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 modelM. Is the market
modelM 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 pi0(Y ) at time 0 through replication.
(c) Recompute pi0(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 inM.
(e) Find the range of arbitrage prices for X using the class M of all martingale
measures for the modelM.
(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 pi0(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 modelM = (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 pi0(Y (α)),
limα→∞ pi0(Y (α)) and limα→3β pi0(Y (α)).