part 1 please

1. For the following parts you need to provide full rigorous proofs for full credit. (a) (1 point) Show that any 2 by 2 market with one risky asset and one riskless asset is complete. (b) (1 point) Show that any 3 by 2 market with one risky and one riskless asset is incomplete. (c) (2 point) In the special case of a 2 by 2 market with one risky and one riskless asset, prove that AF (arbitrage free) implies there is a risk neutral martingale probability Q. 2. Calculate the following: (a) (2 pt) For a 3 by 2 market, M=321111, find an intial price S0 of the risky asset for which this market has a riak-neutral martingale probability Q. (b) (1 pt) Calculate Q in (a). (c) (1 pt) In a complete 2 by 2 market M=[abcd],So=(2a+b2ccd) where a,b,c,d are positive, calculate the hedge X for contingency claim V= (ef). (d) (1 pt) Calculate the initial price of the claim V=(ef) using the hedge in (c). (e) (1 pt) Using c=d=1, and a>b>0, in (c), price a European put on the underlying risky asset with strike K=S0. 1. For the following parts you need to provide full rigorous proofs for full credit. (a) (1 point) Show that any 2 by 2 market with one risky asset and one riskless asset is complete. (b) (1 point) Show that any 3 by 2 market with one risky and one riskless asset is incomplete. (c) (2 point) In the special case of a 2 by 2 market with one risky and one riskless asset, prove that AF (arbitrage free) implies there is a risk neutral martingale probability Q. 2. Calculate the following: (a) (2 pt) For a 3 by 2 market, M=321111, find an intial price S0 of the risky asset for which this market has a riak-neutral martingale probability Q. (b) (1 pt) Calculate Q in (a). (c) (1 pt) In a complete 2 by 2 market M=[abcd],So=(2a+b2ccd) where a,b,c,d are positive, calculate the hedge X for contingency claim V= (ef). (d) (1 pt) Calculate the initial price of the claim V=(ef) using the hedge in (c). (e) (1 pt) Using c=d=1, and a>b>0, in (c), price a European put on the underlying risky asset with strike K=S0