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ASSIGNMENT 1

MATH3075 Financial Derivatives (Mainstream)

Due by 11:59 p.m. on Sunday, 10 September 2023

1. [12 marks] Single-period multi-state model. Consider a single-period market model M = (B, S) on a sample space Ω = {ω1, ω2, ω3}. Assume that r = 3 and the stock price S = (S0, S1) satisfies S0 = 5 and S1 = (36, 20, 4). 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 model M. Is the market model M arbitrage-free? Is this market model complete?

(b) Find the replicating strategy for the contingent claim Y = (10, 2, −6) and com-pute its arbitrage price π0(Y ) at time 0 through replication.

(c) Recompute π0(Y ) using the risk-neutral valuation formula with an arbitrary martingale measure Q from the class M.

(d) Check whether that the contingent claim X = (5, 4, −1) is attainable in M.

(e) Find the range of arbitrage prices for X using the class M of all martingale measures for the model M.

(f) Suppose that at time 0 you have sold the claim X for 2 units of cash. Show that there exists a hedge ratio ϕ such that the wealth V1(2, ϕ) at time 1 strictly dominates the payoff X, meaning that V1(2, ϕ)(ωi) > X(ωi) for i = 1, 2, 3.

2. [8 marks] Static hedging with options. Consider a parametrised family of contingent claims with the payoff Y (α) at time T given by the following expression

Y (α) = min (α, β + 2|β — ST| — ST)

where a real number β > 0 is fixed and the parameter α is an arbitrary real number such that α ≥ 0.

(a) For any fixed α ≥ 0, sketch the profile of the payoff Y (α) as a function of ST ≥ 0 and find a decomposition of Y (α) 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 Y (α) may depend on the value of the parameter α.

(b) Assume that call and put options with all strikes are traded at time 0 at some finite prices. For each value of α ≥ 0, compute the arbitrage price π0(Y (α)) at time t = 0 for the claim Y (α) using the prices at time 0 of call and put options and a suitable decomposition obtained in part (a).

(c) For any α > 0, examine the sign of an arbitrage price of the claim Y (α) in any (not necessarily complete) arbitrage-free market model M = (B, S) with a finite state space Ω. Justify your answer.

(d) Consider a complete arbitrage-free market model M = (B, S) defined on some finite sample space Ω. Show that the arbitrage price of Y (α) at time t = 0 is a monotone function of the variable α ≥ 0 and find the limits limα→0 π0(Y (α)), limα→∞ π0(Y (α)) and limα→3β π0(Y (α)).