ECMT3150: Assignment 2 (Semester 2, 2025)
Lecturer: Simon Kwok
Due: 5pm, 24 October 2025 (Friday)
All numerical answers are to be rounded to 3 decimal places.
1. Consider a single-period …nancial market consisting of a risk-free money account and
a risky stock NVDA Inc. (code: NVDA). Suppose a share of NVDA is priced at $100
now (time 0). At the end of the period (time 1), its price either goes up to $120 or
down to $80. The risk-free interest rate is 0:1 per period (so that $1 deposited in the
risk-free money account at time 0 will accumulate to $1:1 at time 1). There is no
restriction on the borrowing or lending capacity. Let p be the probability that NVDA
price goes up under the risk-neutral probability measure Q.
(a) [3 marks] Show that p = 0:75. [Hint: The discounted stock price process is a
martingale under Q.]
(b) [4 marks] Suppose a European call is written on NVDA at time 0 with strike
price $100 and expiring at time 1. Find its fair price at time 0.
(c) [5 marks] It is possible to replicate the payo¤ of the European call in part (b)
with a combination of risk-free money account and NVDA shares. Construct
the replicating portfolio by clearly specifying the long/short position for the risk-
free money account and NVDA shares. [Hint: Suppose the replicating portfolio
contains D units of risk-free money account and shares of NVDA. Then …nd D
and . Note that positive (negative) numbers represent a long (short) position.]
(d) Suppose the market price of a European call is $15 at time 0.
i. [4 marks] Explain how you can make a riskless pro…t of $300,000.
ii. [4 marks] Name and discuss two restrictions in real life that will prohibit
you from earning the riskless pro…t in part (i).
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2. Robert just graduated from USyd with ECMT3150 under his belt and took up an
exciting analyst role at J.P. Morgan. He was allocated to the forex derivative desk.
Robert’s …rst assignment was to study the derivatives for the forex market between
two currencies: AUD and CAD.
As a starting point, Robert models the risk-neutral distribution (denoted Q) of next
period’s price of AUD/CAD using the following one-period model. Suppose the time-0
price of AUD/CAD is S0. At time 1, the price S1 moves up to S0u with probability q,
and moves down to S0 1u with probability 1 q. It is assumed that u > 1, and that the
risk-free interest rate is zero.
As a convention, the price of AUD/CAD represents how much an Australian dollar is
worth in terms of Canadian dollars.
(a) [4 marks] Show that q = 1
u+1
. (Hint: recall that St is a martingale under Q.)
(b) [4 marks] Suppose S0 = 1. Compute the time-0 price of an at-the-money Euro-
pean call option written on AUD/CAD with strike price 1 and maturing at time
1. Express your answer in terms of u.
(c) [4 marks] Suppose S0 = 1. Compute the time-0 price of an at-the-money Euro-
pean call option written on CAD/AUD with strike price 1 and maturing at time
1. Express your answer in terms of u. (Hint: AUD/CAD priced at St is equivalent
to CAD/AUD priced at 1
St
.)
Robert now moves beyond the one-period model by considering a continuous-
time di¤usion model for AUD/CAD under Q. Let S(t) denote the price process
of AUD/CAD. (Note that the notation is switched from St to S(t) as we are
transiting from discrete-time to continuous-time framework.) The S(t) is modelled
according to the following stochastic di¤erential equation (sde):
dS(t) = S(t)dW (t);
where represents the instantaneous volatility.
(d) [6 marks] Let Y (t; S(t)) = S(t)1 denote the price process of CAD/AUD. Using
Itô’s lemma, derive the sde of Y (t; S(t)).
(e) [4 marks] Mimi, Robert’s manager, was pleased with Robert’s di¤usion model.
Mimi asked Robert to run a stress test by evaluating the impact of extreme events
on Australian economy. Suppose AUD/CAD experienced a ‡ash crash, i.e., S(t)
falls rapidly towards zero at certain time t. Discuss what would happen to the
price dynamics of CAD/AUD by referring to the instantaneous drift and volatility
of Y (t) you derived in part (d).
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