find the limits lim (+3K TO(X(L)), TO(X(L)) and lim 1470 70(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 N. Justify your answer. lim to 2. [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 (2|K ST] + K ST, L) where a real number K > 0 is fixed and L is an arbitrary real number such that L>0. = (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 X(L) may depend on the value of the variable L. (b) Assume that call and put options are traded at time 0 at finite prices. For each value of L > 0, find a representation of the arbitrage price To(X(L)) of the claim X(L) at time t 0 in terms of prices of call and put options at time O using the decompositions from part (a). (c) Consider a complete arbitrage-free market model M= (B, S) defined on some finite state space 2. Show that the arbitrage price of X(L) at time t O is a monotone function of the variable L > 0 and find the limits lim (+3K TO(X(L)), TO(X(L)) and lim 1470 70(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 N. Justify your answer. lim to