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Title: Mechanics of Options Markets Chapter 8


1
Mechanics of Options MarketsChapter 8
2
OPTIONS ARE CONTRACTS Two parties Seller
and buyer A contract Specifying the rights and
obligations of the two parties. An underlying
asset a financial asset, a commodity or a
security, that is the basis of the contract.
3
Assets UnderlyingExchange-Traded Options(p. 190)
  • Stocks
  • Foreign Currency
  • Stock Indices
  • Futures
  • Options
  • Bonds

4
OPTIONS BASICS A contingent claim The options
value is contingent upon the value of the
underlying asset Two Types of Options A
Call THE RIGHT TO BUY THE UNDERLYING ASSET A
Put THE RIGHT TO SELL THE UNDERLYING ASSET
5
CALL Buyer ? holder ? long. In exchange for
making a payment of money, the call premium, the
call buyer has the right to BUY a specified
quantity of the underlying asset for the exercise
(strike) price before the options expiration
date.
6
PUT Buyer ? holder ? long. In exchange for
making a payment of money, the put premium, the
put buyer has the right to SELL a specified
quantity of the underlying asset for the exercise
(strike) price before the options expiration
date.
7
Call Seller ? writer ? short. In exchange for
receiving the calls premium, the Call seller
has the obligation to SELL the underlying asset
for the predetermined exercise (strike) price
upon being served with an exercise notice during
the life of the option, I.e., before the option
expires.
8
Put Seller ? writer ? short. In exchange for
receiving the put premium, the Put seller has
the obligation to BUY the underlying asset for
the predetermined exercise (strike) price upon
being served with an exercise notice during the
life of the option, I.e., before the option
expires.
9
  • The two main types of Options (PUTS and CALLS)
  • American Options
  • exercisable any time before expiration
  • European Options
  • exercisable only on expiration date

10
OPTIONS NOTATIONS S The underlying assets
market price K - The exercise (strike)
price t The current date T The expiration
date T -t The time till expiration c, p
European call, put premiums C, P
American call, put premiums
11
Options definitions using the above
notation LONG CALL On date t, the BUYER of a
call option pays the calls market price, ct, Ct,
and holds the right to buy the underlying asset
at the strike price, K, before the call expires
on date T. (or on T, if the call is European).
Thus gt the call holder expects the price of
the underlying asset, St, to increase during the
life of the option contract.
12
SHORT CALL On date t, the SELLER of a call
option receives ct, Ct, and must sell the
underlying asset for K, if the option is
exercised by its holder before the option expires
on date T. Thus gt expects the price of the
underlying asset, St, to remain below or at the
exercise price, K, during the options life. This
way the writer keeps the premium.
13
LONG PUT On date t, the BUYER of a put option
pays pt, Pt, and holds the right to sell the
underlying asset for K before the put expires on
date T. Thus gt expects the market price of the
underlying asset, St, to decrease during the life
of the put.
14
SHORT PUT On date t, the SELLER of a put
receives pt, Pt, and must buy the underlying
asset for K if the put is exercised by its holder
before the put expires on date T. Thus gt
expects the market price of the underlying asset,
St, to remain at or above K during the life of
the put. This way the put writer keeps the
premium.
15
A numerical example LONG CALL C(S
47.27shareK 45/shareT-t .5yrs) Ct
5.78/share On date t, the BUYER of this call
pays the calls market price, 5.78/share, and
holds the right to buy the underlying asset at
the strike price, K 45/share, before the call
expires at T, half a year from now (at T, if the
call is European). Thus gt the call holder
expects the price of the underlying asset, St
47.27/share, to increase during the life of the
option contract.
16
A numerical example SHORT CALL C(
S47.27shareK 45/share T-t .5yrs) Ct
5.78/share On date t, the SELLER of this call
receives 5.78/share and must sell the underlying
asset for K 45/share, if the option is
exercised by its holder before the option expires
at T, half a year from now. Thus gt hopes the
price of the underlying asset, currently St
47.27/share, remains below or at the exercise
price, K 45/share, during the options life of
half a year and hence, keep the premium ct
5.78/share
17
A numerical example LONG PUT p(S 47.27shareK
45/shareT-t .5yrs) pt 2.25/share On
date t, the BUYER of this put pays the market
price of pt 2.25/share and holds the right to
sell the underlying asset for K 45/share
before the put expires half a year from now, at
T. Thus gt expects the market price of the
underlying asset, St 47.27/share, to decrease
during the half a year life span of the put.
18
A numerical example SHROT PUT p(S 47.27shareK
45/shareT-t .5yrs) pt 2.25/share On date
t, the SELLER of this put receives the market
premium pt2.25/share and must buy the
underlying asset for K 45 if the put is
exercised by its holder before the put expires
half a year from now at T. Thus gt expects the
market price of the underlying asset, St
47.27/share to remain at or above K 45 during
the life span of the put and to keep the premium,
pt 2.25/share.
19
47.27 K 45 t now
T .5yr S
20
More terminology Premium The option Market
Price Premium Intrinsic value extrinsic
value Intrinsic value Calls Max0, St - K)
0 Puts Max0, K - St) 0 Extrinsic value
(time value) Premium Intrinsic value
21
At-the-money St K In this case the intrinsic
value for both calls and puts is zero St - K
K - St 0 and the premium consists of the
Extrinsic (time) value only. PREMIUM 0
extrinsic value
22
In-the-money Calls Puts St gt K
St lt K or St K gt 0 K St
gt 0 The Intrinsic value of an option that
is in-the money is positive.
23
Out-of-the-money Calls Puts St
lt K St gt K or St - Klt 0
K St lt 0 In this case the intrinsic value is
zero and the premium consists of the extrinsic
(time) value only. PREMIUM 0 extrinsic value
24
The next table shows the market prices (premiums)
of calls and puts on IBM On Friday NOV 30 2007
t When IBM was trading at St 105/share. Notice
that there where options traded for several
expiration dates and for a wide range of strike
prices. Blanks mean that the option did not trade
on NOV 30 2007 OR did not exist.
25
S105 CALLS CALLS CALLS CALLS CALLS CALLS PUTS PUTS PUTS PUTS PUTS PUTS
K DEC07 JAN08 APR08 JUL08 JAN09 JAN10 DEC07 JAN08 APR08 JUL08 JAN09 JAN10
55 48.90
60 43.40 48.30 .05 1.05
65 39.00 40.10 52.60 .10 .30 .50 1.75
70 33.70 32.20 37.60 48.20 .15 .55 .70 1.90
75 33.28 30.80 32.30 33.50 .20 .60 1.10 2.40
80 23.30 25.60 28.60 28.80 32.38 .25 .75 1.85 3.97
85 18.90 20.20 24.40 24.90 25.70 .05 .30 1.40 2.45 4.38
90 17.36 18.00 18.40 20.40 22.90 27.00 .10 .60 1.95 3.40 5.70 8.70
95 11.40 11.60 14.70 18.10 18.25 .30 1.30 3.20 5.90 6.80
100 6.30 8.20 11.00 13.60 18.00 21.50 .90 2.30 4.80 6.10 8.90 11.83
105 2.90 4.80 8.20 10.55 13.60 2.35 4.10 6.90 8.00 11.00
110 .95 2.70 5.90 7.70 11.80 16.80 5.50 6.90 9.10 9.80 12.80 18.50
115 .20 1.31 3.95 6.75 10.30 9.80 10.60 10.80 15.00 15.60
120 .05 .60 2.60 5.20 7.10 12.50 16.30 15.00 14.50 18.00 18.50 23.50
125 .25 1.65 3.70 10.30 16.90 18.80 24.80
130 .15 1.15 1.80 4.30 9.60 22.60 22.20 27.50 31.20
135 .10 .80 1.75 33.10
140 .37 .95 3.10 7.20 32.70 38.70
145 .30 .80
150 .15 1.65 4.70
160 1.20 4.00
170 .50
26
  • Options Markets
  • OTC options
  • Over the counter (OTC)
  • Meaning
  • Not on an organized exchange.
  • 2. Exchange traded options
  • An organized exchange
  • Options clearing corporation (OCC)

27
WHEN OPTIONS ARE TRADED ON THE OTC TRADERS
BEAR Credit risk Operational risk Liquidity
risk
28
Credit Risk Does the other party have the means
to pay? Operational Risk Will the other party
deliver the commodity? Will the other party pay?
29
Liquidity Risk. Liquidity the speed (ease) with
which investors can buy or sell securities
(commodities) in the market. In case either party
wishes to get out of its side of the contract,
what are the obstacles? How to find another
counterparty? It may not be easy to do that. Even
if you find someone who is willing to take your
side of the contract, the other party may not
agree.
30
THE Option Clearing Corporation (OCC)(p. 198) The
exchanges understood that there will exist no
efficient options markets without contracts
standardization and an absolute guarantee to
the options holders that the market is
default-free, so they have created the
OPTIONS CLEARING CORPORATION (OCC) The OCC is
a nonprofit corporation
31
THE OPTION CLEARING CORPORATION PLACE IN THE
MARKET
EXCHANGE CORPORATION
OPTIONS CLEARING CORPORATION
CLEARING MEMBERS
NONCLEARING MEMEBRS
OCC MEMBER
BROKERS
CLIENTES
32
The OCCs absolute guarantee The holders of
calls and puts will always be able to exercise
their options if they so wish to do!!!
33
The absolute guarantee The OCCs absolute
guarantee provides traders with a default-free
market. Thus, any investor who wishes to engage
in options buying knows that there will be no
operational default.
34
The OCC Also, clears all options trading.
Maintains the list of all long and short
positions. Matches all long positions with short
positions. Hence, the total sum of all options
traders positions must be ZERO at all times.
35
The OCC Maintains the accounting books of all
trades. Charges fees to cover costs Assigns
Exercise notices Given the OCCs guarantee, the
market is anonymous and traders only have to
offset their positions in order to come out of
the market. The OCC has no control over the
market prices. These are determined by traders
supply and demand.
36
The OCC The OCCs absolute guarantee together
with matching all short and long trading makes
the market very liquid. 1 traders are not
afraid to enter the market 2 traders can quit
the market at any point in time by OFFSETTING
their original position.
37
CLIENT A BUY ORDER INFO TRADERS LONG SHORT OPTIONS PRICE OPEN INTEREST VOLUME
INFO TRADERS LONG SHORT OPTIONS PRICE
BROKER INFO TRADERS LONG SHORT OPTIONS PRICE
INFO TRADERS LONG SHORT OPTIONS PRICE
OCC MEMBER INFO TRADERS LONG SHORT OPTIONS PRICE
PRICE INFO TRADERS LONG SHORT OPTIONS PRICE
THE TRADING FLOOR TRADE INFO TRADERS LONG SHORT OPTIONS PRICE THE OCC
PRICE INFO TRADERS LONG SHORT OPTIONS PRICE
OCC MEMBER INFO TRADERS LONG SHORT OPTIONS PRICE MARGINS
INFO TRADERS LONG SHORT OPTIONS PRICE
BROKER INFO TRADERS LONG SHORT OPTIONS PRICE MARGINS
INFO TRADERS LONG SHORT OPTIONS PRICE
SELL ORDER CLIENT B INFO TRADERS LONG SHORT OPTIONS PRICE MARGINS
38
  • OFFSETTING POSITIONS
  • A trader with a LONG position who wishes to get
    out of the market MAY
  • Exercise, or
  • open a SHORT position with equal number of the
    same options.
  • Example Suppose
  • LONG 5, SEP, 85, IBM puts p0 4/share
  • This position must be offset by
  • SHORT 5, SEP, 85, IBM puts p1 3/share
  • Cash flows -2,000 1,500 -500.

39
OFFSETTING POSITIONS A trader with a SHORT
position who wishes to get out of the market MUST
open a LONG position with equal number of the
same options. Example Suppose SHORT 25,
JAN, 75, BA calls c 7/share This position
must be offset by LONG 25, JAN, 75, BA
calls c 5/share Cash flows 17,500 - 12,500
5,000.
40
THE OCC Standardization Contract size the
number of units of the underlying asset
covered in one option. Exersice
prices Mostly, increments of 2.5, 5.00
and 10.00. Exercise notice and assignment
procedures Delivery sequence.
41
  • THE OCC Standardization
  • Expiration dates Saturday, immediately
    following the third Friday of the
    expiration month.
  • The basic expiration cycles
  • JAN ? APR ? JUL ? OCT
  • FEB ? MAY ? AUG ? NOV
  • MAR ? JUN ? SEP ? DEC

42
A Review of Some Financial Economics Principles
  • Arbitrage A market situation whereby an investor
    can make a profit with
  • no equity and no risk.
  • Efficiency A market is said to be efficient if
    prices are such that there exist no arbitrage
    opportunities.
  • Alternatively, a market is said to be inefficient
    if prices present arbitrage opportunities for
    investors in this market.

43
  • Valuation The current market value (price) of
    any project or investment is the net present
    value of all the future expected cash flows from
    the project.
  • One-Price Law Any two projects whose cash flows
    are equal in every possible state of the world
    have the same market value.
  • Domination Let two projects have equal cash
    flows in all possible states of the world but
    one. The project with the higher cash flow in
    that particular state of the world has a higher
    current market value and thus, is said to
    dominate the other project.

44
  • The Holding Period Rate of Return (HPRR)
  • Buy shares of a stock on date t and sell
  • them later on date T. While holding the
  • shares, the stock has paid a cash dividend in
  • the amount of D/share.
  • The Holding Period Rate of Return HPRR is

45
Example St 50/share ST 51.5/share DT-t
1/share T t 73days.
46
  • Risk-Free Asset is a security of investment
    whose return carries no risk. Thus, the return
    on this security is known and guaranteed in
    advance.
  • Risk-Free Borrowing And Landing By purchasing
    the risk-free asset, investors lend their capital
    and by selling the risk-free asset, investors
    borrow capita at the risk-free rate.

47
  • The One-Price Law
  • There exists only one risk-free rate in an
    efficient economy.
  • Proof By contradiction. Suppose two risk-free
  • rates exist in a market and R gt r. Since both are
  • free of risk, ALL investors will try to borrow at
    r
  • and invest the money borrowed in R, thus assuring
  • themselves the difference. BUT, the excess demand
  • For borrowing at r and excess supply of lending
  • (investing) at R will change them. Supply
    demand
  • only when R r.

48
Compounded Interest (p. 76)
  • Any principal amount, P, invested at an
  • annual interest rate, R, compounded
  • annually, for n years would grow to
  • An P(1 R)n.
  • If compounded Quarterly
  • An P(1 R/4)4n.

49
  • In general
  • Invest P dollars in an account which pays an
  • annual interest rate R with m compounding
  • periods every year.
  • The rate in every period is R/m.
  • The number of compounding periods is nm.
  • Thus, P grows to
  • An P(1 R/m)mn.

50
  • An P(1 R/m)mn.
  • Monthly compounding becomes
  • An P(1 R/12)12n
  • and daily compounding yields
  • An P(1 R/365)365n.

51
  • EXAMPLES
  • n 10 years R 12 P 100
  • 1.Simple compounding, m 1, yields
  • A10 100(1 .12)10  310.5848
  • 2.Monthly compounding, m 12, yields
  • A10 100(1 .12/12)120   330.0387
  • 3.Daily compounding, m 365, yields
  • A10 100(1 .12/365)3,650 331.9462.

52
  • Notations
  • The annual rate R will be stated as Rm in order
  • to make clear how many times a year it is
  • compounded.
  • For the annual rate is 10 with quarterly
  • compounding, the corresponding formula is
  • An P(1 .10/4)4n
  • For the same annual rate with monthly
  • compounding the corresponding formula is
  • An P(1 .10/12)12n

53
  • DISCOUNTING
  • The Present Value today, date t, of a future cash
    flow, FVT, on a future date T,
  • is given by DISCOUNTING

54
  • DISCOUNTING the general case
  • Let cji, j 1,2,3,m, be a sequence of m cash
  • flows paid in year i, i 1,2,3,,n.
  • Let Rm be the annual rate during these years.
  • DISCOUNTING these cash flows yields the
  • Present Value

55
  • CONTINUOUS COMPOUNDING
  • In the early 1970s, banks came up with the
  • following economic reasoning Since the
  • bank has depositors money all the time, this
  • money should be working for the depositor
  • all the time! This idea, of course, leads to the
  • concept of continuous compounding.
  • We want to apply this idea to the formula

56
CONTINUOUS COMPOUNDING As m increases the time span of every compounding period diminishes CONTINUOUS COMPOUNDING As m increases the time span of every compounding period diminishes CONTINUOUS COMPOUNDING As m increases the time span of every compounding period diminishes
Compounding m Time span
Yearly 1 1 year
Daily 365 1 day
Hourly 8760 1 hour
Every second 3,153,600 One second
Continuously 8 Infinitesimally small
57
  • CONTINUOUS COMPOUNDING
  • This reasoning implies that in order to impose
    the concept of continuous time on the above
    compounding expression, we need to solve

This expression may be rewritten as
58
  • Recall that the number e is

X e 1 2 100 2.70481382 10,000 2.7181459
2 1,000,000 2.71828046 In the limit
2.718281828..
59
  • Recall that in our example
  • n 10 years.
  • R 12
  • P100.
  • So, P 100 invested at a 12 annual rate,
  • continuously compounded for ten years will
  • grow to

60
Continuous compounding yields the highest
return Compounding m Factor Simple 1 3.105
848208 Quarterly 4 3.262037792 Monthly 12 3.
300386895 Daily 365 3.319462164 Continuously
8 3.320116923
61
Continuous Discounting (p. 77)
This expression may be rewritten as
62
Continuous Discounting
This expression may be rewritten as
63
Recall that in our example P 100 n 10
years and R 12 Thus, 100 invested at an
annual rate of 12 , continuously compounded for
ten years will grow to 332.0117.
Therefore, we can write the continuously
discounted value of 320.0117
64
  • Equivalent Interest Rates (p.77)
  • Rm The annual rate with m compounding
    periods
  • every year.

65
  • Equivalent Interest Rates (p.77)
  • rc The annual rate with continuous
    compounding

66
  • Equivalent Interest Rates (p.77)
  • Rm The annual rate with m
  • compounding periods every year.
  • rc The annual rate with continuous
    compounding.
  • Definition Rm and rc are said to be
    equivalent
  • if

67
  • Equivalent Interest Rates (p.77)

68
  • Equivalent Interest Rates (p.77)
  • The same method applies to any two rates with
    different periods of compounding. Thus, if we
    have Rm1 and another Rm2 then the relationship
    between the two rates is

69
  • Risk-free lending and borrowing
  • Treasury bills are zero-coupon bonds, or pure
    discount bonds, issued by the Treasury.
  • A T-bill is a promissory paper which promises its
    holder the payment of the bonds Face Value (Par-
    Value) on a specific future maturity date.
  • The purchase of a T-bill is, therefore, an
    investment that pays no cash flow between the
    purchase date and the bills maturity. Hence, its
    current market price is the NPV of the bills
    Face Value
  • Pt NPVthe T-bill Face-Value
  • We will only use continuous compounding

70
  • Risk-free lending and borrowing
  • Risk-Free Asset is a security whose return is a
    known constant and it carries no risk.
  • T-bills are risk-free LENDING assets. Investors
    lend money to the Government by purchasing
    T-bills (and other Treasury notes and bonds)
  • We will assume that investors also can borrow
    money at the risk-free rate. I.e., investors may
    write IOU notes, promising the risk-free rate to
    their buyers, thereby, raising capital at the
    risk-free rate.

71
  • Risk-free lending and borrowing
  • LENDING
  • By purchasing the risk-free asset,
  • investors lend capital.
  • BORROWING
  • By selling the risk-free asset, investors borrow
    capital.
  • Both activities are at the
  • risk-free rate.

72
  • We are now ready to calculate the current value
    of a T-Bill.
  • Pt NPVthe T-bill Face-Value.
  • Thus
  • the current time, t, T-bill price, Pt , which
    pays FV upon its maturity on date T, is
  • Pt FVe-r(T-t)
  • r is the risk-free rate in the economy.

73
  • EXAMPLE Consider a T-bill that promises its
    holder FV 1,000 when it matures in 276 days,
    with a risk-free yield of 5 Inputs for the
    formula
  • FV 1,000 r .05 T-t 276/365yrs
  • Pt FVe-r(T-t)
  • Pt 1,000e-(.05)276/365
  • Pt 962.90.

74
  • EXAMPLE Calculate the yield-to -maturity of a
    bond which sells for 965 and matures in 100
    days, with FV 1,000.
  • Pt 965 FV 1,000 T-t 100/365yrs.
  • Solving for r
  • Pt FVe-r(T-t)

75
  • SHORT SELLING STOCKS (p. 97)
  • An Investor may call a broker and ask to sell a
    particular stock short.
  • This means that the investor does not own shares
    of the stock, but wishes to sell it anyway. The
    investor speculates that the stocks share price
    will fall and money will be made upon buying the
    shares back at a
  • lower price. Alas, the investor does not own
    shares of the stock. The broker will lend the
    investor shares from the brokers or a clients
    account and sell it
  • in the investors name. The investors obligation
    is to hand over the shares some time in the
    future, or upon the brokers request.

76
  • SHORT SELLING STOCKS
  • Other conditions
  • The proceeds from the short sale cannot be used
    by the short seller. Instead, they are deposited
    in an escrow account in the investors name until
    the investor makes
  • good on the promise to bring the shares back.
    Moreover, the investor must deposit an additional
    amount of at least 50 of the short sales
    proceeds in the escrow account. This additional
    amount guarantees that there
  • is enough capital to buy back the borrowed shares
    and hand them over back to the broker, in case
    the shares price increases.

77
  • SHORT SELLING STOCKS
  • There are more details associated with short
    selling stocks. For example, if the stock pays
    dividend, the short seller must pay the dividend
    to the broker. Moreover, the short seller does
    not gain interest on the amount deposited in the
    escrow account, etc.
  • We will use stock short sales in many of
    strategies associated with options trading. In
    all of these strategies, we will assume that no
    cash flow occurs from the time the strategy is
    opened with the stock short sale until the time
    the strategy terminates and the stock is
    repurchased.

78
  • SHORT SELLING STOCKS
  • In terms of cash flows per share
  • St is the cash flow/share from selling
    the stock short thereby, opening a SHORT
    POSITION on date t.
  • -ST is the cash flow from purchasing the
    stock back on date T (and delivering it to
    the lender thereby, closing the SHORT
    POSITION.)

79
  • Options Risk-Return Tradeoffs at expiration
  • PROFIT PROFILE OF A STRATEGY
  • A graph of the profit/loss as a function of all
    possible market prices of the underlying asset
  • We will begin with profit profiles at the
    options expiration I.e., an instant before the
    option expires.

80
  • Options Risk-Return Tradeoffs at Expiration
  • 1. Only at expiration ? T-t 0
  • No time value! Only intrinsic value!
  • The CALL at Expiration
  • is exercised if the CALL is
  • in-the money ST gt K
  • and the Cash flow/share ST K.
  • expires worthless if the CALL is
  • out-of-the money ST ? K
  • and the Cash flow/share 0.
  • Algebraically
  • Cash Flow/share Max0, ST K

81
  • Options Risk-Return Tradeoffs at Expiration
  • 1. Only at expiration ? T-t 0
  • No time value! Only intrinsic value!
  • The PUT at Expiration
  • is exercised if the PUT is
  • in-the money ST lt K
  • and the Cash flow/share K - ST.
  • expires worthless if the PUT is
  • out-of-the money ST gt K
  • and the Cash flow/share 0.
  • Algebraically
  • Cash Flow/share Max0, ST K

82
  • All parts of the strategy remain open till the
    options expiration.
  • 4. All parts of the strategy are closed out at
    options expiration.
  • 5. A Table Format
  • The analysis of every strategy is done with a
    table of cash flows.
  • Every row is one part (leg) of the strategy.
  • Every row is analyzed separately.
  • The cash flow of the entire strategy is the
    vertical sum of the rows.

83
  • The algebraic expressions of cash flows per
    share ICF(t) CF at Expiration(T)
  • Long stock -St ST
  • Short stock St - ST
  • Long call -ct Max0, ST -K
  • Short call ct - Max0, ST -K
  • Long put -pt Max0, K- ST
  • Short put pt - Max0, K - ST
  • The profit/loss per share is the cash flow at
    expiration plus the initial cash flow of the
    strategy, disregarding the time value of money.

84
  • The algebraic expressions of P/L per share at
  • expiration
  • P/L per share at Expiration
  • Long stock -St ST
  • Short stock St - ST
  • Long call -ct Max0, ST -K
  • Short call ct - Max0, ST -K
  • Long put -pt Max0, K- ST
  • Short put pt - Max0, K - ST

85
  • 6. A Graph of the profit/loss profile at
    expiration
  • The P/L per share from the strategy as a
    function of all possible prices of the underlying
    asset at expiration.
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