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Unformatted text preview: For the: problems of Chapter43., consider the following two 13th 3.0. extracts Oilg, with fixed Costs of $20! barrel and variable costs 0f $3fbarr l. I! The oneyear forward price of oil is $42fha1‘1'el, The leveeyear can
4%. Che year oil Option prices are shown in the table helew: Stuke Call Price Put Page
35 ' $8.085 $1.359
40 $5.106 $3 185
42 $4.174 $4 174
45 $303: $5 915
50 $1 71 1 $9 397 2.  Compute the estimated profit in one year if 30.. buys a put Option wit. a strike of $40(_$_50). Draw 1
proﬁt diagrams for both options, and include the unhedged preﬁt. Des ribe the difference in the pt‘Ofit diagrams of the $40~strike hedged and the $50Jst1‘ike hedged peitieh. 3. If Plastics Corp. dees scathing to manage all price risk, what is its pro. ehe year fmm new? per
barrel 01“ oil? If it wants to hedge the price of oil, should Plastics Corp buy or sell oil forwards? e their oil price exposure with future so'that they are completely ' 7
e, how manyeil futures shehld they'uy per unit of resin pI‘educed?
w"? Construct graphs illu tratihg both unhedged and _ '. '  4. If Plastics Corp. chooses te‘hedg
immune to changes in the oil prie
What is its estimated profit one yew; from he
hedged pesitions. ' 7. Seppos
Find the strike price for an oil _ put that wouid generate a zeroms cdntinuouely compouﬂded interest mike 0f 4%,
one year. The oil forward prim is $42. Use the fellowng Table far Questiems 4.9 and 4.10: Extratﬁen Schedule (Millions of Barreie) Scenarm A Scenarm B (b) What is the eerreletion coefficient between eil price and the ex
What about the correlanen coefﬁciem between 0i '
difference. if any. W. Compute me variance—miniming hedge mtie for 8.0.? s revenues fer Seenarie C. What is the
reducuon m the standard deviatien BEE 3.033 revenge? " A 1'1 Swer 50m 911' 7 7
_ . Chapter 4' Introduction to Risk Maeagement 1. The following tame summarizes the unhedged "and hedged prefit calenhtiene: Half'FHII‘“ ﬂ Unhedged Profit on 1/2 She 1 Net Income on {Oil Price per Barrel _
Forward Centrfet ﬂedged Pmﬁt ' in One Year Tetal Cast Profit
$25 $20 ' ——$3 ' . $8.50 $5.50
$30 ‘ * $2.8 +$2 $6 ' ' $8
$35 _ $28 ' +557  $3.50  P $10.50
$40 $28 +$12 $1 $13.00
" $45 $28 ' +$17 451.50 $1550
550 $28 +$22 —$4 $13.00 r T_ . m_._.— _.__... #HLMH_J_HMF_—u—g—Iﬂu—_A_Fu_nl—I —_.._._ '—'—III——I——I'Irl uru—a—a,—In_n_—__ —u—.—.—.—
3' '1'
'3 i1 EDI CE
ri.
.—..—_'F‘ ‘‘‘ _I_II. m .—.ln nay—Hm .
ﬂu—u—u—IaI_ #WWH I.— Wlmmmqum I F f I _M_'_'_ ________ ____m I___';l1—“_m._u _ _ .\.\.!r 21—“! ——rI wwwww aI.u—Ia—u..n—u——_\.—u.— _u_un— aaaaa ﬂmHIIu—l—lLﬂri—I—IJ—wnn—J ﬂ'umJﬂmuﬁpiMII—ni’I—ﬂlmrl 78 2. McDonald Please note that the continuously compounded
effective annual interest rate is esp(0.04) — .._ $3.315. The following tab for the other put is similar hedging strategies. ' Derivatives Markets, Second Edition rate of interest is given: it is 4%. Therefore. the " 0.0408. Ia this exercise, to calcul te overall profits.
Onstrike put. it is: 3.185 >4 1.0408. 2
he $40.00—strike n at. The calculation 1e shows the profit calculations for t
rams for both . The ﬁgure on the next page compares the profit dia Profit on Long Oil Price Unhedged $40.00—Strike Put Net Income on in One Year Total Cost Proﬁt Option Pot Pie mm ﬂedged Profit
$25 $28 —$3 $15 $33 5 $8.685
$30 $28 +$2 $10 $33  5 $8.685
$35 $28 +$7 $5 $33 5 $8.685
$40 $28 +$12 0 $33 5 $8,685
$45 $28 +9517 0 $3.3 5 $13685
$50 $28 +3322 0 $3.3 5 $18.685 Proﬁt diagram $4t3etrike put 60 5E! 30 40 higher oil prices than the difference for the $40 + unhedged proﬁt
%2 hedged preﬁt 40
Oil price ﬂ p59. 35 40
Cat price 50' D 20 lease, the unhedged positin yields less proﬁt than
.  he protection on the downide. Also note that the
he unhedged position and the 50strike hedged position is larger for
strike. We receive in return a higher profit in the low oil price region with the $50 oil price.  . m..."—   —. .
._"“_.. r a _ l i . _ . 1" —. ﬁwHI—hrmjr—pp 'I.Jr—\. 9999 IIII—IIIIhl'IrII—Iﬁ—u—ni—I'I—h“ —..J Iona—mi r 'r'il‘rlﬁhmm ”'_"'— ‘ ' ' 1 1 ' LJJ Answer Sectioa ' _ 7‘) he variablr oil costs per onii of resin  produce two units of resin. T
c sis. minus oilvariable coSts. Plastics uses one barrel of oii to
minus fixed are thus 0.5 x pun. We can write out the profits as revenue
and minus nonoil variable cost: = 04:410..“ $15 #957 ~— $3.30 W 0.5m“i
:2 $4.70 ~— 0.1m“ ' 1" unit of l‘esio of Plastics l orp. depends on the oil price. PE. iasties The following graph shows how the profit pe Proﬁt diagram of unheaged pﬂﬁiiiﬁﬁ ._n... “lap—l—III'I '——I——I' —I—_—— —ll''I'—I—l—'I'II—I\. 5 r—w— H—MﬂT—ﬂc—Wtﬁ—"w'“*‘THW..__. .. f _ ' H uohedged proﬁt ‘
E T t ._ _  .. __ __
4 i_. ‘7— . _:
l l w + i
3i” + i i + l "i . 1
i '“' 2 i  i: i' I ':.
i. ii— i 1 l' ‘ i E ‘l g
EL l l
{3 §~ “
I l i 
I —+ l
l f l
i .3
‘1 a 1
l i ~1
1 + _
‘El' i + i
l * l
i + l
L 'i
3 i +
i
i
4 ..__._.. .._____ .1_.._.._._... a .... ._i . ____. . _ E .1 _ __ E.  3
D 18 2D“ SCI 45} Si} ED ﬂ! ' *ﬂ The higher the oil price. the lowet the profits of Plastics Corp. Therei re. Plastics Corp. needs to buy futut'es to hedge their esposure. that the profit of Plastics Corp. can be esci‘ibed as: i 3 $4.70 — (ll >< p“ii ' iiiiisiiss 0 of a futures contract to completey remove the oil pr 4. We have seen in question 4.3 Therefore. we will need to buy 1/ 1 ice risk
from Plastics Corpfs profits. ' Profit on 1/10
'Uhhedged Profit Long Forward ﬂedged ’mﬁt s25 . $2.20 is} .70  s0. 0
$30 $1.70  ds120 — $00
$35 $1.20 saw s0. .
$40 $0.70 —s0.20  s0. 0
$45 $0.20 ' $0.30 s0. 0 s50 _ _ I $0.30 $0.80 __ s0. 80 McDDnald ' Derivaz‘iws Markets, Sacend Edition M We: abtain the following profit diagrams: Pmﬁt diagram {if unhedged Waitinn and hedged Wait1cm 5 1 . :
+ unhedged pmﬁi '
+ J“ I 9% hedged pmﬁt
4 ++
T+
%.
3 +4
.1...
+
+
2 ++
+
+
.1..— 1 ++
E _+
E'iﬁt+++:&+ﬁvi++vr+ﬂ€$7h~llﬁv$ﬁi$$iiﬁ§c$1¢7§+++++$i~¥+$+ﬁ¥
u. .—}_ ed ' Profit on 1/10 0113 Oil Price Unhedg in One year Proﬁt $35.00Strike Call
$25 $.20 0 ' 30.8415
$30 $1.70 " 0 $08415
$35 $1.20 0 $08415
$40 $0.70 $0.50 $03415
$45 $0.20 $1 #00 $03415
$50 —$O.30 $ 1. .50 _ $03415 WWW Anzawur Scamn ' '3! We obtain the following pmﬁt diagrmm: OH price Plastics Corp. will sell a 20113:; which maans that they buy 2
{ion pramium position, and find it; future: value. We haw gative value I am; we haw a cash inflow. to compute for. each case the net Up ($1.71 1 ~— $3.185) >< 1.0408 :2 w$i534¢ Note: that tha FIE: Oil Price in Unhedged Profit on .1116 Pnjfit on 1/10
Short 40 PM Long One Year Profit
"$25 $2.20 451.50
$30 $1.70 H$I .00
$35 $1.20 #33050
$40 $0.70 a $0
$45 $0.20 $0
$550 45030 $0 “We”. r—— J_ ...__.___.,._._ _i M 46
{iii {an ca 0 0
0
0
.0
0. 50 Call he can he and gel! the, put leg. WE haw: _ Nt ﬂedged
Pren .ium Profit
—$04.!53 $0.853
“$0 153 $0.853
#330 153 $0,853
—$0 153 330.853
~—$0 .153 $01,353
#320 153 —$0. 147 8'2 McDeaald ' Derivatives Markets. Second Editicn Proﬁt diagram: Preﬁtdiagram $40 eat. $5042.31! cotter
4 W—Wwﬁ—r— A gecd starting point is t
chapter. We can see that the $45wsttike call costs $3.032... Since the put prices are decreasing in the strike price. we k
strike marginain lower than $40. Ttiai and Bt‘i‘Ot‘ results in
the premium on the bought call and sold put cancel each ether cut. (a) The hedged proﬁt can be rewritten. as:
P: N P +£H~N6)PG #HXF H“ H" The last term is known and has :10 variance.
rule for variance. with b = (H — N C;) VtIFIZcIX—t bY) = (12 Var (X) + b1 Var (Y) + Zechv (X. Y) (b) If we differentiate equation. 4.4 with respect to H and set it equal to zero. 201 “was: +2N..po‘.ﬁa = 0 N pa 6 N pct
T “___ 1,! 1.‘ G __ 'gfr
42> H—NGm— “ 2" #N6— 1 “
06 0G (C) We need to plug the tiptimal hedge fade, Ha:
Simplifying yields; new that we need t
a strike price of $3,662. At that strike. r 6: Obtain: If, n. ___ Equation 4.4 is a direct appii aticn (3f the feiicwittg ...
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