nba694class05 - -_ Equity Derivatives and Related Pr W...

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Unformatted text preview: -_ Equity Derivatives and Related Pr W oduCts Class 5: Introduction to Credit Derivatives] Practical Considerations in _ Using Options _ Mark Zurack . ; September 26, 2006 n-.- Wumfimmegmmmvommwvmm -.-~:,-¢\.-. .5; A >. a r 4 Stephen Deletto from Bear Stearns will discuss the Credit Derivatives Market, focusing on Credit Default Swaps. ' Options pricing in the real world violates assumptions made in most models. The three most important are the liquidity impact of hedging, the risk—free rate and volatility used In the second half 0f the class, we explain why transactions - costs and intereSt rates are important to options pricing and what interest rate is relevant for a particular tranSaction. We also introduce stock rebates and explain how they work At the end of class, we introduce how traders look at the relationship between implied and historical volatility. 0608082-C0r-S How Options Traders Make (or Lose) - '- Jmfimmwwmiwmrmw)émdb¢fiEl'fivfiv,"71'c w .._-- -, _ ,. _ ., The i Delta Hedging PlrocesS Option traders neutralize their . exposure when Most option traders do not. takedirectional views (i.-e., bullish or bearish) .ma’fmgamarffetbr When creating or hedging options positions; Instead, they utilize an buying or selling the option model to create an offsetting hedgeto neutralize the underlying stock _ _ n _ underlying pnce risk of the option. This hedge can. be executed with'the underlying stock (or index) Since the amount of stock required by the'hedge changes as the stock price changes, the-process of replicating an option is Often referred to as “dynamic hedging" or “delta hedging”. It requires making the foilbwing adjustments. over time: Buy- Call Seli Stock Seil More stock Buy Stock SeliCall . i I Buy Stock I Buy More Stock Sell Stock Buy Put Buy Stock : Seli Stock I Buy More Stock sen Put ' " Se‘il Stock ‘ "Buy Stock Sell More Stock OSGSGSZ-COr-S YnamiCCreation (Hedging) of. * Payoff _ ‘ ea: “:53;ng We VOW/i3; so W w you aw 0a make Movie 39 Ion-i” low and me“; ) 91,. h, :3 or; To replicate option payoffs, option traders hold an offsetting “delta” position in stock or futures, shifting their holdings as the underlying price moves on; gamma Davfd- 50H .o‘qhuflj 31440. 4° 3°. . 7 Eiffel“ C4“ “1‘41 was 0;: david will Lug S0 flan/E: 9%? risk d6 hill-e more-,- Stock (¢O(b¢ck)j Hm s: SAM-5570052 $25) haul J60 Lug-'SJVICIL when 17' moi of) Ciao-n3}, {A 481%“ pen chomja, M 54190]: FHCQ _ Gran/lag 'f‘. :6“ whim 30:3 clown/- bar; 145% {all (on: g 06mg.th how motel/1 aab'ushMErrl‘ i1€€cl£cl 3 oeososzm-s he? loge: Mme? fix . i ,/""\ I " . If ,-"""\ ‘ 5&le options 1': {genius voialfmjl 5' ‘- Ilfiiol‘t gammm - 1‘ IN wi” 3%” Op‘h'bn @ Beware he, {hanks H' [,0ng £349 hgtfflttpvfificq HOW Options Traders Make_._..(0r Lose) — . u—su: iiwtilfmkfi'mymifibu‘cfllm-Jtvpemmfrauzfl'aw 9: m, l r . . - . In praCtice,'Dynamic Hedging means that if. you establish a bullish position (e.g.-,} IOng a call 0r,Sh0rt a, put); you will need to sell an appropriate amount 0f the underlying stockto eliminate the risk of a price Change. The amount of stoCk needed‘for the hedging portfolio is knOWn as the" “Delta” Or the “Hedge Ratio”. Conversely, if- you establish a bearish position (e.g., longa put or short a call), you will need to buy deltatime-s the number of the shares underlying the option The seller (“writer”) of the. option (both puts and calls) has an expected loss arising from thehedging processbecause the option writer will typically adjust the hedging portfolio by buying stock into a rising market and selling stock into a falling market In a sense, the option price paid 'upfront from the buyer to the writer ' can be Viewed as the present value of the expected cost of this . “dynamic__hedging'f’;process until the maturity or exercise of the opfion ' 2' 0608082-Cor-5 . a Option’s Model — WWW.cboe.co . trading tools, Volatility optimizer, y I ' ‘ E _ (1 j , Optlons Calculator . . use to Price _oi9+i1ms 1 in ‘3???” mimi‘mr/ “'44” Sfim'wt {We is”, Sfijlfig-EUMD; Fncezloo, JMke:[Ol_ PE)“ Ma = FLEy/ 'DTerQIJ v4 5,20% )3}: 5,57, J D,” Was-[,3 Lets pricethe optiogyou Created a spreadSh'eet on CW '54” WS- “5 JWMM‘ 3"” I I 304 _ ' r Examples: ' ' I FAAPL,’=September37.5 Calls, AAPL 2‘ 38.25, volatility = 40%, 2 ' ' -2=month LIBOR‘='5.25%, noydividends , . . _t a AAPL, September 35 Puts, -’AAPL' '——'-' 38:25, Volatility = 41%, 2—month LIBOR' _'='_ 5.25%, nO'dividends - I SPX, Two-month .1350 Calls, Two-month 1250 Puts SPX .= 1,300, 2—monthj-LIBOR = 5.25%, Dividend Yield = 1.8% S&P 500 Two-monthfu‘tures trading at Fair Value At—the—money (ATM) Implied Volatility = 12% 25 Delta Put/Call Skew = 5 pts ‘ ' II "I SPX, Six-month and two-year ATM Calls, same inputs as above on _ Index," nd Dividend Yield, assume flat LIBOR.curve,twelve_-month 'ATM'SPX Implied Volatility = 14.7% 5 0608082-C0r-S _"“"‘ .u‘, x ,i x act and Options Pricing WW‘w Mme-maze. Mo * When a trader prices a stock option, in most cases he/she must delta hedge that option by purchasing or selling stock ' . As a result, the trader must estimate the liquidity impact of trading the hedge in order to price the option Example—=1: An options trader was asked to offer 10,000 contracts of Apple (AAPL), September $37.5-calls which have a delta of 0.6." AAPL was trading at " $38.25 need in hedge poo/oooth: (lgooo Callback ref, {M Ska/er) 0P Ara/ole _;I__ How manyshares of AAPL needed to be traded? 7' .IIPurchas'edor‘wld? ” - ' - , . '_ cast at mail/r3 905* I If the estimated liquidity shortfall of trading the stock was 50bps, . _. when may, owns \ what stock price should the trader s’e to price the option? k y I‘ " - (at, +0 N m.- JLteH’W) " -_ ' _ _, (f.ooE)C3’& 2:) f Ban/cf heme-+0 3d ohfioq=pmce: 3,15 - i - __ r0 'Lj I Note difference in fair value of option With and wrthout LICIUICIIty Impact. ' 0608082-Cor'5 Rho —-Different Transactions Require - - a , _ ._ 'flfifsk—Fwe "mm care LIGOK Per Different Interest Rates ,W W i 3944,803 41, (4-9de up HMS" {sf-ock CuStCmei'Traneeoti-onI " ' Optidns'Market'Mekeri_' Relevantlnterest Rate Buy Index Calls - * Buys futures, ETFs or LIBOR +/— expected ' smoke r . . 'mi'sprioingforfutures or ' ' ' ” V ' ” - cost of-funding stocks or ETFs Buy Stock Calls _ _ _‘ Buysstook : Cost of funding stocks Buy Index Puts V _ Sells'futures, ETiFs or I LIBOR +/- expected I Ex—' 10000 Am pm, claim“; 46” m?“ ‘ stocks short . miSpricing for futures or (Short I t ' ' rebate On Short sale ~ ’2_.__'_:?_--B-uy Stock-Puts ' _ ;’ Sellsstook short _. a Rebate on short sale ‘ How would you price Example-1 if the trader’s funding rate is LIBOR plus 30 bps? :- Question”______F_£_., 0608082-C0r-5 'r ‘ =' ‘ .5 ‘t e on a Short Sale? ‘4 Mug.» w fiwmw a: The Opt-ions trader must borrow the security in order to facilitate a A 3 short Sale '- P SS'Ume A: 0: - _ .- .L‘Tbjoz Mt {compo $00,000 JW 0F A741”!— geli m usher? Permian +1) “3-000 Puts on Options Trader The borrower USUaIIy receives income from the short sale, that is referred to as a Rebate 1 , .A shOrt seller is required to__reim'burSe the lender for any dividends paid during the time the short position is in place 0608082-C0r—S ortSale ' ' W1M$gWMWHflE=>MV1RJ$ 9 w?“ ea'm in’rewext 0” C4311 WM “5— “Hahn”? amt. 0": momej Offibv‘s Tracie» may W Wk I mm; Jé‘giifi'fibg vagina, doesn'? lose mole/moi: uses l‘éiii'u-I x0019 ' . +"".—7—-."_“: -- " Stockim. ._ " ‘_——“““'?' . Cash "lbw? barrow" _, [ham-qu amt 010 [Way-5’, [ens/ea Foch)!“ More 970 'miiHaL/Q ms)? (200 (or: (25‘ annexed +0 {YA/3’) A” exqflfie) MéSf- Sh'ck jean/53:6??? aha/543%? Icon: ; I _ I _ . _ I i has (0.51194 1096031525 [card EOVVDLU) (IFAJZ/IV car/1th an? for}; 3" D par _ (a Cw be hoe/ax! LjJHuol-rm"? elem/a:th ngp_ when I Vatui‘rfi thJWS, need: ‘fb use dhqerfdf VISIU‘FWE 5;? 44,141. Offigq (m be (“l/24 a F 0608082~Cor-5 “in; a: M COM H: (‘6 m1] 1%?er 0608082—Cor—5 vv - a»? “.9 Milk”? M Date Ram? Rate where lender is able to reinvest cash Spread lender wishes to make which depends on the supply/demand for individual securities in the market E. g. — IBM 120 mm available inventory I Large amount of Sup-plyforces lenders to compete in order to get stock but on“ loan (pay- higher rebates) E. g. — PALM, 9 available inventory l Stocks demandedby hedge funds, used to hedge convertibles, recently come to market,_or closely held can be difficult to borrow Spread broker wishes to'make which depends on'the cost of using their balance sheet and‘their ability to find a lender 1 . Term- - usually overnight Options Trader’s rebate = Lender return on cash — Lender spread 10 0608082-C0r-S A f \ An Example -v w - WW-aszmmmamwfim‘swx- yum: new-aw _-: -- '1'.‘ 'The' optionstradet in Example—2 can borrow a large cap stock at toprate which is LIBOR minus 15 bps or 5.10%. _ How much. does-the lender‘earn on this transaction in a a. year? If the-trader Was: pricing a p" t optiononthat stock, what interest rate shOuldi'he/she Use? 1’] 0503032-C0r-5 HistOri'cal . _- Volatility: 'Where )7 is the mean of the returns, xi are the returns, and N is the number of returns If an asset-has a 20% annualized volatility, _-t_hen over many years, we would expect 2/3 (1 standard deviation)-of the observations to. be i20% around the mean return " , This shows that a $100 stock with an annual volatility of 20% would be expected to move <$1._25 per day around $100, 2/3*o_f_ the time Traders generally-lack at two measures of Volatility when pricing options: I Implied volatility -— The volatility estimate which matches the observable market price of the option with its theoretical value ' I Historical volatility —- The annualized standard deviation of daily or weekly price changes over some period of time. Traders tend to use 1- and 3—month historical volatility to price short-dated options and 1- to 3—year volatiiity to ‘ price long dated options 12 The History of ActualS&P 500 ' m - erwm ffnik‘fihflflxWildfmifif‘e : r .~ _ issuable Distribution of 20 _ ' One-Year S&P 500 Realized Volatilities 2004 a 2005 I .- 13 ......... -- 60°/o . 50% .16 . a: ____________________________________________________________________________________________ .. g 40% 14 > 1-12 """""""" ": "" """"""""""""""""""""""""" "1 g a; g 30% £10 ------------- -- - --' ------------------------------------------------------------------ 3 a 0‘ ‘3 8 . ------------------------------------------------------------------------------------ -- ._ c _ - 2002 g 20% “- / 6 ---------------------------------------------------------------------------------- C I: < 4 ' T """""""" ' ' """"""""""""""""""""""""""""""""""""""" "'i 10% - 2 . . _ _ . _ _ _ . _ _ _ _ _ _ . . . . . . . _ . . _ 2.9.8.7. __________ .1332 ____ .2??? Median: 14.0 0% o 1929 1939 1949 .1959 1969 1979 1989 1999 4 8 12 16 20 24 28 32 36 40 44 48 _ 52 . Source: Goldman Sachs I _ . ' ' . ' Historical Volatility (%)' ' _ l- _ -_ . r I _ ' ' I Source: Goldman Sachs Yearly data from 1929 to 2005 Questions: What is your best estimate of annual volatility in the S&P 500 in 2006? What is your best estimate of annual voiatility in the S&P 500 from 2006—9? Which estimate is more likely to be accurate? 13 OGDSGSZ-Cor—S f ‘1 _/ ‘1 I‘m OSGSGBZ-COr-S VIX Close d ' Volatility Index .- mun mammsmrm: kiwi-away: arr u The VIX measures implied volatility for 30-day S&P 500 index options I At the beginning of July, the VIX stood at 13.0% with the S&P 500 at 1275 '- I' This implies the-"following: in- points in pct 1—day VIX 10.5 0.82 1-week VIX 23.0 ' 1.80 1—month VIX 48.5 3.80 Calculation: VlX dividend by SQRT of the number of periods in a year. multiplied by index level to move into points. assumes fonrvard price about the same as index level I Futures and options on VIX trade '14 Why Implied . Volatility Differs from Actual " ” Volatility. _ o Gap 01" Jump Riekfg; _ a Mean Reversion _' g}. _ 0 Supply& Demand“ __ ' o Risk/Liquidity -- 4 Premium Questions: How much of the spread between implied and historical volatility at the end of the chart can be attributedto mean re version; how much to hedging risks? - ' 0608082-Cor—5 .40, 5: .35 - Index Options Trade at Implied Volatility alizedm‘leatility - u madman» 9mm: azzfiaumfiuehrzk —— 1-y’r'ATM Implied Vaiatility _-—- I-Vij" Realized Volatility ' 14.68 10.35 :0 I .. L . . . _- . . 5/94 5/95 5/96-5/97 .5l98n 5/99 5100 5/01 5/02--‘ 5/03 '5/04 ‘5/05 Weekly data from May ’94 to Dec ‘05 Source: Goldman Sachs 15 r- \ .- ' ‘\. Options Valuation Without a Model-An ,.- .- .. WWWdfi-xmm:Q'L'v-megaluxmmu-nu 1., .-; .7 , - . .-~-v:« Assumptions: Stock = 100 _ - “A M Strike Price = 101 w Y0}qu a 3,—Month LIBOR = 5.5%]yr Dividend Yield = 1.5%lyr 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. - e 3. Determine what the option is worth for each 2 dollar range using the midpoint of the range as the ending price of the stock. 4. Value the option. 0608081-Cor-4 l...\ 02m: m 3035: Emficcmo: 53: m 3mm: 90 Em Burma Emnm mg m mflmnama Qmszo: o." Em 565:? 2 <8 Gum ._uzom_<_0_m._.€10m_ 339a 38. 5—33... «mcmfi 3.8 m3 8:. mxmm_ 5;: RES Em 905mg? Em, Em magic mfiomx 30m <5: cm 63 Em: 0.: mgcm“ 6 Em 61$: <0: magma. _: Ea mxmBEm. 8 23 Em 9032:? 90 Em magso mnoox 96m $3an .3 03m 90 Em ozm: 36229 23 Em E0335 0* Em msaso man—A 30m cmio "mmm Em: Em 38235: <mEm 9n Em 362m. mag m:ng :03 Em." Em v3.8.9.3 9fl Em mango mfionx 30m cmEo _mmm Em: Em SEEBCE <mEm Qn Em 362m; _ 4 .. . “9.: ......nowmwofiEmOEas- Em, ._m.wcaun::m..._. . 3%»Em??? Ragga _. .w .‘ Em‘. .068 am. 6. Em ” _ mmfliomsmEum , hm amx mmommm M...” . . $3 o 88 mo_oo&__ mwmoo mwmoo Doom; mo 88 mwmoo mwuoo ooomm mooooo «wwoo wwo oo ooomu mo oooo aflooo mm; 00 ooomm wo oooo 93 oo mmmoo 0.95M mo oooo mmwoo wow oo o_o._mo mooooo mmmoo wmfloo oommo mooooo moNoo woo oo oomhm mo oooo emooo mm: oo ochwo wooooo 9o; ,8 wow oo oomwm eooooo momoo momoo ooomm wobooo momoo moNoo o owow mooooo moNoo moooo o 3mm mobooo moo oo fioeoo oowow mooooo 9.6.. oo fiomoo 0,38 moouoo fowoo m4 om oo ooam wobmmm moowoo 94 ow oo o ,owom mo, wmom woofloo fioo oo oooNL 903me w; oo .oo o; .3 oo o. omwm oo hwmm ejaoo mjwoo oofiwo won—won mimoo mimoo 0,0me eo ALE mfim oo m3u,oo oommo we moo» mfiwoo mfiooo ooémo wo mmom m4 3 oo $34 ,8 o o3» wo Mao“? 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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 University (Engineering School).

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nba694class05 - -_ Equity Derivatives and Related Pr W...

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