Research School of Finance, Actuarial Studies and Statistics
Test 1
Semester 2, 2022
FINM2002 DERIVATIVES / FINM7041 APPLIED DERIVATIVES
Writing Time:
90 minutes
Exam Conditions:
Wattle-based
Permitted Materials:
Any
Instructions to Students:
1. It is not permitted to disclose, disseminate, reproduce, or publish any portion of this test in any manner.
2. This exam consists of a total of THREE questions with subparts. Questions are of unequal value, with marks
indicated for each question. Please answer all questions.
3. All answers must be typed and provided within the Wattle exam interface. Only answers submitted and
typed within the answer space will be read and graded.
4. Incorrect choices for multiple-choice questions will be penalized.
5. Please show all working for marks. A number without intermediate steps and explanations will be awarded
a mark of zero.
6. Emailed answers will not be accepted.
7. Unless stated otherwise in a question, please keep all decimal places for interim numerical answers, final
answers should be rounded up to 4 decimal places. Failure to do so will result in penalties.
8. Please remember to save your answers often by clicking either “next page” or “previous page” as per the
discussion on how to prepare for Test 1 in Lecture 4, including how to work with Wattle quiz interface.
Total Marks = 57
This test is redeemable and counts towards 20% of your final grade for the course.
QUESTION 1. THIS QUESTION HAS 4 SUBPARTS, PLEASE ANSWER ALL SUBPARTS. (Total 8 marks)
I. Which of the following is(are) correct? Select all that apply. (2 mark)
a. The price of an ASX SPI 200 futures contract is trading at 3,710. It follows that each contract can be used
to hedge a portfolio worth $3,710.
b. We need to pay fees to store gold securely in a bank vault. We therefore expects the price of a futures
contract on gold to consider this cost and it should be lower than the price of a futures contract on a
non-dividend asset without any storage costs, but is otherwise identical.
c. The spot price and the forward price of an asset are both prices for purchasing or selling the asset, the
difference is in the timing of the transaction.
d. When the Treasury term structure is flat, we expect futures and forward to have the same value.
Solution:
Index futures & contract multiplier, L2 recording, slides 26-28
Carry cost adjustment, positive (negative) carry cost must be added back into (subtracted from) the
pricing formula so we are indifferent between holding the forward vs. holding the underlying, L2
recording, slide 32
Forward + futures value and understanding Treasury Rates, L2 recording, slide 11, L3, slides 5 & 8
II. Which of the following is(are) correct? Select all that apply. (2 marks)
a. Alex buys 15 October corn futures contracts from Nick, and then sells 3 October corn contracts to Xinyi.
Boyu buys 7 October corn contracts from Clare. Assume no other transactions took place, the current
open interest is 22 and the trading volume is 25.
b. Stock brokers earn significant fees from lending their clients’ stocks, this gives them incentives to
facilitate stock lending that is required in covered short selling.
c. Vincent Kosuga managed to short the onion market after taking control of 98% of onions in the U.S. by
directly short selling onions that he owned.
d. Company X can borrow USD at 5% p.a. and borrow GBP at 4% p.a., while Company Y can borrow GBP at
4.5% p.a., and borrow USD at 4.9% p.a. It follows that Y’s comparative advantage is in the GBP.
Solution:
L1 recording, slides 23-24:
OI = # of contracts outstanding, Trading volume = # of contracts traded in one day
Alex-Nick trade: 15 open interest & 15 volume
Alex-Xinyi trade: 0 change in OI with Xinyi buying existing contracts & 15+3 = 18 volume
Boyu-Clare trade: 15+7 = 22 open interest & 18+7 = 25 volume
W1 recording, slides 4 to 5, short selling and mechanics
W1 recording, slides 10 to 11 + discussion forum post [Workshop 1] Short Futures + How did Kosuga profit
from short futures?
The trader doesn’t need to directly short the asset, Kosuga was short onion futures, not shorting onions
directly.
Currency swap, W3, T4 Q5
USD (X-Y) = 5-4.9 = 0.1
GBP (X-Y) = 4-4.5 = -0.5 => X’ comparative advantage is in the GBP because it’s 0.5% better off
rather than 0.1% worse off in the USD.
III. Which of the following is(are) incorrect? Select all that apply. (2 marks)
a. If the long position earns a profit of $10, we know that the corresponding short position should also earn
a profit of $10
b. An in-the-money call has positive intrinsic value.
c. A convertible bond is usually issued by small, growing companies as a means to borrow money at a
lower interest rate compared to a conventional bond.
d. To hedge an equity portfolio from broad market movements, we could short futures on an index that is
representative of the broad market.
Solution:
Zero-sum gain, short + long = 0.
L4
W2 recording, slides 16-18.
IV. Which of the following is(are) correct? Select all that apply. (2 marks)
a. It is a good idea to read the question first, then answer based on what the question asks for.
b. There are three questions with multiple subparts in the test, it’s a good idea to scan through all the
questions first so we understand what to expect and how to plan our time in the test.
c. It is a good idea to move on to the next question if we find ourselves stuck on a 10-mark question, but
there are more questions and more marks waiting for us.
d. We should work on our own tests independently, make sure it’s our own work, and not do anything that
could be considered as violations to the ANU Academic Integrity Rules.
Solution:
A freebie question containing course administrative information and test/exam-taking tips discussed in L4
preparation for Test 1.
QUESTION 2. COMPLETE THE QUESTION WITH ALL ITS SUBPARTS. (Total 21 marks)
2002 VERSION
I. It’s August, a bond with semiannual coupons and matures in one year is selling for $80, the next coupon of
$2 is due in October. The risk-free interest rate is 2% p.a. continuously compounded for all maturities. The 5-
month forward contract on this bond with physical delivery is selling for $76.5. Are there arbitrage
opportunities to be exploited? If so, lay out the opportunity in full with detailed explanations and exact cash
flows at each step. (9 marks)
SOLUTION
T2, Q5, copy & paste from tutorial solutions with modifications
(t=0)---------(t=2, coupon)-------------------(t=5, maturity)
1. Semi-annual coupon paid every 6 months, if the next coupon due in 2 months, the one after would be due
in 2+6 =8 months and out of scope for this forward. We only worry about the $2 due in 2 months.
2. Interest rate constant for all maturities, both the 2-mo. and 5-mo. rates are 2% p.a., continuously
compounded.
D or PV of coupon at t=0: (Cash coupon)e^(-rT) = 2e^(-0.02x2/12) = $1.993344432 (0.5 mark, no need to show
this answer, as long as F_fair is correct, the 0.5 is given)
F_fair= (S0-D)e^(rT) = (80 – D)e^(0.02x5/12) = $78.65942713 (1 mark)
F_market < F_fair, the forward is undervalued.
“Buy low, sell high” by setting up the following scheme that is cost-less to the arbitrageur at t=0:
1. Long forward to purchase the bond at T at F_market (1 mark)
2. Short the bond and keep the position until T (0.5 mark), invest the proceeds in two parts such that D =
PV($coupon) is invested for the duration until the coupon payment (0.5 mark) and invest $(80 – D) for duration
T (0.5 mark)
t = 0,
* no initial net cash flow (cash inflow from the short position is perfectly offset by the cash outflow for the two
investments) (1 mark).
t = 2 mo.,
* receive $2 from the 2-mo. investment, use to pay the $2 coupon due on the short bond => zero net cash flow
at this point (1 mark).
t = T = 5 mo.,
* buy the bond from the forward and pay F_market, use bond to close short position (1 mark).
* T-mo. investment matures and give (80 – D)e^(0.02x5/12) = $78.66 (1 mark).
=> earn net cash inflow of 78.65942713 – F_market = $2.159427135, our arbitrage profit (1 mark) (the size of
difference between the F_market and F_Fair from our first check).
7041 VERSION
I. It’s August, the EUR-CHF spot exchange rate is 0.95 Swiss Francs per Euro. A Swiss trader notes the risk-
free interest rate at home is 2.5% p.a. continuously compounded for all maturities, and the Euro risk-free
rate is 3% p.a. continuously compounded for all maturities. The 5-month forward contract on Euro with
physical delivery is selling for 0.935 Swiss Francs. Are there arbitrage opportunities to be exploited? If so, lay
out the opportunity in full with detailed explanations and exact cash flows at each step. (9 marks)
SOLUTION
T2, Q6, copy & paste from tutorial solutions with modifications
r = Domestic interest rate: 2.5%
rf = Foreign interest rate: 3%
F_fair= S0e^((r-rf)T) =0.95e^((0.025-0.03)x5/12) = 0.94802289 (2 marks)
F_market < F_fair, the forward is undervalued. “Buy low, sell high” by setting up the following scheme that is
cost-less to the arbitrageur at t=0:
1. Enter a long position in the forward to buy 1 Euro for F_market in T (1 mark)
2. Borrow e(-rf*T) Euros (1 mark) for T at the rf interest rate, sell the Euro proceeds at the spot to receive e(-
rf*T)(0.95) = 0.93819891 swiss francs (1 mark), invest the proceeds for T at the domestic interest rate.
t = 0
* there is no initial net cash flow (1 mark).
t = T
* we pay F_market and receive 1 Euro from the forward contract, use it to repay the Euro loan that has grown
to exactly 1 Euro now (1 mark).
* Swiss Franc investment now worth
(e^(-0.03x5/12))(0.95)(e^(0.025x5/12)) = 0.95e^((0.025-0.03)*5/12) = 0.948022894 swiss francs (1 mark)
=> net cash inflow of 0.948022894 – F_market = 0.013022894 swissfrancs per euro (1 mark), our arbitrage
profit.
II. It’s August, wheat is currently trading at $380/tonne. Wilson Farms expects to harvest wheat in January
2023 and decides to take positions in ASX Jan ‘23 Eastern wheat futures to hedge against price volatility. The
futures is trading at $387/tonne. In October, the futures price has increased to $390/tonne while the spot
price is $385/tonne. In January 2023, the contract trades at $400/tonne just before the last trading day on
contract.
(a) Describe the type of hedge used and the position entered in August. (2 marks)
(b) Calculate the basis in August and then in October. (2 marks)
(c) Assume the market is free of arbitrage in January 2023, the farm closes its futures positions and cash
settles just before the last trading day while it simultaneously trades on the spot market. Assume the
underlying is a perfect match for the farm, explain the steps of how the hedge is closed, show cash flows
at each step, and calculate the net effective price for the farm. (5 marks)
(d) Suppose the farm instead used forward contracts that required physical delivery and each contract is on
100 metric tonnes. Based on past observations, the farm expects 1000 metric tonnes of wheat might be
harvested. Should the farm hedge with 10 contracts? Why or why not? (3 marks)
Tutorial 2, Q7 + LIVE Q&A with Will in Lecture 2
(a) Farmer expects harvest and needs to sell wheat, short hedge (1 mark)
=> short wheat futures (1 mark)
(b)
August basis = S_AUG – F_AUG = $380 – $387 = -$7 (1 mark)
October basis = S_OCT – F_OCT = $385 – $390 = -$5 (1 mark)
(c) L1 slide 38
Market free of arbitrage => futures price converged to spot price in January (1 mark, Lec1 s30, Lec2 s38)
S_JAN = F_JAN = $400
The short Jan ‘23 position is closed out by long Jan ‘23 futures in January at F_JAN = $400 (1 mark), resulting
in a financial loss of $387 – $400 = -$13/tonne (1 mark)
The farmer’s income of selling wheat spot is $400/tonne (1 mark), it follows that the effective cash flow is
(+400) + (-13) or an income of $387/tonne. (1 mark)
(d) LIVE Q&A with Will in L2
10 contracts would fully hedge the wheat production. So, no, the farm should not hedge the expected
production fully. (1 mark). As we learned with Will, Wheat production is subject to effects of climate
changes and elements, to fully hedge expected harvest may create production risk whereby the farm is
subject to failure to deliver penalties if they harvested less than expected (both in terms of quantity and
quality) (2 marks).
QUESTION 3. THIS QUESTION HAS 2 SUBPARTS, PLEASE ANSWER ALL SUBPARTS. (Total 28 marks)
I. Amazon (NASDAQ: AMZN) has never paid a dividend and has no plans to do so. It announced a 20-for-1
stock split to be taken into effect in June. AMZN was trading at $2,785/share in June just prior to the stock
split, and October European call options on 100 AMZN shares with strike $2800 were trading at $11. The
risk-free interest rate is 3% p.a. continuously compounded for all maturities.
(a) Are there any arbitrage opportunities in which you earn an immediate profit? If there is, describe
activities and cashflows for t=0 ONLY. (6 marks)
(b) Would you answer to (a) change for an otherwise identical October American call option trading at the
same price? Why or why not? (2 marks)
(c) Describe adjustments to the call option after the stock split, support with calculations. (2 marks)
SOLUTION
(a)
T4, Q2, copy & paste from tutorial solutions with modifications
AMZN non-dividend stock, check upper and lower bounds for the call
The upper bounds for c<=S0 holds (0.5 mark)
The lower bound given by c>=max(S0-Xe^-rT, 0)
c>=max(S0-Xe^-rT, 0)?
 max(2785-2800e^(-0.03x4/12),0) =12.86047 (1 mark)
 LHS We should buy low (LHS) sell high (RHS)
t=0, simultaneously short the RHS and use the proceeds to long the LHS
Short the RHS: short (S0-Xe^-rT)  recall from Lecture 4, slide 22, the RHS is made up of portfolio B (1 stock)
and part of portfolio A (cash)  short RHS means short the stock + short a negative amount of money (i.e.
Xe^-rT ) short the stock + invest Xe^-rT
a. short the stock and receive $2785, hold the position for T (1 mark)
b. invest Xe^-rT for 6 months (1 mark)
Long the LHS
c. long one European call @ $11 (1 mark)
 (2785)-(Xe^-rT)-11 = $1.860466 profit earned immediately (1 mark)
(b) L4, slide 15 + Discussion Forum post [Lecture 4] Lower bounds American stock option
No, an American call on non-dividend paying stock should never be exercised early, so it is equivalent to its
European counterpart. (2 marks)
(c) L4, slides 11 & 12
N = 100, m = 1, n = 20
The strike should be adjusted as: mX/n = 1x2800/20 = 140 (1 mark)
The number of shares per option is increased to: nN/m = 20*100/1 = 2000 (1 mark)
II. Company X borrowed $100M for 5 years at a floating rate but wishes to switch to a fixed rate. Company Y
borrowed the same amount for the same period at a fixed rate, but wishes to switch to a floating rate. The
following rates are available to X and Y, respectively. Both approached FINM Bank for advice on how to
transform their exposures and to potentially receive a better deal.
Fixed Rate (p.a.) Floating Rate (p.a.)
Company X 3.8% SOFR+1%
Company Y 3% SOFR+1.1%
(a) As the head of FINM’s swap desk, you have designed a swap that gives 60% of the gains to Y and the rest to
X, and charged 15bps p.a. as commission. All payments within the swap are exchanged via FINM. The floating
rate is paid all the way across. What does X receive via FINM within the swap? What does Y receive via FINM
within the swap? Explain and support with calculations. (7 marks)
(b) Payments are exchanged every 6 months in the swap from (a), and there is 15 months left in the swap. The
current SOFR is 10% p.a. for all maturities, compounded semiannually. The 6-month SOFR on the last payment
date was 11% p.a. What is the value of the swap for Y? Explain with supporting calculations. (11 marks)
(a)
T3, Q5 + T4, Q4, copy & paste from tutorial solutions with modifications
Step 1 – Determine the relative comparative advantage
The difference (i.e., spread) between X and Y’s rates:
Fixed-rate market: 3.8-3=0.8% (0.5 mark)
Floating-rate market : SOFR+1-(SOFR+1.1)=-0.1% (0.5 mark)
Step 2 – Determine the gains from swap (i.e., trade)
The total gains by both X and Y from the swap is the difference of differences:
Maximum savings: 0.8%-(-0.1%) =0.9% (0.5 mark)
Gains net of fees: 0.9% - 0.15% = 0.75% (0.5 mark)
Step 3 – Decide on gains allocation
As a result of the swap:
 X should be (0.75%*0.4)=0.3% better off than on its own: 3.5% (1 mark)
 Y should be (0.75%*0.6)=0.7% better off than on its own: SOFR +0.65% (1 mark)
X currently borrowing floating at SOFR+1%
Within swap:
- X should receive SOFR +1% (1 mark)
(floating is paid all the way across the swap and net out the floating payment) (1 mark)
- pays 3.5% to FINM.
- FINM nets fees from the fixed payment and passes 3.5-0.15=3.35% to Y
- Y receives 3.35% from FINM. (1 mark)
- Y pays SOFR + 1% to FINM
(No need to show this for part (a), but for part (b) and to check, Y gets 3.35 and pays SOFR+1% and borrows
3% on existing obligations, so it nets with 3.35-(SOFR+1%)-3% = -(SOFR+0.65%) as wished)
(b)
T3, Q6 + [Lecture 3] Interest Rate Swap Valuation
Coupon payment dates left (start from the last date): 15 mo., (15-6)=9 mo., (9-6)=3 mo.
The next payment date is in 3 months. (1 mark)
Within the swap, Y receives fixed 3.35% and pays floating SOFR+1% (1 mark)
Value = B_fixed – B_floating
Discount rate conversion:
rc = nln(1+rn/n) = 2ln(1+0.1/2) = 0.097580328 (1 mark)
B_fixed:
Fixed coupon = 3.35%/2 x 100M =1.675M (1 mark)
1.675e^(-0.097580328*3/12)+ 1.675e^(-0.097580328*9/12)+(100+1.675)e^(-0.097580328*15/12)
=93.1911 M (2 marks)
The next floating coupon is set in advance, paid in arrears, so it’s set based on the reference floating rate of
the previous payment date, SOFR =11%. From (a), the floating payment should be SOFR+1%, so the size of the
coupon expected is 12% p.a. (1 mark)
B_floating:
Next floating coupon value = 12%/2 x 100M = 6M (1 mark)
(6+100)e^(-0.097580328*3/12) = 103.4454 M (2 marks)
Value = B_fixed – B_floating = 93.1911 – 103.4454 = - 10.2543M (1 mark)