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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