MATH3075代写、Python/Java语言编程代做

日期：2024-09-19 10:03

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 ?nite sample space ? = {ω1, ω2, ω3}. We assume that the

money market account B equals B0 = 1 and B1 = 4 and the stock price S = (S0, S1)

satis?es 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 model M. Is the market

model M arbitrage-free? Is this market model complete?

(b) Find the replicating strategy (?) for the contingent claim X = (5, 1, ?3)

and compute the arbitrage price π0(X) at time 0 through replication.

(c) Compute the arbitrage price π0(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 in M.

(e) Find the range of arbitrage prices for Y using the class M of all martingale

measures for the model M.

(f) Suppose that you have sold the claim Y for the price of 3 units of cash. Show

that you may ?nd a portfolio (x, ?) with the initial wealth x = 3 such that

V1(x, ?) > Y , that is, V1(x, ?)(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 ?xed and L is an arbitrary real number such that

L ≥ 0.

(a) For any ?xed L ≥ 0, sketch the pro?le of the payoff X(L) as a function of ST ≥ 0

and ?nd 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 ?nite prices. For

each value of L ≥ 0, ?nd a representation of the arbitrage price π0(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 model M = (B, S) de?ned on some

?nite state space ?. Show that the arbitrage price of X(L) at time t = 0 is a

monotone function of the variable L ≥ 0 and ?nd the limits limL→3K π0(X(L)),

limL→∞ π0(X(L)) and limL→0 π0(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 model M = (B, S) de?ned on

some ?nite state space ?. Justify your answer.