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Finance: FINA2322
Creation date：2024-05-27 15:51:42
Belonging category：金融Finance
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Finance: FINA2322
Derivatives
There are two parts in the exam.
Part I. 15 Multiple Choice (True or False) Questions (2 point each, 30 points in total)
Part II. 5 Short Questions (70 points in total)
Total Pages of the Exam Paper (including this cover page): 9
• The exam is open book.
• You have to write down your UID on the Answer Book.
• Make sure you show the steps and equations to score full credit.
• Leave 4 d.p. for the answers.
Academic Integrity Statement:
By taking this examination, students agreed to the following Academic Integrity Statement.
A. I acknowledge that University examinations require all students to respect the highest
standards of academic integrity. For the examination I am about to take, I make the following
pledge: 1. All the work will be my own, and I will not plagiarize from anyone else; 2. I will
not obtain or seek to obtain an unfair advantage by communicating or attempting to
communicate with any other person during the examination; neither will I give or attempt to
give assistance to another student in taking the examination;
B. I understand that students who are suspected of violating this pledge are liable to be
referred to the Disciplinary Committee, and maybe subject to disciplinary action such as
suspension of studies or expulsion from the University.
Not for circulation
Not to be taken away
2
Part I: Multiple Choice/True or False Questions
MC Question 1
A borrower should take long position in Eurodollar futures to hedge the interest rate risk.
A. True
B. False
MC Question 2
A synthetic borrowing with fixed interest rate can be created by taking short position in long-
term ZCB and a long position in short-term ZCB.
A. True
B. False
MC Question 3
The yield to maturity of the ZCB is the same as the par coupon rate of the Treasury Bond.
A. True
B. False
MC Question 4
A swap requires investor to pay everything at the initiation of the swap.
A. True
B. False
MC Question 5
Out-of-the-money call has strike price lower than out-of-the-money put, for two options with
the same underlying asset.
A. True
B. False
MC Question 6
If you do not want to early-exercise the American put option, you should hold the option until
expiration.
A. True
B. False
MC Question 7
When the underlying asset becomes more volatile, the cost of early exercising increases.
A. True
B. False
MC Question 8
The 1-year standard deviation of a stock with uncorrelated monthly returns is the half of the
2-year standard deviation of the same stock.
A. True
B. False
MC Question 9
As strike price increases, the delta of a call option increases.
A. True
B. False
3
MC Question 10
As time to maturity increases, the magnitude of Psi becomes larger.
A. True
B. False
MC Question 11
Gamma is the same for both call and put options.
A. True
B. False
MC Question 12
For a put option, rho is not negative.
A. True
B. False
MC Question 13
Assume put premium=$5, S = $50, K=55, r = 5% p.a. continuously compounded, the stock
pays a 3.0% p.a. continuous dividend and the time to maturity is 1 year. What is the implied
volatility?
A. 13.62%
B. 23.62%
C. 33.62%
D. 43.62%
MC Question 14
The implied volatility usually reverts to its historical mean. If VIX historical mean is 20, and
the current level is 15, which of the following is not a good strategy now? Assume the
options are all S&P 500 index options. (This question is motivated by the guest lecture)
A. Long a straddle
B. Long a VIX ETF
C. Long a S&P 500 put option
D. Long a butterfly spread
4
MC Question 15
Accumulators are financial derivative products sold by an issuer (seller) to investors (the
buyer) that requires the buyers to buy shares of some underlying security at a predetermined
strike price, settled periodically. This allows the investor to "accumulate" holdings in
the underlying security over the term of the contract; this then constitutes a structured
product. The strike price is usually a discount to prevailing market price. Usually, the
contracts include a kick-out feature that causes the contract to expire if shares rise above a
certain level.
For example, the current stock price is $90. The investor accumulates 100 shares a month at a
fixed price of $80 a share. The tenor of the contract is 12 months. The knock-out price is
$105.
How many of the following statements are correct? We assume interest rates and dividends to
be zero when computing profits. (This question is motivated by the guest lecture.)
1. If the stock price stays constant at $90, the total share purchased is 1200.
2. If the stock price keeps increasing and reaches $105 during the fourth month and falls
to $90 in 12 months, the total share purchased is 300.
3. If the stock price keeps increasing and reaches $105 during the fourth month and falls
to $90 in 12 months, the total profit is $3,000.
4. If the stock price fluctuates between $80 and $100 and is at $100 in 12 months, the
total share purchased is 1,200.
5. If the stock price fluctuates between $80 and $100 and is at $100 in 12 months, the
total profit is $24,000.
6. If the stock price keeps falling and is at $70 in 12 months, the total profit is -$12,000.
A. 3
B. 4
C. 5
D. 6
5
Part II: Short Questions
Short Question 1: FRA (9 marks)
The following table shows the zero coupon bond prices matures at different time. The face
value of the zero-coupon bond is 1.
Maturity
(months)
Zero Coupon Bond
Price
3 0.980
6 0.957
9 0.935
12 0.909
15 0.885
(a) Suppose you need to borrow 1 million 9 months later. If you would like to fix the
interest rate by that time so that you can borrow the 1 million for 6 months by entering a
FRA, what should be the effective fixed interest rate? (3 points)
(b) Suppose 6 months later, the 3-month and 9-month spot rate is 2% and 7% effectively.
What is the value of FRA in part (a)? (3 points)
(c) At the time of borrowing, the 6-month spot rate is 6% effectively. What is the
settlement amount of the FRA if settled at initiation? (3 points)
6
Short Question 2: Swaps (15 marks)
The table shows the information in March.
Maturity June September December
Treasury Zero-Coupon Bond
Price
0.97 0.94 0.90
Interest Rate Swap (i) (ii) (iii)
Oil Forward Price (iv) (v) (vi)
Oil Swap Price 105 107 109
(a) For (i) to (vi), use the information in the table and construct the set of fixed rates of
the interest rate swaps and the forward prices for oil for 3 through 9 months. The
swap settlements occur every quarter. (6 marks)
(b)
i) Consider the 9-month oil swap. 3 months later, the oil price is $105/barrel. If cash
settlement occurs, what is the payoff of the fixed price payer at t=3 months on a
1,000-barrel swap agreement? (3 marks)
ii) After the settlement above, what is the value of the swap? Given that the 3-month
and 6-month interest rate at that time is 3% and 6% effectively. Assume the
dividend yield (lease rate) of oil is negligible. (6 marks)
7
Short Question 3: Binomial Tree (17 marks)
Currently, the exchange rate between HKD and AUD is HKD 5.5 per AUD. The annualized
volatility is 20%. The HKD denominated continuously compounded risk-free rate is 1% p.a..
while the AUD denominated continuously compounded risk-free rate is 5%. Use a 2-step
binomial tree to price a European call option that has a strike price of 5.5 HKD per AUD
and expires in 6 months.
(a) Consider the European call option: (8 marks)
i. What is the value of u?
ii. What is the value of Sud?
iii. What is the value of call option at node uu?
iv. What is the value of call option at node d?
v. What is the call option premium at t=0?
(b) What is the option premium of an otherwise identical European put? (4 marks)
(c) What is the option premium of an otherwise identical American put? (5 marks)
8
Short Question 4: Black-Scholes (24 marks)
Currently, the Twentycent stock price is $380. The annualized volatility is 25%. The
continuously compounded risk-free rate is 1%. The stock does not pay dividend in the next 3-
month. Assume the option price follows the Black-Scholes formula.
(a) Consider a European put option that has a strike price of $380 and expires in 3
months, i.e. T = 91/365. (5 marks)
i. What is the value of the put option premium under Black-Scholes formula?
ii. What is the delta for this put option?
iii. If the stock price changes to $381 immediately, what is the approximate
change of the put price based on delta?
iv. What is the percentage change in the option price for a 1% change in the stock
price?
v. If the Sharpe ratio of the stock is 0.6. What is the Sharpe ratio of the call
option?
(b) Suppose you long a 380-strike European straddle with 3 months to expiration. What is
your profit if the stock price is $400 in 3 months? (5 marks)
(c) Suppose the exchange-traded Twentycent stock options only have the strike price of
$360, $370, $380, $390, $400 with 3-month time to maturity and S=$380. You
believe the price will be $300 in one month. Among the options above, what position
(long or short) of which option (strike, call or put) generates the largest return? (Steps
are not required for this part) (4 marks)
(d) You are an option market maker. Suppose you sell a 380-strike put with 3 months to
expiration and delta-hedge the positions. What is your overnight profit if the stock
price tomorrow is $370? (5 marks)
(e) You are an option market maker. Suppose you sell a 380-strike straddle with 3
months to expiration and delta-hedge the positions. What is your overnight profit if
the stock price tomorrow is $370? (5 marks)
9
Short Question 5: FRN (5 marks)
The U.S. Treasury began issuing Floating Rate Notes (FRNs) in January 2014. Unlike usual
bonds with fixed coupon rates, FRNs pay varying amounts of coupon quarterly until maturity.
The coupon rate rises and falls, and it equals the 3-month treasury bill rate 3 months ago. For
example, in September, the coupon rate is the 3-month treasury bill rate from June to
September. By nature of their security design, FRN prices fluctuate far less than those of
other Treasury securities. Since its introduction, FRNs have become one of the most popular
types of Treasury debt.
When it was first issued, traders on the market tried to figure out how to price the FRNs. As a
trader, you want to apply your knowledge in derivatives to the problem.
In the beginning of January, what should be the price of a 4-quarter FRN that matures in the
end of December with a face value of 100?
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COMSM0093 COMP281 AFM 333 IBUS1001 CP2501 Math 105A VE320 TEC101 SOCY2166 COM240 EFIM20033 ENGI4467 EFIM20034 EC 979 BE432 7EJ502 CSSE6400 COMP90025 Econ 312 ENG2031 ECO0006 EXERSCI 207 CAN209 ECO 137W FT312 ECO 2199 MATH32052 BDW4213 APS 360 SAS 6 CHME0017 PO92Q Economics 01AV MATH39512 MATH 5486 COMP8620 ISE102 ECON235 ECON 206 FI 345 PHAY0020 MATH0099 MATH0085 EFB106 PHY 134 MAE115 circuits Chemistry CSE12 MATH-UA 233 PHAS0038 COMP2140 CONMGNT 7049 COMP151101 ECON 485 ACX5903 ENGN6536 ARCH1162 Law PHAS0042 CCF1C03 MATH 351 Phys 139 AG942 MGT001371 APH106 ECON1051 MA4AM MA3Z7 NUMBER THEORY REST0006 SESS0026 STAT0023 TABL 2741 CMP4008Y EFIM10015 Acct 560 PSYC735 GEOG 5 ACCT3104 CSEC5616 PHIL1012 ITLS6111 ECON8022 5CCM226A compX123 MA4XJ R001 EWB LLP120 COMP643 ENG 103 ECON 23950 ACCT 207 FINC 616 Bio99 Math 137A Psychology BCPM0054 MATH20212 EC3450 CENV6141 PSY 205 CE335 CS365 Calculus ACSD INFT6800 MCIT INFO6030 PHYS 111LV H63EEJ ECON2151 ENGI 2181 MTRX1701 PLIT08009 CS240 Math 270 CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 ​COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 ​BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 ​LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ​ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 ​FINC5090 ​MATH1052 ​SciComp Elec4622 ​FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 ​MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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