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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. 0608082C0rS 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, theprocess 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 OSGSGSZCOrS YnamiCCreation (Hedging) of. *
Payoff _ ‘ ea: “:53;ng We VOW/i3; so W w you aw
0a make Movie 39 Ioni” 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
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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‘cﬂlmJtvpemmfrauzﬂ'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 deltatimes 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' 0608082Cor5 . 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”,
SﬁjlﬁgEUMD; 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, Twomonth .1350 Calls, Twomonth 1250 Puts
SPX .= 1,300, 2—monthjLIBOR = 5.25%, Dividend Yield = 1.8%
S&P 500 Twomonthfu‘tures trading at Fair Value
At—the—money (ATM) Implied Volatility = 12%
25 Delta Put/Call Skew = 5 pts ‘ ' II "I SPX, Sixmonth and twoyear ATM Calls, same inputs as above on
_ Index," nd Dividend Yield, assume flat LIBOR.curve,twelve_month 'ATM'SPX Implied Volatility = 14.7% 5 0608082C0rS _"“"‘ .u‘,
x ,i x act and Options Pricing WW‘w Mmemaze. 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 $37.5calls 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. ' 0608082Cor'5 Rho —Different Transactions Require   a , _ ._ 'ﬂﬁfsk—Fwe "mm care LIGOK Per
Different Interest Rates ,W W i 3944,803 41, (49de up HMS" {sfock CuStCmei'TraneeotionI " ' Optidns'Market'Mekeri_' Relevantlnterest Rate
Buy Index Calls  * Buys futures, ETFs or LIBOR +/— expected
' smoke r . . 'mi'sprioingforfutures or
' ' ' ” V ' ”  cost offunding 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_.__'_:?_Buy StockPuts ' _ ;’ Sellsstook short _. a Rebate on short sale ‘ How would you price Example1 if the trader’s funding rate is LIBOR
plus 30 bps? : Question”______F_£_., 0608082C0r5 'r ‘ =' ‘ .5 ‘t e on a Short Sale? ‘4 Mug.» w ﬁwmw a: The Options trader must borrow the security in order to facilitate a A 3 short Sale ' P SS'Ume A: 0:  _ . .L‘Tbjoz Mt {compo $00,000 JW 0F A741”!— geli m usher? Permian +1) “3000 Puts on Options
Trader The borrower USUaIIy receives income from the short sale, that
is referred to as a Rebate 1 , .A shOrt seller is required to__reim'burSe the lender for any
dividends paid during the time the short position is in place 0608082C0r—S ortSale ' ' W1M$gWMWHﬂE=>MV1RJ$ 9 w?“ ea'm in’rewext 0” C4311 WM “5— “Hahn”? amt. 0": momej Ofﬁbv‘s Tracie» may W Wk
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0608082~Cor5 “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 Supplyforces 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 0608082C0rS A f \ An Example v w  WWaszmmmamwﬁm‘swx yum: newaw _:  '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. doesthe lender‘earn on this transaction in a
a. year? If thetrader Was: pricing a p" t optiononthat stock, what interest rate shOuldi'he/she Use? 1’] 0503032C0r5 HistOri'cal
. _ Volatility: 'Where )7 is the mean of the returns, xi are the returns, and N is the number of returns If an assethas 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 generallylack 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 shortdated 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 _ ' OneYear S&P 500 Realized Volatilities
2004 a 2005 I . 13 .........  60°/o . 50% .16 .
a: ____________________________________________________________________________________________ ..
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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 OGDSGSZCor—S f ‘1 _/ ‘1 I‘m OSGSGBZCOrS VIX Close d ' Volatility Index . mun mammsmrm: kiwiaway: arr u The VIX measures implied volatility for 30day 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
1week 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?  ' 0608082Cor—5 .40, 5: .35  Index Options Trade at Implied Volatility alizedm‘leatility  u madman» 9mm: azzﬁaumﬁuehrzk —— 1y’r'ATM Implied Vaiatility
_— IVij" Realized Volatility ' 14.68
10.35
:0 I .. L . . . _ .
. 5/94 5/95 5/965/97 .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 ModelAn ,. . .. WWWdﬁxmm:Q'L'vmegaluxmmunu 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 7375, 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. 0608081Cor4 l...\ 02m: m 3035: Emﬁccmo: 53: m 3mm: 90 Em Burma
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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).
 Fall '06
 ZURACK,MARK
 Implied volatility, historical volatility

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