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Unformatted text preview: ' Equity :Derivatives . .f _ Pmd W5 were» : e Class 4: _
Options luatibn ' Mark Zu'rac'k
_ September 19,2006  V' MOWERWa‘mbizﬁ}xw9nmcum.:5»,n.a':'.—:;. , r u .:  » .  I To provide you the mathematical underpinnings of options valuation in
an intuitive way To explain basic properties of options like intrinsic value, time premium,
time and insurance value, and an option’s delta and gamma Put/call‘parity and forward prices are discussed to show how arbitrage
'  ' forbes affect put and call prices as well as many strategies I then show how a stock’svolatility and forward price affects its
distribution of future priceswhich is‘the basis of its valuation Travis Jordan from Deutsche Bank will discuss how Hedge Funds use
Equity Derivatives in financing transactions ' 0608081Cor—4 \.  . mmrmm: a}, mm‘mﬁw“ an :. Options Pricing ' Comments9f.fan'OPtiOn:Price ' 
" " I=The:'differen_cet5between the market valueof'thei’underlying; seCurity'and _ rthe exercisegpric‘e or strike'_'pri_'ce fora cal_;optio_n'. __ I m'one o.utéof:the'mone .,_h_av_e no'intrinsicivaluei if .  * "  . The amoentf'.exf_c_e'_eding the intrinsic'value‘tha’t a buyer is Willingto pay
'_ "gfor‘thej leverage evailable in "the option linstead of owning the stock
; _'0'utri_ght*and theinsurance chare which limits loSSestothe'ogtion’s
' “premium only; __Relat__edtOtheyplatility ofthe.,;underlying' stock . can} Value ' 'IntrinsiC'Value '= Time premium  Expiration Value  Underlying
Price ' Strike Price 0603081COf4 a an Option Price .‘ usuwwwmmxwxmWmmswr. .«.:=~J=‘«.vr:d.tu . WitthMGN trading at 48.36, the Nov 45 strike callshave an intrinsic value of
3.36 andtime premium of 0.54, using the offered side of the market. 
451' sawﬁe =‘I'X. BEQ‘mJn @) “1 no 5054 Source: Bloomberg __ Data as of 11/04/02._ ASK #9574552  I ' I ' aﬁozzamow
s70 Components WW may“, Stock = $50 Strike = $45 3—month Interest Rate = 0.5% or 2% per year 3month OptionStruCk at $45 '= $7 ' No Dividend "Expected * ' ' ' 7' '
oiavidwi.(w1guctg=, ago/veer , I ' is theintrinsic value? ‘30 4555' '
2. What is the time value? Q 75 a Q n w I); ;. ﬁguwhat benefit doesthe investOr’accr‘ue by not having to
(W i '_'_""_pay:for the stock .in full? 507 :93. . (WW005) = .085 5’ 0’ 2 2’  7' I 1 4. What is the cost ofinsurance in the option? av 0.22 = 1,78”
3.31m,“ ‘qu value ' ' "blme Vwmﬁlevemaxi insurance. _ _
i ..  _  _ Would .your__answer toquestion 3 and 4 change if the Cw aim) ’ stock h'asa10/g yield,”'with dividend; aid quarterly? WW
 ' 7   [email protected]_ o.o.°r . y" . 5&5: \ 0608081Cor—4 . (q) l 4 What Happens t0 an OptiOhWhen the an « rmmmwmwmﬁ mum» Liw'utt nasalq: . ,_ . _ _, __  Nothingif,out—of—themoney; if inthemoney, moves .At Expiration. ~ dollar for dollar with stock  Priorto '  . . I
Expiration; Moves less than the stock moves, that amount IS referred to as the delta. Factors influencing delta: ' 'II'Whether the option has intrinsic value
 1' "Change in time premium I mama (hidollar's 11¢ Atos (ml met“? 4/90, ﬁlm oreM T050
jive: good 'lndicafm 0F Wm Jfocfc Ir warm iv: Jhcrf‘
am? all {that} oJl/«w {mch 3194c curve anljjlwf exFirali'M
A dagFoo how.mmdq Jack meals: 1h aro’rr at»
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A a mutant? mgr arrm will Wm primMons, v._...., . _ _
 w..." m, . 0608081C0r—4 ,3 \. DELTA verage .o‘.1.;:~n..~d:<ixt6;um:mxW . n . Investors can use the delta on an optionto determine which strikegives By buyingthe Nova22:5 strike callsinstead of the Nov 20 Calls, the bullish
SGP investor can earn an incremental 81.4 cents for each dollar that SGP
stock moves up:  1.851.35 = 5.29 5;29_X 29.9 .= 158.2 cents (compared to76.8 cents)
A . ' r SUMMARY PAGE
7 '[IHERII‘JEPLDLIEH CHEF
‘32? Bid .327: 'Eiiai  ' : CALL l j. n t I El _ r._ ‘ '  .3: .3; iE b.1nt 113 a a _ ' .dﬁ
Source: Bloomberg I I delta = *3 “Wm 4‘ 5‘!) Call fv‘bﬂéé’ Data as Ofﬂmmz. Olafﬁn Ming £7,353 will “NJ 78135 ﬂu (“all . LKC ”‘
A _ d Wt— gs' ﬁne: a!
KY 2,23 wcgmj ax $.33.) 3‘!) F4 buy abo ’3: mad! 66 0605081£oi~3 ’ I
which o‘lpon Prim/(Ads most [eve/032; $9.7..S‘ S 5
7 me 3" 7/409.) much delfm 09°17“: T I ﬁt 6 {oak f) ‘00 09w? r4 5 “Er 711'. (x Changeover ' Wyezswmmm delh chan . afdodc Frag cola/13m owthe D in a
r
,me
mm
a
d the stock f Gamma equals the
price 0 change in $1 change _ “Few”WL UW' .WCAMJ . i" .10va fd’ rye/yﬂ con Ll: short mole], '6)ij an”,
3hort’ 0608081COr4 The Linkage Between Calls and Puts — {— ‘tills how mach; 1%? pwf‘xflqomlot ice. “7942465 +9 Cw“ Cam m 1+0 limo! paras' Poincﬁ‘m “a (“H “933’” ‘ﬁ'[‘ I . Jlf'l'vflowf‘iDn
Defines the relationshipbetween the price of the stock, the
price of the call, and the price of the put. Based on ability for traders to engage “riSk¥free” arbitrage
._ ._ when call and put prices with the same Strike prices trade away
'3 7 fair value. . _ ' I‘I‘RiSk—free arbitrage” strategy involves either(buying stock, selling acall and buying a put)with the same strike price
(referred to asafOrward conversion), or(selling stock, buying a
call and selling aput)with the same strike price (referred to as a reverse conversion) ' IPUtecallfparity also enablesinvestors to'create synthetic Call ,  pUt, and stock positions with combinations of stock, cash, and
ropﬁons Véimi‘emenb 01c End'ch gariﬁ' «to (Merit _ I®0Phohi have J‘gwke MelinaI'D I I
® 54"“ J54“ FVRE Mi, {foe/e [JJ/cew) £5»va J'fDLk_  I How Riskfree Arbitrage Defines the Cal's um u mawxm wsmQmswwawm. Arbitrage _ ..;_.When the call on a; stock is tradingat too higha
' I premium relative to the put with the same strike
"price, ‘buy the stock, sell the call, and buy the put.
' This strategy isCommonly'kn'own as aforward
 “conversion, ) 003 what, we acm,_ tag a pear .
' ' " ii. MM; new” it 79"”. ' " «Stock I I = $120
_ " 3gmon'th' Call _ = $5.50, struck at$120 firm «ﬁfgglggj:jg“gjed +6 I 3month Put": _=I$4.00,lstruck at$120 '
 creme!“ met    . ExpectedDividend. ,= $0 . . '. r giw ‘4'?”th . . .
hf wok 4’ , PKG/“e F” f?“ ' / Initial Cost of Forward Conversron :1 : $120.00  $5.50 + $4.00 = $11850 0608031Cor—4 ,_ nversion PayOff at Expiration ~ I : mm WMQZ'MWdidwﬁmh mmmmnmzmvz: : .1» If stock price is below $120, the put is exercised for $120 1 31f stock price is‘aboveg $120jr the investor must deliver
stock against call assignment and receive $120
I Return equals: $120/$118.50 ~— 1 = 1.27% for 3 months
  _ or 5.08% peryear ‘ _ ,  when m aim/em. ems, I If riskfree rate is. below “508%, this strategy dominates Strategy I w‘ll aria? it ' 10ther risk—free inVestments ‘ P90? 1 I W ﬁght,“ {5— above (WW“t r10: Payoff in $ . Pigg)i"$ ._ _ _  30 "'""' " Stock + Put  Call
"' " ' Short Call
— Long Put 80 90 100 110 120 130 140 150 160
Price “pinned” — {oer'5 Pinned is M4814 Jr‘ock (mile. 9085 1" 1%? “4°”? if} WW‘L’WO
0608031—Cor—4 . ' v r ‘ .
Pm ask as low } 0'19 nyk 711d 2x191? ('4 ﬁma’d 5””le “yam In 444:3 (laxS "mg5k PW” M4SQC’HD" Reverse COnverSlon ' Wm:a..:z M‘s mwmxmt and securities in order to outperform holding stock "
Call is C9164? ﬁlm'sz 1% HimFM— . Strategy I I Sell stock, buy call, sell put end reinvest“ cash in. risk free .__Stock _ I ' = $120 '  ' 3~month Call = $5.50, struck at $120 3month Put ' = $4.00, struck at $120
Expected Dividend = $0 ' Initiél Credit = $120.00 _'$'5_'.50 +.$4.00 = $118.50 Risk free reinvestment rate = 6%]yr or 1.5%/qtr
(um? ) . ‘ Payoff in $ mmm4 no M‘H’W wind haffﬂ’km’ 75 stock/.3149 Lug: ﬂue 11 ,r'"\ . . ,ﬂ\ ‘ k ' , = P—,‘ PMi'ICQll [90411:] 51°“ not was? r’ ’ ﬂock Md @1028 low + ’ 2 E ‘ we gmmﬁ/reueuse oastuz’rsrbn has n‘l‘le‘Fnee arbiméq (Awwe‘f Dividends ‘ ‘  vatﬁm'mmwmimm mimbuﬁlM mﬁ‘axwg; <: Adds return to forward conversion, takes away return from a
reverse conversion '  ' Stock = $120 _ _
3month Call “ = $5.50," struCkiat $120
34month Put " = $4.00;r struck at? $120
Riskfree rate __ = 6%/yr or 1,5%/qtr Stoek ays a50¢ per share dividend ever‘the next quarter J33? veg,» Payoff in $ / ' Return: 120.50 Kama133 We Jhck m +14}; (we, (4.73“ 118.50
mhﬁeéﬁ (142' = 1.690/olqtror6.76%lyr make 6.317. Mom Hum rel/era
drﬁidenoi .
. / (in/16194;: fr Value at puma/ti Conu'. 12
0608081Cor~4 {412/1 +1790 WT 96ml J] NU?!“ Comur
Puf’ an“ Pa/ﬁj does not 8x131” ‘1mwmmskMImgmaeumem‘xaxw :rM» ced atFair' Assumptions StOCk  7 = $120
W .v:  ATM Call (@ﬂemonej) = $5.50
 _ _ATM Put ' .  = $4.00 _
Expected Dividend = $0.50
Risk free Irate ' = 6.75%lyear or 1.69%/qtr F'orWard Conversion Initial Investment = . $120 —$5_.50 + $4.00 ="$118.50
Reverse Conversion CaSh‘ Receipt = '
" $120 — $5_.50 +’$4.00 = $118.50 Payoff in Return = 1.69010 "Indtntiezent bah/em Kc Mai 10 13
0608081C0r—4 '  I
5"“ ,2 1+1; (1’. Met? 1, .i foosvHoq YT now MJMJ 61f 'pM'f/Cau pm‘hj Positions Based on PutCall Parity
. It I Mm“ a BM
  '_ Stook_—“Ca + Put =. [(Strike'+ Div) ,:"(1+ risk free rate (r))] I I." "Synthetic Long Stock I
+ Stock: Call —' Put + [(Strike +Div) / (1 + r)] I Synthetic Short Stock _. . 7 .. __ I_ I Stock = Put— Call — [(Strike + Div) /’ (1 + r)] .' Long Call (Portfolio InsuranCE)‘
" ' Stock + Put = [(Strike + Div) /’ (1 + r)] + Call
 Synthetit: Shl'ortl.f.col_lj _ ' I I I '  _ ~— Put — Stock = éCalla[(Strike_+ Div) / (1 + r)] ' _' SVnthetic Long Put ‘ " ' __ ~ ' 5' . Call — Stock =‘ Put —'[(Strike + Div) / (1 + r)] :7. II Synthetic Short Put (BuyaandeWr,ite) Stock— can = [(Strike + Div) /' (1 + r)] PUt i when” P”?! “NM “1“” W i”. “VFW” 14
owmlﬂw‘; a: ,Fu/lchbq of: pp (7!er .1”th Price OPHMS cioﬁr mm! 40 be. simcké) 5767‘!“ Iowa; Fajita“ Iva/Hy \ in mam wsammmnntmmyEQJ Dam on Strike price where an investor would be willing to pay the same amount for a call as a put. By inference, the stock
7 ‘ price that options investors believe is most likely to occur
. __ sinceit is the price level where they will pay the same _
" amount for upside participation and downside participation Calculatio StOck 4 can 4 Put = Strike +'Div ‘ i" _W Tommi stud: ,StoCk I. ' ' .. = Strike‘+ Div
' " '  1+r ' [(1+r) x Stock] — Div. " = Strike  _ [(1+00169) x 120] — .50: Strike . ., _ _ = 45% at {4453 J‘lﬁke Fritz) ‘oph'm ram/W may FIE/21.33 is ' 'f’iqoecl'td ﬁlm price, 51C Shack at: ﬁt?! strike, SDZ chance 71°50 abwc “mad _ 121.53 + .50 — 1 =' 159%
Conversron  120 M15. 6’”, 50% bf); 30 h—a
0608031‘Cor4 {harm PWCFSM. . kmomh J'f’bck Fina. Could £9, {33am Jerk/j 9AA know Fr'xce
by degtiiHwi know) ‘H’lt oFH'eq opi‘ién ' _ OptionsVamation i .. “mmwmvmwmﬁzmtmmxwing::22; :3. ‘ Option Pricing Discounts FutureStocikrPrices
The shaded area H 4   '  ' ~  ' ‘ I ' ir‘i' wi’ii’a , 41"
shows the range of '    Pl‘lce [email protected] cenie/jt flow “1 OF M
stock prices for 4.3 cumviii 7328’ r 0 which the call option 3k 7‘ _ h k C dam am
has valueat will aSSUer 57 c Fm. 3% .
.190 Volatility provides an estimate
of the range of prices at *expiry expiration and the likelihoodof each price. _4 ‘ ’°3'”‘°"‘”“@ whim“ H 1 Must asuuwe Jomeﬂafnj ‘60 3H"
Questions Oph‘bn‘ 33 garage/Ht value 07C 4” [90:51 Isle, J‘th [antes r dismhuh M
1) “Maﬁa: the ' : f h, l: .~= l H 4" C
probability of the 13 0 V: 5:3, .Id 9‘: (paged, j
stock reaching $135 ' Forward Price at 105 ’ " i
or higher in one WW: Ho”?
_ _ _ . — — —  F — " ' " ' ” ' d  ‘ 'S 2) What is the Initial Stock ' & ‘Fr‘ice
probability ofthe me at 10° stock reaching $75 or ceriiv 0? (it 5 f = (o s A Call Strike Price at so Iowerin oneyear?  i—O' : 30! up H7313: ' dowel6'57}? Fajo'F; hon/Walla
1—Year Call 0 tion Gating“ mic dishibxtl’al : _ VI Strike Price: ' $80 g'mck Price ,3 gem“ 75' “4 4o_ =
' Volatility: _ 5;; "30% if}; ’42:,9“0’?” ‘iil‘lml" I Dollar Volatility: $30 (30%*$100 stock price) 10 '
I IntereSt Rate: 6% Probability of Price Range
. Dividend Yield: 1% 16 0603081Cor4 want wide. dikhﬁbmﬁM / beam bfjjpn Rm P05)HV€ owf'CoiMESJ inchfFExeqt
gr thdHW 0608081—Cor4  an wmamﬁh’vs‘i')hb% 2K . . I The theoretical value of an option dependson the variables
" below, the table shows effects of increasing each variable on
option price  .  .. _Ca_“S' M
Current Stock Price Higher_'"" Lower
Strike Price Lowert" Higher
Time to Maturity _ Higher Higher* * Not true for far in~the~money European Put Options 17 . Whereii'isit'he mean Of the returns, xi
: returns ts. mmmw meuar «L?A.m.::+a:.u:.eon. : ' l .. . . .  , ‘ :?X._.HiStloric_a_I. Z'I‘E'Xi—R]2._
' ' Volatilityté, ' i.=._1 are the returns, and N is the numberbf Ifan asset has a 20% annualized volatility, then over many years, we would
expect 2/3 (1 standard deviation) of the" observations to be ':I:20% around the _ mean return _ _ afoul/1de '; wlaHlirjv/MM @564 be  £3 da'ﬁll’u‘h‘bn at,“ kWDoO “Q. I
Fmbdbihhés m orﬁbns inc/W
91557345 060803 1’Cor4 This Shows thata $100 stock with an annual” Volatility tomove <$1.25 per day around $100, 2/3 of the time
Traders generally look at'two measureszof volatility. when pricingoptions: If Implied volatility — The volatility estimate which matches the observable
market price of the option with its theoretical value of ' 20% womd  be expeCted Histori_cal.volatility"— The annualized standard deviation ofdaily or weekly
price changes over some period oftime; Traders tend to use 1— and 3month
historical vdlatility to price short—dated optibns and 1 to 3—year volatility to
price long dated options ' 18 (D
m
.9.
o
X
H
> 13 10'
1104 VIX is most no 7104 PM a} Mensa 949 of: muted Volasmj ' VIX — The Implied VOIatiIi mwtausmm‘owWJMH . This implies the folloiNing: Index 8 I 14000 Oth‘VIS' are 394, [35“! 71004 IKE/w, "I The VIX measures imglied Volatil ty i
for 30.da S&P 500 index 0 tions 'At thebeginning of July, the VIX StOOd at'13.0%with the S&P 500
at 1275 mm .in. pct ‘I/
i 1day VIX 10.5 ' 0.82 7’ W
1week VIX 23.0 1.80
lmonth VIX 48.5 . 3.80
Calculation: VIX diVidend by SQRT of the
number of periods in a year, multiplied by Index
level to move into points, assumes forward
price about the same as index level
Futures and options on VIX trade 19 0608081C0r4 Options Valuation Without a Model—An Example  '
Assumptions:
Stock = 100
Strike Price = 101 in and, ole/8’0? r’lmt‘r‘m“ 5" WW"
3—Month LIBOR = 5.5%/yr 3 9“”6‘“? 3’5"” "“M 4" gm”:
Dividend Yield = 1.5%]yr
Volatility = 20%/yr
Time until Expiration = 3 months
Questions:
1. What is the forward price of the stock at the expiration date of the option?
2. Using a range between 73129 and working in increments of 2, that is 73—75, 75—
77, etc., determine the probability of the stock trading between every two points
in the range. Additionally, find the the probability of the stock trading below 73
and above 129.
3. Determine what the option is worth for each 2 dollar range using the midpoint of 4. Value the option. the range as the ending price of the stock. 20 ...
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