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**Unformatted text preview: **' Equity :Derivatives . .f _ Pmd W5 wereÂ» : e Class 4: _
Options luatibn ' Mark Zu'rac'k
_ September 19,2006 - V' MOWER-Waâ€˜mbizï¬}x-w9nmcum.:5Â»,n.a':'.â€”:;.-- , r u .: - Â» . - 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 optionâ€™s delta and gamma Put/callâ€˜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 stockâ€™svolatility and forward price affects its
distribution of future prices-which isâ€˜the basis of its valuation Travis Jordan from Deutsche Bank will discuss how Hedge Funds use
Equity Derivatives in financing transactions ' 0608081-Corâ€”4 -\-. - .- mmrmm: a}, mmâ€˜mï¬wâ€œ an :. Options Pricing ' Comments-9f.fan'OPtiO-n:Price -' --
" " I=The:'differen_cet5between the market valueof'theiâ€™underlying; seCurity'and _ rthe exercisegpricâ€˜e or- strike'_'pri_'ce for-a -ca|l-_;optio_n'. __ I m'one o.utÃ©of:the'-mone .,_h_av_e no'intrinsicivaluei if . - * " - . The amoentf'.exf_c_e'_eding the intrinsic'valueâ€˜thaâ€™t a buyer is Willingto pay
'_ "gforâ€˜thej leverage evailable in "the option linstead of owning the stock
; _-'-0'utri_ght*and the-insurance chare which limits loSSes-tothe'ogtionâ€™s
' â€œpremium only; __Relat__edtO-theyplatility of-the.,;underlying' stock . can} Value ' 'IntrinsiC'Value '= Time premium - Expiration Value - Underlying
Price ' Strike Price 0603081-COf-4 a an Option Price -.-â€˜- usuwwwmmxwxm-Wmmswr. .Â«.-:=-~J=â€˜Â«.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' saw-ï¬e =â€˜I'X. BEQâ€˜mJn @) â€œ1 no 5054 Source: Bloomberg __ Data as of 11/04/02._ ASK #9574552 - I ' I ' aï¬ozzamow
s70 Components WW mayâ€œ, Stock = $50 Strike = $45 3â€”month Interest Rate = 0.5% or 2% per year 3-month Option-StruCk at $45 '= $7 ' No Dividend "Expected * ' ' ' 7' '
oiavidwi.(w1guctg=, ago/veer , I -' is theintrinsic value? â€˜30 4555' '
2. What is the time value? Q 7-5 a Q n w I); ;. ï¬guwhat benefit does-the investOrâ€™accrâ€˜ue by not having to
(W i '_'_""-_pay:for the stock .in- full? 50-7 :93. . (WW-005) = .085 5â€™ 0â€™ 2 2â€™ - 7' I 1 4. What is the cost ofinsurance in the option? av 0.22 = 1,78â€
3.31m,â€œ â€˜qu value ' ' "blme Vwmï¬levemaxi insurance. _ _
i .. -- _ - _ Would .your__answer toquestion 3 and 4 change if the Cw aim)- â€™- stock h'asa10/g yield,â€'with dividend; aid quarterly? WW
- ' 7 - - [email protected]_ o.o.Â°r- . y" . 5&5: \ 0608081-Corâ€”4 . (q) l 4 What Happens t0 an OptiOhWhen the an Â«- rmmmwmwmï¬ mumÂ» Liw'utt nasal-q: . ,_ . _ _, __ - Nothing-if,outâ€”ofâ€”the-money; if in-the-money, 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 (hi-dollar's 11Â¢ Atos (ml metâ€œ? 4/90, ï¬lm ore-M T050
jive: good 'lndicaf-m 0F Wm Jfocfc Ir warm iv: Jhcrfâ€˜
am? all {that} oJl/Â«w {mch 3194c curve anljjlwf exFirali'M
A dag-Foo how.mmdq Jack meals: 1h aroâ€™rr atÂ»
hÃ©dgz lie-HOV) - M 7'
A a mutant? mgr arr-m will Wm prim-Mons, ---v--._...., . _ _
- w..." m, . 0608081-C0râ€”4 ,3 -\. DELTA verage .oâ€˜.1.;:~n..~d:<ixt6;um:mx-W .- n . Investors can use the delta on an optionto determine which strikegives By buyingthe Nova-22:5 strike calls-instead 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 to-76.8 cents)
A . ' r SUMMARY PAGE
7 '[IHERIIâ€˜JE-PLDLIEH CHEF
â€˜32? Bid .327: 'Eii-ai - ' : CALL l j. n t I El _ r._ â€˜ ' - .3: .3; i-E- b.1nt 113 a a _ ' .dï¬
Source: Bloomberg I I delta = *3 â€œWm 4â€˜ 5â€˜!) Call fvâ€˜bï¬‚Ã©Ã©â€™ Data as Ofï¬‚mmz. Olaf-ï¬n Ming Â£7,353 will â€œNJ 78135 ï¬‚u (â€œall . LKC â€â€˜
A _ d Wtâ€” gs' ï¬ne: a!
KY 2,23 wcgmj ax $.33.) 3â€˜!) F4 buy abo â€™3: mad! 66 0605081Â£oi~3 â€™ I
which oâ€˜lpon Prim/(Ads most [eve/032; $9.7..Sâ€˜ S 5
7 me 3" 7/409.) much delfm 09Â°17â€œ: T I ï¬t 6 {oak f) â€˜00 09w? r4 5 â€œEr 711'. (x Changeover -' Wye-zswmmm 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 _ â€œFewâ€WL UW' .WCAMJ . i" .10va f-dâ€™ rye/yï¬‚ con Ll: short mole], '6)ij anâ€,
3hortâ€™ 0608081-COr-4 The Linkage Between Calls and Puts â€” {â€” â€˜t-ills how mach; 1%?- pwfâ€˜xflqomlot ice. â€œ7942465 +9 Cwâ€œ Cam m 1+0 limo! par-as' Poincï¬â€˜m â€œa (â€œH â€œ933â€™â€ â€˜ï¬'[â€˜ I . Jlf'l'vflowfâ€˜i-Dn
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 â€œriSkÂ¥freeâ€ arbitrage
._ ._ when call and put prices with the same Strike prices trade away
'3 7 fair value. . _ ' Iâ€˜Iâ€˜RiSkâ€”free arbitrageâ€ 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
ropï¬ons VÃ©imiâ€˜emenb 01c End'ch gariï¬' Â«to (Merit _ IÂ®0Phohi have Jâ€˜gwke Melina-I'D I I
Â® 54"â€œ J54â€œ FVRE Mi, {foe/e [JJ/cew) Â£5Â»va J'fDLk_ - I How Risk-free Arbitrage Defines the Cal's um u mawxm wsm-Qmsww-awm. Arbitrage _ ..;_.When the call on a; stock is trading-at too higha
' I premium relative to the put with the same strike
"price, â€˜buy the stock, sell the call, and buy the put.
' This strategy isCommonly'kn'own as aforward
- â€œconversion, ) 003 what, we acm,_ tag a pear .
' ' " ii. MM; newâ€ it 79"â€. ' " Â«Stock I I = $120
_ " 3gmon'th' Call _ = $5.50, struck at$120 firm Â«ï¬fgglggj:jgâ€œgjed +6 I 3-month Put": _=I$4.00,lstruck at$120 '
- creme!â€œ met - -- - . Expected-Dividend. ,= $0- . . '. r giw â€˜4'?â€th . . .
hf wok 4â€™ , PKG/â€œe Fâ€ f?â€œ ' -/ Initial Cost of Forward Conversron :1 : $120.00 - $5.50 + $4.00 = $118-50 0608031-Corâ€”4 ,_ nversion PayOff at Expiration ~ I -: mm WMQZ'MWdidwï¬mh mmmmnmzmv-z: :- .1Â» If stock price is below $120, the put is exercised for $120 1 31f stock price isâ€˜aboveg- $120jr the investor must deliver
stock against call assignment and receive $120
I Return equals: $120/$118.50 ~â€” 1 = 1.27% for 3 months
- -- _ or 5.08% peryear â€˜ _- , - when m aim/em. ems, I If risk-free rate is. below â€œ508%, this strategy dominates Strategy I wâ€˜ll aria? it ' 10ther riskâ€”free inVestments â€˜ P90? 1 I W ï¬ght,â€œ {5â€” above (WW-â€œt r10: Payoff in -$ . Pigg)-i"-$ ._ _ _ - 30 "'""' " Stock + Put - Call
"' " ' Short Call
â€” Long Put 80 90 100 110 120 130 140 150 160
Price â€œpinnedâ€ -â€” {oer-'5 Pinned is M4814 Jrâ€˜ock (mile. 9085 1" 1%? â€œ4Â°â€? if} WWâ€˜Lâ€™WO
0608031â€”Corâ€”4 . ' v r â€˜ .
Pm ask as low } 0'19 nyk 711d 2x191? ('4 ï¬maâ€™d 5â€â€le â€œyam In 444:3 (lax-S "mg-5k PWâ€ M4SQCâ€™HD" Reverse COnverSlon ' Wm:a..:z Mâ€˜s mwmxmt and securities in order to outperform holding stock "
Call is C9164? ï¬lm'sz 1% Him-FMâ€” . Strategy I I Sell stock, buy call, sell put end reinvestâ€œ cash in. risk free .__Stock _ I ' = $120 ' - ' 3~month Call = $5.50, struck at $120 3-month Put ' = $4.00, struck at $120
Expected Dividend = $0 ' InitiÃ©l Credit = $120.00 -_'$'5_'.50 +.$4.00 = $118.50 Risk free reinvestment rate = 6%]yr or 1.5%/qtr
(um? ) . â€˜ Payoff in $ mmm4 no Mâ€˜Hâ€™W wind haffï¬‚â€™kmâ€™ 75 stock/.3149 Lug: ï¬‚ue 11 ,r'"\ . . ,ï¬‚-\ â€˜ k- ' , = Pâ€”,â€˜ PMi'ICQll [90411:] 51Â°â€œ not was? râ€™ â€™- ï¬‚ock Md @1028 low + â€™ 2 E â€˜ we gmmï¬/reueuse oastuzâ€™rsrbn has nâ€˜lâ€˜leâ€˜Fnee arbimÃ©q (Awweâ€˜f Dividends â€˜ â€˜-- -- vatï¬m'mmwmim-m mimbuï¬l-M m-ï¬â€˜axwg; <: Adds return to forward conversion, takes away return from a
reverse conversion ' - ' Stock = $120 _ _
3-month Call â€œ = $5.50," struCkiat $120
34month Put " = $4.00;r struck at? $120
Risk-free rate __ = 6%/yr or 1,5%/qtr Stoek ays a50Â¢ per share dividend everâ€˜the next quarter J33? veg,Â» Payoff in $ / ' Return: 120.50 Kama-133 We Jhck m +14}; (we, (4.73â€œ 118.50
mhï¬eÃ©ï¬ (142' = 1.690/olqtror6.76%lyr make 6.317. Mom Hum rel/era
drï¬idenoi .
. / (in/16194;: fr Value at puma/ti Conu'. 12
0608081-Cor~4 {412/1 +1790 WT 96ml J] NU?!â€œ Comur
Pufâ€™ anâ€œ Pa/ï¬j does not 8x131â€ â€˜1mwmmskMImgmae-ume-mâ€˜xaxw :rMÂ» ced atFair' Assumptions StOCk - 7 = $120
W .v: - ATM Call (@ï¬‚emonej) = $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 â€”$5_.50 + $4.00 ="$118.50
Reverse Conversion CaShâ€˜ Receipt = '
" $120 â€” $5_.50 +â€™$4.00 = $118.50 Payoff in Return = 1.69010 "Indtntiezent bah/em Kc Mai 10 13
0608081-C0râ€”4 ' - I
5"â€œ ,2 1+1; (1â€™. Met? 1, .i foosvHoq YT now MJMJ 61f 'pM'f/Cau pmâ€˜hj Positions Based on Put-Call Parity
. It I Mmâ€œ a BM
- - '_ Stook_â€”â€œ-Ca|| + Put =. [(Strike'+ Div) ,:"(-1+ risk free rate (r))] I I." "Synthetic Long Stock I
+ Stock: Call â€”' Put + [(Strike -+-Div) / (1 + r)] I Synthetic Short Stock _. . 7 .. __ I_ I Stock = Put-â€” Call â€” [(Strike + Div) /â€™ (1 + r)] .' Long Call- (Portfolio InsuranCE)â€˜
" ' Stock + Put = [(Strike + Div) /â€™ (1 + r)] + Call
- Synthetit: Shl'ortl.f.col_lj _ ' I I I ' - _ ~â€” Put â€” Stock = Ã©-Calla[(Strike_+ Div) / (1 + r)] ' _' SVnthetic Long Put â€˜ " ' __ ~ ' 5' . Call -â€” Stock =â€˜ Put â€”'[(Strike + Div) / (1 + r)] :7. II Synthetic Short Put (BuyaandeWr,ite) Stockâ€” can = [(Strike + Div) /' (1 + r)]- PUt i whenâ€ Pâ€?! â€œNM â€œ1â€œâ€ W iâ€. â€œVFWâ€ 14
owmlï¬‚wâ€˜; a: ,Fu/lchbq of: pp (7!er .1â€th Price OPHMS cioï¬r mm! 40 be. simckÃ©) 5767â€˜!â€œ Iowa; Fajitaâ€œ Iva/Hy \ in mam wsammmnntmmy-EQJ 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 â€˜ 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 â€˜ i" _W Tommi stud: ,StoCk I. ' ' .. = Strikeâ€˜+ Div
' " ' - 1+r ' [(1+r) x Stock] â€” Div. " -= Strike - _ [(1+0-0169) x 120] â€” .50:- Strike . ., _ _ = 45% at {4453 Jâ€˜lï¬ke Fritz) â€˜oph'm ram/W may FIE/21.33 is ' 'fâ€™iqoecl'td ï¬lm price, 51C Shack at: ï¬t?! strike, SDZ chance 71Â°50 abwc â€œmad _ 121.53 + .50 â€” 1 =' 159%
Conversron - 120 M15. 6â€™â€, 50% bf); 30 hâ€”a
0608031â€˜Cor-4 {harm PWCFSM. . kmomh J'fâ€™bck Fina. Could Â£9, {33am Jerk/j 9AA know Fr'xce
by degtii-Hwi know) â€˜Hâ€™lt oFH'eq opiâ€˜iÃ©n ' _ OptionsVamation i .. â€œmmwmvmwmï¬zmt-mmxwing:-:22-; :3. â€˜- Option Pricing Discounts Future-StocikrPrices
The shaded area H 4 - - ' - ' ~ - ' -â€˜ I ' irâ€˜i' wiâ€™iiâ€™a , 41"
shows the range of ' - - - Plâ€˜lce [email protected] cenie/jt flow â€œ1 OF M
stock prices for 4.3 cum-viii 7328â€™ r 0 which the call option 3k 7â€˜ _ 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 likelihood-of each price. _4 â€˜ --â€™Â°3'â€â€˜Â°"â€˜â€â€œ@ whimâ€œ H 1 Must asuuwe Jomeï¬‚afnj â€˜60 3H"
Questions Ophâ€˜bnâ€˜ 33- garage/Ht value 07C 4â€ [90:51 Isle, Jâ€˜th [antes r dismhuh M
1) â€œMaï¬a: the ' : f h, l: .~= l H 4" C
probability of the 13 0- V: 5:3, .Id 9â€˜: (paged, j
stock reaching $135 ' Forward Price at 105 â€™ " i
or higher in one WW: Hoâ€?
_ _ _ . â€” â€” â€” - F â€” " ' " ' â€ ' d - â€˜ 'S 2) What is the Initial Stock ' & â€˜Frâ€˜ice
probability ofthe me at 10Â° stock reaching $75 or ceriiv 0? (it 5 f = (o s A Call Strike Price at so Iowerin oneyear? - iâ€”O' : 30! up H7313: ' dowel-6'57}? Fajo'F; hon/Walla
1â€”Year Call 0 tion Gatingâ€œ mic dishibxtlâ€™al : _ VI Strike Price: ' $80 g'mck Price ,3 gemâ€œ 75' â€œ4 4o_ =
'- Volatility: _ 5;; "-30% if}; â€™42:,9-â€œ0â€™?â€ â€˜iilâ€˜lm-l" I Dollar Volatility: $30 (30%*$100 stock price) 10 '
I IntereSt Rate: 6% Probability of Price Range
. Dividend Yield: 1% 16 0603081-Cor-4 want wide. dikhï¬bmï¬M / beam bfjjpn Rm P05)HVâ‚¬ owf'CoiMESJ inchfFExeq-t
gr thdHW 0608081â€”Cor-4 - an wmamï¬hâ€™vsâ€˜i')hb% 2-K- . . I The theoretical value of an option dependson the variables
" below, the table shows effects of increasing each variable on
option price - . - .. _Ca_â€œS' -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 Â«L?A.m.-::+a:.u:.eon. : ' l .. .- . . - , â€˜ -:?X._.HiStloric_a_I. Z'Iâ€˜E'Xiâ€”R]2._
' ' VolatilitytÃ©, ' i.=._-1 are the returns, and N is the numberbf If-an 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 - Â£3 da'ï¬llâ€™uâ€˜hâ€˜bn at,â€œ kWDoO â€œQ. I
FmbdbihhÃ©s m orï¬bns inc/W
91557345 060803 1â€™Cor-4 This Shows thata $100 stock with an annualâ€ 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 â€” The volatility estimate which matches the observable
market price of the option with its theoretical value of ' 20% womd - be expeCted Histori_cal.volatility"â€” The annualized standard deviation ofdaily or weekly
price changes over some period of-time; Traders tend to use 1â€” and 3-month
historical vdlatility to price shortâ€”dated optibns and 1- to 3â€”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 â€” The Implied VOIatiIi mw-tausmmâ€˜owWJMH .- This implies the folloiNing: Index 8 I 14000 Oth-â€˜VIS' are 394, [35â€œ! 71004 IKE/w, "I The VIX measures imglied Volatil ty i
for 30-.da S&P 500 index 0 tions 'At the-beginning of July, the VIX StOOd at'-13.0%-with the S&P 500
at 1275 mm .in. pct â€˜I/
i 1-day VIX 10.5 ' 0.82 7â€™ W
1-week VIX 23.0 1.80
l-month 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 0608081-C0r-4 Options Valuation Without a Modelâ€”An Example - '
Assumptions:
Stock = 100
Strike Price = 101 in and, ole/8â€™0? râ€™lmtâ€˜râ€˜mâ€œ 5" WW"
3â€”Month LIBOR = 5.5%/yr 3 9â€œâ€6â€˜â€œ? 3â€™5"â€ "â€œM 4" gmâ€:
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 73-129 and working in increments of 2, that is 73â€”75, 75â€”
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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- Fall '06
- ZURACK,MARK
- Strike price