BUFN 761 Derivative Securities–FinalExam 2017

BUFN761DerivativeSecurities–FinalExam
R. H. Smith School of Business
Prof. JulienCujean
MOCKEXAM
Total Points: 120
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be sure to explain or show your computations.
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1


Exercise 1. Below is a pro?t diagram for a position. This position involves trading
optionswith6?monthsmaturityonastockwithcurrentstockprice$50,payingdividends
atrate?=1%. Thecontinuously-compoundedrisk-freerateis5%.
10
8
Slope=1
Slope=?1
6
4
2
Slope=?2
0
?2
?4
?6
?8
?10
?12
?14
Slope=2
Zero-Pro?tPoint=$43.1391
?1610
20
30
40
50
60
70
80
90
StockPricein6Months
a) (5points)Inthetablebelow, report thenumberofoptions(foreachstrikeprice)
neededtoconstructthediagram—ifyouneedtobuynoptionswrite+nandifyou
needtowritenoptionswrite?n. Useputoptionsonly. Therelevantslopesare
displayedonthediagram.
Strike Price
20
30
40
50
60
70
80
Number of Puts
2


b) (5points)Whatistheinitialcash-?owrequiredtoestablishthisposition(i.e.,what
isthepremiumofthisposition)?
c) (5 points) Consider the put positions in a): suppose you set your position in the
40?strike,the50?strike,andthe60?strikeputstozero,keepingotherpositions
unchanged. Drawthepayo?diagramoftheresultingstrategy.
10
8
6
4
2
0
?2
?4
?6
?8
?1010
20
30
40
50
60
70
80
90
StockPricein6Months
d) (5points)Supposethestockpricevolatilityis0: Whatisthepayo?ofthestrategy
undera)andd)respectively? Whatifthestockpricefallsto$30in6months?
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Exercise 2. Supposeyouwanttopriceacalloptionwithstrikeprice$95and1?year
to maturity. The risk-free rate is 8% and the underlying stock pays dividends at a
continuously-compoundeddividendyieldof8%. Thestockpriceevolvesaccordingtothe
2?period binomialtree:
S2uu=156.25
S1u=125
S2ud=100
S0=100
S2du=100
S1d=80
S2dd=64
a) (2points)Whatistheannualizedvolatilityusedtocomputethetree?
b) (5points)PricetheEuropeancalloptionbycompletingthetreebelow.
C2uu=blablabla
C1u=blablabla
C2ud=blablabla
C0=blablabla
C2du=blablabla
C1d=blablabla
C2dd=blablabla
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c) (2.5 points) What is the risk-neutral probability that the option ends up in-the-
money?
d) (2.5 points) If the call had a knock-in barrier set at $140, what would the risk-
neutralprobabilitybethatthecallgetsknocked-inatmaturity?
e) (2.5 points) What is the risk-neutral probability that the running minimum
mink?{0,1,2}Sk overthelifeoftheoptionis$100?
f) (2.5 points) What is the risk-neutral probability that the arithmetic average
P
1
2
2
k=1
Sk ofthestockpriceis$90?
5


g) (5 points) Suppose the Wall Street Journal quotes a price of C0 = $15 for the
European calltoday—assumetheoptioniscorrectlypricedattheendofthe?rst
period. In the table below, explain the transactions you undertake to arbitrage
current option mispricing. In the third column, show that your strategy has zero
cash-?owbothintheup-anddown-node.
Cash Flows
Time 0
Period 1
u
Transaction
d
ConverttoyenConverttoyen Convertto
ConverttoyenConverttoyen Convertto
ConverttoyenConverttoyen Convertto
ConverttoyenConverttoyen Convertto
Total
Converttoy Converttoy
Converttoy Converttoy
Converttoy Converttoy
Converttoy Converttoy
6


h) (5 points) Price the American call option by completing the tree below. Circle
thenode(s)atwhichyouexercise.
C2uu=1.36731
C1u=1.36731
C1u,NO=1.36731
C1u,EX=1.36731
C2ud=1.36731
C0=1.36731
C0,NO=1.36731
C0,EX=1.36731
C2du=1.36731
C1d=1.36731
C1d,NO=1.36731
C1d,EX=1.36731
C2dd=1.36731
Exercise3. SupposeyouareintheBlack-Scholesworldandyouconsideraputwith
strikeprice$105anda4?monthsmaturity. Thenon-dividend-payingstockhascurrent
value$100andvolatility?=0.5. Thecontinuously-compoundedrisk-freerateis7%.
a) (2.5points)Computetheputpremium.
b) (2.5 points) Compute the premium for the 105?strike 4?months call using the
put-callparity.
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SupposetheWall StreetJournal quotesaputpremiumof$14today—assumethe
putiscorrectlypricedtomorrow. Considernow2possiblescenarios: thestockprice
tomorrowis$120—scenarioA—or$95—scenarioB. Ignore interest expenses.
c) (7.5points)Inthetablebelow,showthatshort-sellingtheputandtakingadelta-
replicating position in shares does not allow you to make an arbitrage pro?t.
Completeline1byreportingthemark-to-marketpro?tsontheshortput. Online
2, indicate your position in the stock (?rst column), the number of shares traded
(secondcolumn), andthedailypro?tsontheseshares(thirdcolumn). Online3,
reportyourtotal daily pro?ts.
Position (number)
Today
Daily Pro?ts
Tomorrow
Transaction
ScenarioA
ScenarioB
1. Sell 1put
?1
Converttoy
Converttoy
Converttoy Converttoy
Converttoy Converttoy
Converttoy Converttoy
2. ConverttoyenConvert
ConverttoyenConvert
3. Total
ConverttoyenConverttoyen Converttoy
Converttoy Converttoy
d) (5 points) To obtain an arbitrage pro?t, you need your strategy to be gamma-
neutral. Todoso,consideracorrectly-pricedat-the-money6?monthscall: com-
putethenumberofcallsyouneedtobuyorsellforyourarbitragestrategytobe
gamma-neutral.
8


e) (7.5 points) In the table below, show that a gamma-neutral delta-replicating
strategygeneratesanarbitragepro?t. Completeline1byreportingthetotaldaily
pro?ts computed under c). On line 2, indicate your position in the call (?rst col-
umn), the number of calls traded (second column), and the daily pro?ts on these
calls (third column). Buying the call distorts your delta-replicating strategy—on
line3,indicatethenumberofadditionalsharesyouneedtotradetoo?setthecall
distortion(secondcolumn)andthepro?tsyoumakeontheseshares(thirdcolumn).
Online4,reportyourtotal daily pro?ts.
Position (number)
Today
Daily Pro?ts
Tomorrow
Transaction
ScenarioA
ScenarioB
1. ??replicatingstrategy
2. ConverttoyenConvert
ConverttoyenConvert
3. ConverttoyenConvert
ConverttoyenConvert
——
Converttoy
Converttoy
Converttoy
Converttoy
Converttoy Converttoy
Converttoy Converttoy
Converttoy Converttoy
Converttoy Converttoy
Converttoy Converttoy
4. Total
ConverttoyenConverttoyen Converttoy
Converttoy Converttoy
Exercise 4. Supposeapublicly-traded?rmXYZ,currentlytradingate35,surprises
market-makers by announcing a special one-time dividend of e5 paid in 1 month from
now—e.g.,in1998,Daimler-Benzannouncedadividendtentimeslargerthantheusual
payout. Assume the continuously-compounded rate on a euro-denominated bond is 3%
andconsideraforwardcontractthatexpiresin4months.
a) (3points)Computetheno-arbitrageforwardpriceof?rmXYZ ineuros.
9


b) (2points)Suppose market-makersdecidetoignoretheone-timedividendpaidby
XYZ. Computetheforwardpricetheywouldaccordinglyquoteineuros.
c) (6points)Thequotedpriceyoucomputedunderb)impliesanarbitrageopportu-
nity. Inthetablebelow, explainthetransactionsyouundertaketoobtainazero
pro?t at time 0 and a positive pro?t in 4 months—state each transaction
in the ?rst column and report the associated cash-?ow in the relevant columns.
Report your total cash-?ow at the bottom of each column. All transactions
are undertaken in euros.
Cash Flows
Transaction
Time0
ssss
Time1month
Time4months
Shortstock(1unit)
ssss
ssss
ssss
ssss
ssss
ssss
ssss
ssss
ssss
ssss
ssss
ssss
ssss
ssss
ssss
ssss
Shortstock(1unit)ssss ssss
Shortstock(1unit)
Shortstock(1unit)
Shortstock(1unit)
ssss
ssss
ssss
Shortstock(1unit)ssss ssss
Shortstock(1unit)
Shortstock(1unit)
ssss
ssss
Total
S0e ?F0,T >0 S0e ?F0,T >0 S0e ?F0,T >0
rT rT rT
d) (5 points) XYZ is a German ?rm and you are a US-based investor—you wish
to convert your 4?months arbitrage pro?ts under c) into dollars. Suppose the
current$/e?exchangerateis1.33andthecontinuously-compoundedrateondollar-
denominatedbondsis5%. Whatisthedollar valueofyourarbitragepro?tsin4
months?
10


Suppose nowthatXYZ producesacommoditythatcurrentlytradesat30$. The
Wall StreetJournal quotesa3?monthsforwardpriceof33$forthiscommodity.
e) (2points)Computetheimpliedleaserateforthiscommodity.
f) (5 points) Suppose that, in this commodity market, storage is optimal and that
the highest no-arbitrage forward price is 38$. Compute the implied continuously-
compoundedrateofstoragecostsandtheconvenienceyield.
g) (2 points) If the lease rate and the risk-free rate are constant over time, is this
commoditymarketincontangoorinbackwardation ? Explain.
11


Exercise 5.
Consider an at-the-money call and an at-the-money put both with a
T?yearmaturity. Thenon-dividend-payingstockhascurrentvalueS0 andvolatility?.
Thecontinuously-compoundedrisk-freerateisr.
a) (3 points) Assuming that S0 = 70, ? = 0.6, r = 5%, and T = 2, compute the
Black-Scholespremiumforboththeputandthecall.
Youcurrentlyhavenoviewonthefuturedirectionofthemarket—youwishtoenter
aT?yearcontracttodaythatallowsyoutodecidewhetherthecontractwillbea
putoracallin0<t<T yearfromnow. Thiscontractiscalledachooseroption.
b) (6points)Usingput-callparity,showthatthechooseroptionisworth
Ct+max(Ke?r(T?t)?St,0)
intyear,whereCt denotestheBlack-Scholescallpriceattimet.
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c) (6points)Findanexpressionforthecurrentvalueofthechooseroption.
d) (3points)Assumingt=1andusingtheparametervaluesundera), computethe
currentpriceofthechooseroption.
e) (6points)Whatistherisk-neutralprobabilitythatyouendupchoosingaputin1
year?
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