CS计算机代考程序代写 MATH3075/3975

MATH3075/3975

Financial Derivatives

School of Mathematics and Statistics
University of Sydney

Semester 2, 2020

Tutorial sheet 9

Background: Chapter 4 – European Options in the CRR Model.

Exercise 1 Consider the CRR model M = (B, S) with the horizon date
T = 2, the risk-free rate r = 0.1, and S0 = 10, S

u
1 = 13.2, S

d
1 = 9.9. Let X

be a European contingent claim with the maturity date T = 2 and the payoff
at maturity given by the formula

X =
(

min(S1, S2)− 10
)+
.

(a) Find the martingale measure P̃ for the market model M = (B, S).

(b) Show explicitly that X is a path-dependent contingent claim.

(c) Let Ft = FSt = σ(S0, . . . , St) for t = 0, 1, 2. Compute the arbitrage
price (πt(X), t = 0, 1, 2) using the risk-neutral valuation formula, for
t = 0, 1, 2,

πt(X) = Bt EP̃

(
X

BT

∣∣∣Ft).
(d) Find the replicating strategy (ϕt, t = 0, 1) for the claim X and check

that the wealth process V (ϕ) of the unique replicating strategy for X
coincides with the price process π(X) computed in part (c).

Exercise 2 We take for granted the CRR call option pricing formula

C0 = S0

T∑
k=k̂

(
T

k

)
p̂k(1− p̂)T−k −

K

(1 + r)T

T∑
k=k̂

(
T

k

)
p̃k(1− p̃)T−k

where k̂ is the smallest integer k such that

k log
(u
d

)
> log

(
K

S0dT

)
.

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Assume that the initial stock price equals S0 = 9, the risk-free interest rate
is r = 0.01 and the stock price volatility equals σ = 0.1 per annum. Use the
CRR parametrization for the parameters u and d, that is, set

u = eσ

∆t, d =
1

u
,

with the time increment ∆t = 1 (year).

(a) Compute the arbitrage price C0 of the European call option with strike
price K = 10 and maturity date T = 5 years.

(b) Compute the prices Cu1 and C
d
1 at time t = 1 for the same option using

a suitable version of the CRR call option pricing formula.

(c) Find the hedge ratio for the option at time 0.

Exercise 3 (MATH3975) Consider any arbitrage-free multi-period model
M = (B, S) where B is deterministic and S is an F-adapted process defined
on the finite probability space (Ω,F ,P) endowed with a filtration F. We
assume that B and S are strictly positive. Let Q̃ be any martingale measure
for the process S/B and let Q̂ be a probability measure equivalent to Q̃
such that the Radon-Nikodym density of Q̂ with respect to Q̃ on F equals L.
Assume that the process L is given by the following expression

Lt =
dQ̂
dQ̃
|Ft :=

B0
S0

St
Bt
, t = 0, 1, . . . , T.

In your answers, you may use results from Exercises 4 and 5 in week 7.

(a) Show that L0 = 1 and L is a strictly positive martingale with respect

to the filtration F under Q̃ so that the probability measure Q̂ is well
defined.

(b) Check that the processB/S is a martingale with respect to the filtration

F under Q̂.

(c) Using the abstract Bayes formula and the expression for the Radon-
Nikodym density L show that for any s > t and an arbitrary Fs-
measurable random variable Y the following equality holds

St EQ̂

(
Y

Ss

∣∣∣Ft) = Bt EQ̃
(
Y

Bs

∣∣∣Ft)

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(d) Assume that the call option with strike K can be replicated in M.
Using part (c), show that the arbitrage price Ct at time t ≤ T can be
represented as follows

Ct = St Q̂(D | Ft)−KB(t, T ) Q̃(D | Ft)

where D = {ω ∈ Ω : ST (ω) > K} and B(t, T ) := Bt/BT .

(e) Show that B(t, T ) is the arbitrage price at time t of the zero-coupon
bond, which pays one unit of cash at time T .

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