ASSIGNMENT 1 MATH3075 Financial Derivatives (Mainstream) Due by 11:59 p.m. on Sunday, 8 September 2024 1. [12 marks] Single-period multi-state model. Consider a single-period market model M = (B, S) on a finite sample space Ω = {ω1, ω2, ω3}. We assume that the money 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 that P(ωi) = pi > 0 for i = 1, 2, 3. (a) Find the class M of all martingale measures for the modelM. Is the market modelM arbitrage-free? Is this market model complete? (b) Find the replicating strategy (ϕ00, ϕ10) 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 formula with an arbitrary martingale measure Q from M. (d) Show directly that the contingent claim Y = (Y (ω1), Y (ω2), Y (ω3)) = (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 martingale measures for the modelM. (f) Suppose that you have sold the claim Y for the price of 3 units of cash. Show that you may find a portfolio (x, ϕ) with the initial wealth x = 3 such that V1(x, ϕ) > Y , that is, V1(x, ϕ)(ωi) > Y (ωi) for i = 1, 2, 3. 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) For any fixed L ≥ 0, sketch the profile of the payoffX(L) as a function of ST ≥ 0 and find a decomposition of X(L) in terms of the payoffs of standard call and put options with maturity date T (do not use a constant payoff). Notice that a decomposition of X(L) may depend on the value of the parameter 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 pi0(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 modelM = (B, S) defined on some finite state space Ω. Show that the arbitrage price of X(L) at time t = 0 is a monotone 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 Ω. Justify your answer. 51作业君版权所有