MATH3075 金融衍生品

MATH3075 Financial Derivatives (Mainstream)

Due by 11:59 p.m. on Sunday, 11 September 2022

1. [12 marks] Single-period multi-state model. Consider a single-period marketmodel M = (B, S) on a finite sample space ? = {肋1, 肋2, 肋3}. We assume that themoney market account B equals B0 = 1 and B1 = 4 and the stock price S = (S0, S1)satisfies S0 = 2.5 and S1 = (18, 10, 2). The real-world probability P is such thatP(肋i) = pi > 0 for i = 1, 2, 3.
   (a) Find the class M of all martingale measures for the modelM. Is the marketmodelM arbitrage-free? Is this market model complete?
   (b) Find the replicating strategy (?00, ?10) for the contingent claim X = (5, 1,?3)and compute the arbitrage price pi0(X) at time 0 through replication.
   (c) Compute the arbitrage price pi0(X) using the risk-neutral valuation formulawith an arbitrary martingale measure Q from M.
   (d) Show directly that the contingent claim Y = (Y (肋1), Y (肋2), Y (肋3)) = (10, 8,?2)is not attainable, that is, no replicating strategy for Y exists inM.
   (e) Find the range of arbitrage prices for Y using the class M of all martingalemeasures for the modelM.
   (f) Suppose that you have sold the claim Y for the price of 3 units of cash. Showthat you may find a portfolio (x, ?) with the initial wealth x = 3 such thatV1(x, ?) > Y , that is, V1(x, ?)(肋i) > Y (肋i) for i = 1, 2, 3.
2. [8 marks] Static hedging with options. Consider a parametrised family ofEuropean contingent claims with the payoff X(L) at time T given by the followingexpressionX(L) = min(2|K ? ST |+K ? ST , L)
   where a real number K > 0 is fixed and L is an arbitrary real number such thatL ≡ 0.
   (a) Sketch the profile of the payoff X(L) as a function of the stock price ST andfind a decomposition of X(L) in terms of terminal payoffs of standard call andput options with expiration date T . Notice that the decomposition of the payoffX(L) may depend on values of K and L.
   (b) Assume that call and put options are traded at time 0 at finite prices. Foreach value of L ≡ 0, find a representation of the arbitrage price pi0(X(L)) ofthe claim X(L) at time t = 0 in terms of prices of call and put options at time0 using the decompositions from part (a).
   (c) Consider a complete arbitrage-free market modelM = (B, S) defined on somefinite state space ?. Show that the arbitrage price of X(L) at time t = 0 is amonotone function of the variable L ≡ 0 and find the limits limL↙3K pi0(X(L)),limL↙﹢ pi0(X(L)) and limL↙0 pi0(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 modelM = (B, S) defined on
   some finite state space · Justify your answer.
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