Options on Stock Indices, Currencies, and Futures Chapter 13

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Options onStock Indices, Currencies, and
FuturesChapter 13
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STOCK INDEXES (INDICES) A STOCK INDEX IS A
SINGLE NUMBER BASED ON INFORMATION ASSOCIATED
WITH A PORTFOILO OF STOCKS. A STOCK INDEX IS
SOME KIND OF AN AVERAGE OF THE PRICES AND THE
QUANTITIES OF THE SHARES OF THE STOCKS THAT ARE
INCLUDED IN THE PORTFOLIO THAT UNDERLYING THE
INDEX.
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STOCK INDEXES (INDICES) THE MOST USED INDEXES
ARE A SIMPLE PRICE AVERAGE AND A VALUE
WEIGHTED AVERAGE.
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STOCK INDEXES - THE CASH MARKET A. AVERAGE PRICE
INDEXES DJIA, MMI N The number of stocks in
the index P Stock market price D
Divisor INITIALLY, D N AND THE INDEX IS
SET AT A GIVEN LEVEL. TO ASSURE INDEX CONTINUITY,
THE DIVISOR IS ADJUSTED OVER TIME.
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EXAMPLES OF INDEX ADJUSMENTS STOCK SPLITS 2 FOR
1 1. 2. Before the split (30 40 50 60
20) /5 40 I 40 and D 5. An instant
later (30 20 50 60 20)/D 40 The
new divisor is D 4.5
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CHANGE OF STOCKS IN THE INDEX 1. 2. Before the
change (31 19 53 59 18)/4.5 40
I 40 and D 4.5. An instant later (31
149 53 59 18)/D 40 The new divisor
is D 7.75
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A STOCK DIVIDEND DISTRIBUTION Firm 4 distributes
66 2/3 stock dividend. Before the
distribution (32 113 52 58 25)/7.75
36.129 D 7.75. An instant later (32 113
52 34.8 25)/D 36.129 The new divisor
is D 7.107857587.
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STOCK 2 SPLIT 3 FOR 1. Before the split (31
111 54 35 23)/7.107857587 35.7351
An instant later (31 37 54 35 23)/D
35.73507 The new Divisor is D 5.0370644.
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ADDITIONAL STOCKS 1. 2. Before the stock
addition (30 39 55 33 21)/5.0370644
35.338 An instant later (30 39 55
33 21 35)/D 35.338 D 6.0275.
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A price adjustment of Altria Group Inc. (MO),
(due to a distribution of Kraft Foods Inc. (KFT)
shares,) was effective for the open of trade on
trade date April 2, 2007. As a result, the new
divisor for the DJIA became  D 0.123051408.  
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VALUE WEIGHTED INDEXES S P500, NIKKEI 225,
VALUE LINE B SOME BASE TIME PERIOD INITIALLY
t B THUS, THE INITIAL INDEX VALUE IS SOME
ARBITRARILY CHOSEN VALUE M.
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The SP500 index base period was 1941-1943 and
its initial value was set at M 10. The NYSE
index base period was Dec. 31, 1965 and its
initial value was set at M 50.
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The rate of return on the index The return on a
value weighted index in any period t, is the
weighted average of the individual stock returns
the weights are the dollar value of the stock as
a proportion of the entire index value.
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THE RATE OF RETURN ON THE INDEX
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THE BETA OF A PORTFOLIO THEOREM A PORTFOLIOS
BETA IS THE WEIGHTED AVERAGE OF THE BETAS OF THE
STOCKS THAT COMPRISE THE PORTFOLIO. THE WEIGHTS
ARE THE DOLLAR VALUE WEIGHTS OF THE STOCKS IN THE
PORTFOLIO.
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THE BETA OF A PORTFOLIO Proof Assume that the
index is a well diversified portfolio. I.e., the
index represents the market portfolio. Let
P denote the portfolio underlying the Index,
I. j denote the individual stock j 1, 2, ,N.
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By definition
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STOCK PORTFOLIO BETA
STOCK NAME PRICE SHARES
VALUE WEIGHT BETA
?P (.044)(1.00) (.152)(.8) (.046)(.5)
(.061)(.7) (.147)(1.1) (.178)(1.1)
(.144)(1.4) (.227)(1.2) 1.06
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A STOCK PORTFOLIO BETA STOCK NAME PRICE
SHARES VALUE WEIGHT
BETA
?P .122(.95) .187(1.1) .203(.85)
.048(1.15) .059(1.15) .076(1.0) .263(.85)
.042(.75) .95
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Sources of calculated Betas and calculation
inputs Example ß(GE) 6/20/00 Source ß(GE)
Index Data Horizon Value Line Investment
Survey 1.25 NYSECI Weekly
Price 5 yrs (Monthly) Bloomberg
1.21 SP500I Weekly Price
2 yrs (Weekly) Bridge Information Systems
1.13 SP500I Daily Price 2
yrs (daily) Nasdaq Stock Exchange
1.14 Media General Fin. Svcs. (MGFS)
SP500I Monthly P ice 3 (5) yrs
Quicken.Excite.com 1.23 MSN
Money Central
1.20 DailyStock.com
1.21 Standard Poors Compustat Svcs
SP500I Monthly Price 5 yrs
(Monthly) SP Personal Wealth
1.2287 SP Company Report)
1.23 Charles Schwab Equity Report Card 1.20 SP
Stock Report
1.23 AArgus Company Report 1.12
SP500I Daily Price 5 yrs
(Daily) Market Guide
SP500I
Monthly Price 5 yrs (Monthly) YYahoo!Finance
1.23 Motley Fool
1.23
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STOCK INDEX OPTIONS ONE CONTRACT VALUE
(INDEX VALUE)(MULTIPLIER) One contract
(I)(m) ACCOUNTS ARE SETTLED BY CASH
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EXAMPLE Options on a stock index MoneyGone, a
financial institution, offers its clients the
following deal Invest A 1,000,000 for 6
months. In 6 months you receive a guaranteed
return The Greater of 0, or 50 of the return
on the SP500I during these 6 months. For
comparison purposes The annual risk-free rate is
8. The SP500I dividend payout ratio is q 3
and its annual VOL s 25.
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MoneyGone offer Deposit A now. Receive
AMax0, .5RI in 6 months. Denote the date in
six month T. Rewrite MoneyGone offer at T
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The expression
is equivalent to the at-expiration cash flow of
an at-the money European call option on the
index, if you notice that K I0. Calculate this
options value based on S0 K I0 T t .5
r .08 q .03 and s .25. Using
DerivaGem c .08137. Thus, MoneyGones promise
is equivalent
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to giving the client NOW, at time 0, a value
of (.5)(.08137)(A) .040685A. Therefore, the
investors initial deposit is only 95.9315 of
A. Investing .959315A and receiving A in six
months, yields a guaranteed return of 8.3
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• STOCK INDEX OPTIONS FOR
• STATIC PORTFOLIO INSURANCE
• Decisions
• How many puts to buy?
• Which exercise price will guarantee
• a desired level of protection?
• The answers are not easy because the
• index underlying the puts is not the same
• as portfolio to be protected.

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The protective put with a single stock
STRATEGY ICF AT EXPIRATION AT EXPIRATION
STRATEGY ICF ST lt K ST K
Hold the stock Buy put -St -p ST K - ST ST 0
TOTAL -St p K ST
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The protective put consists of holding the
portfolio and purchasing n puts.
STRATEGY ICF (t 0) AT EXPIRATION (T 1) AT EXPIRATION (T 1)
STRATEGY ICF (t 0) I1 lt K I1 K
Hold the portfolio Buy n puts -V0 -n P(m) V1 n(K- I1)(m) V1 0
TOTAL -V0 nP(m) V1n(m)(K- I1) V1
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WE USE THE CAPITAL ASSET PRICING MODEL. For
any security i,the expected excess return on the
security and the expected excess return on the
market portfolio are linearly related by their
beta
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THE INDEX TO BE USED IN THE STRATEGY, IS TAKEN
TO BE A PROXY FOR THE MARKET PORTFOLIO, M.
FIRST, REWRITE THE ABOVE EQUATION FOR THE INDEX I
AND ANY PORTFOLIO P
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Second, rewrite the CAPM result, with actual
returns
In a more refined way, using V and I for the
portfolio and index market values, respectively
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NEXT, use the ratio Dp/V0 as the portfolios
annual dividend payout ratio qP and DI/I0 the
index annual dividend payout ratio, qI.
The ratio V1/ V0 indicates the portfolio required
protection ratio.
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For example
The manager wants V1, to drop down to No less
than 90 of the initial portfolio market value,
V0 V1 V0(.9). We denote this desired level
by (V1/ V0). This is the managers decision
variable about the amount of protection to obtain
by the protective puts.
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1. The number of puts is
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2. The exercise price, K, is determined by
substituting I1 K and the required level, (V1/
V0) into the equation
and solving for K
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We rewrite the Profit/Loss table for the
protective put strategy
STRATEGY INITIAL CASH FLOW AT EXPIRATION AT EXPIRATION
STRATEGY INITIAL CASH FLOW I1 lt K I1 K
Hold the portfolio Buy n puts -V0 -n P(m) V1 n(K - I1)(m) V1 0
TOTAL -V0 - nP(m) V1n(m)(K - I1) V1
We are now ready to calculate the floor level of
the portfolio V1n(m)(K- I1)
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We are now ready to calculate the floor level of
the portfolio Min portfolio value V1n(m)(K-
I1) This is the lowest level that the portfolio
value can attain. If the index falls below the
exercise price and the portfolio value declines
too, the protective puts will be exercised and
the money gained may be invested in the portfolio
and bring it to the value of V1n(m)K- n(m)I1
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Substitute for n
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To substitute for V1 we solve the equation
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3. Substitution V1 into the equation for the Min
portfolio value
The desired level of protection is made at time
0. This determines the exercise price and
management can also calculate the minimum
portfolio value.
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Conclusion The STATIC Portfolio Insurance
strategy is to BUY puts and hold them for the
entire insurance period.
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EXAMPLE A portfolio manager expects the market
to fall by 25 in the next six months. The
current portfolio value is 25M. The manager
decides on a 90 hedge by purchasing 6-month puts
on the SP500 index. The portfolios beta with
the SP500 index is 2.4. The SP500 index stands
at a level of 1,250 points and its dollar
multiplier is 100. The annual risk-free rate is
10, while the portfolio and the index annual
dividend payout ratios are 5 and 6,
respectively. The data is summarized below
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Solution Purchase
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The exercise price of the puts is
Solution Purchase n 480 six-months puts with
exercise price K 1,210.
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CONCLUSION Hold the portfolio. Purchase 480,
6-months protective puts on the SP500 index,
with the exercise price K 1210. The portfolio
value, currently 25M, will not fall below
22,505,000 during the insurance period of the
next six months.
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Example Portfolio insurance for 3 months
Solution Purchase
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The exercise price of the puts is
Solution Purchase n 10 three-months puts with
exercise price K 960.
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CONCLUSION Hold the portfolio. Purchase 10,
3-months protective puts on the SP500 index,
with the exercise price K 960. Insurance The
portfolio value, currently 500,000, will not
fall below 450,000 in three months.
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Table 13.2
?P 2. V0 500,000. n 10 puts ?P 2. V0 500,000. n 10 puts ?P 2. V0 500,000. n 10 puts
V1/V0 V1 K
1.14 570,000 1,080
1.06 530,000 1,040
.98 490,000 1,000
.90 450,000 960
.82 410,000 920
.74 370,000 880
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A SPECIAL CASE In the case that a) ß 1
and b) qP qI, the portfolio is statistically
similar to the index. In this case
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Assume that in the above example ßp 1 and qP
qI, then
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Example (page 295,6) ßp 1 and qP qI, then
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Dynamic Portfolio Insurance