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nba694class04 - ' Equity :Derivatives . .f _ Pmd W5 were»...

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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' MOWER-Wa‘mbizfi}x-w9nmcum.: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 prices-which is‘the basis of its valuation Travis Jordan from Deutsche Bank will discuss how Hedge Funds use Equity Derivatives in financing transactions ' 0608081-Cor—4 -\-. - .- mmrmm: a}, mm‘mfiw“ an :. Options Pricing ' Comments-9f.fan'OPtiO-n:Price -' -- " " I=The:'differen_cet5between the market valueof'thei’underlying; seCurity'and _ rthe exercisegpric‘e or- strike'_'pri_'ce for-a -ca|l-_;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 the-insurance chare which limits loSSes-tothe'ogtion’s ' “premium 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 -.-‘- usuwwwmmxwxm-Wmmswr. .«.-:=-~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' saw-fie =‘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 3-month Option-StruCk at $45 '= $7 ' No Dividend "Expected * ' ' ' 7' ' oiavidwi.(w1guctg=, ago/veer , I -' is theintrinsic value? ‘30 4555' ' 2. What is the time value? Q 7-5 a Q n w I); ;. figuwhat benefit does-the investOr’accr‘ue by not having to (W i '_'_""-_pay:for the stock .in- full? 50-7 :93. . (WW-005) = .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 - - .@33_ o.o.°r- . y" . 5&5: \ 0608081-Cor—4 . (q) l 4 What Happens t0 an OptiOhWhen the an «- rmmmwmwmfi mum» Liw'utt nasal-q: . ,_ . _ _, __ - Nothing-if,out—of—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 11¢ Atos (ml met“? 4/90, film ore-M T050 jive: good 'lndicaf-m 0F Wm Jfocfc Ir warm iv: Jhcrf‘ am? all {that} oJl/«w {mch 3194c curve anljjlwf exFirali'M A dag-Foo how.mmdq Jack meals: 1h aro’rr at» hédgz lie-HOV) - M 7' A a mutant? mgr arr-m will Wm prim-Mons, ---v--._...., . _ _ - w..." m, . 0608081-C0r—4 ,3 -\. DELTA verage .o‘.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 '[IHERII‘JE-PLDLIEH CHEF ‘32? Bid .327: 'Eii-ai - ' : CALL l j. n t I El _ r._ ‘ ' - .3: .3; i-E- b.1nt 113 a a _ ' .dfi Source: Bloomberg I I delta = *3 “Wm 4‘ 5‘!) Call fv‘bfléé’ Data as Offlmmz. Olaf-fin 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 -' 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 _ “Few”WL UW' .WCAMJ . i" .10va f-d’ rye/yfl con Ll: short mole], '6)ij an”, 3hort’ 0608081-COr-4 The Linkage Between Calls and Puts — {— ‘t-ills how mach; 1%?- pwf‘xflqomlot ice. “7942465 +9 Cw“ Cam m 1+0 limo! par-as' Poincfi‘m “a (“H “933’” ‘fi'[‘ I . Jlf'l'vflowf‘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 “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 Melina-I'D I I ® 54"“ J54“ FVRE Mi, {foe/e [JJ/cew) £5»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, ‘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 3-month Put": _=I$4.00,lstruck at$120 ' - creme!“ met - -- - . Expected-Dividend. ,= $0- . . '. r giw ‘4'?”th . . . hf wok 4’ , PKG/“e F” f?“ ' -/ Initial Cost of Forward Conversron :1 : $120.00 - $5.50 + $4.00 = $118-50 0608031-Cor—4 ,_ nversion PayOff at Expiration ~ I -: mm WMQZ'MWdidwfimh mmmmnmzmv-z: :- .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/$118.50 ~— 1 = 1.27% for 3 months - -- _ or 5.08% peryear ‘ _- , - when m aim/em. ems, I If risk-free rate is. below “508%, this strategy dominates Strategy I w‘ll aria? it ' 10ther risk—free inVestments ‘ P90? 1 I W fight,“ {5— above (WW-“t r10: Payoff in -$ . 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 (lax-S "mg-5k 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% Him-FM— . 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 3-month 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'mmwmim-m mimbufil-M m-fi‘axwg; <: Adds return to forward conversion, takes away return from a reverse conversion ' - ' Stock = $120 _ _ 3-month Call “ = $5.50," struCkiat $120 34month Put " = $4.00;r struck at? $120 Risk-free rate __ = 6%/yr or 1,5%/qtr Stoek ays a50¢ per share dividend ever‘the next quarter J33? veg,» Payoff in $ / ' Return: 120.50 Kama-133 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 0608081-Cor~4 {412/1 +1790 WT 96ml J] NU?!“ Comur Puf’ an“ Pa/fij does not 8x131” ‘1mwmmskMImgmae-ume-m‘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 0608081-C0r—4 ' - I 5"“ ,2 1+1; (1’. Met? 1, .i foosvHoq YT now MJMJ 61f 'pM'f/Cau pm‘hj Positions Based on Put-Call 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 = é-Calla[(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 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 ‘ 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+0-0169) 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‘Cor-4 {harm PWCFSM. . kmomh J'f’bck Fina. Could £9, {33am Jerk/j 9AA know Fr'xce by degtii-Hwi know) ‘H’lt oFH'eq opi‘ién ' _ OptionsVamation i .. “mmwmvmwmfizmt-mmxwing:-:22-; :3. ‘- Option Pricing Discounts Future-StocikrPrices The shaded area H 4 - - ' - ' ~ - ' -‘ I ' ir‘i' wi’ii’a , 41" shows the range of ' - - - Pl‘lce W@ cenie/jt flow “1 OF M stock prices for 4.3 cum-viii 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 likelihood-of 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: ' dowel-6'57}? 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‘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. dikhfibmfiM / beam bfjjpn Rm P05)HV€ owf'CoiMESJ inchfFExeq-t gr thdHW 0608081—Cor-4 - an wmamfih’vs‘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_“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 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 - £3 da'fill’u‘h‘bn at,“ kWDoO “Q. I Fmbdbihhés m orfibns inc/W 91557345 060803 1’Cor-4 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 of-time; Traders tend to use 1— and 3-month 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 mw-tausmm‘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 the-beginning of July, the VIX StOOd at'-13.0%-with the S&P 500 at 1275 mm .in. pct ‘I/ i 1-day VIX 10.5 ' 0.82 7’ 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 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 73-129 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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This note was uploaded on 09/25/2007 for the course NBA 6940 taught by Professor Zurack,mark during the Fall '06 term at Cornell.

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nba694class04 - ' Equity :Derivatives . .f _ Pmd W5 were»...

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