# Options on Stock Indices and Currencies

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```					                    12.1

Options on
Stock Indices and
Currencies

Chapter 12
12.2
European Options on Stocks
Paying Dividend Yields
We get the same probability
distribution for the stock price at time
T in each of the following cases:
1. The stock starts at price S0 and
provides a dividend yield = q
2. The stock starts at price S0e–q T
and provides no income
12.3
European Options on Stocks
Paying Dividend Yield
continued
We can value European options by
reducing the stock price to S0e–q T and
then behaving as though there is no
dividend
12.4

Extension of Chapter 8 Results
(Equations 12.1 to 12.3)
Lower Bound for calls:
 qT           rT
c  S0 e                Xe
Lower Bound for puts

 rT             qT
p  Xe                  S0 e
Put Call Parity

 rT                 qT
c  Xe            p  S0 e
12.5
Extension of Chapter 11
Results (Equations 12.4 and 12.5)
c  S0e  qT N (d1 )  Xe  rT N (d 2 )
p  Xe  rT N (  d 2 )  S0e  qT N (  d1 )
ln( S0 / X )  (r  q   2 / 2) T
where d1 
 T
ln( S0 / X )  (r  q   2 / 2) T
d2 
 T
12.6

The Binomial Model
S0u
ƒu
S0
ƒ
S0d
ƒd

f=e-rT[pfu+(1– p)fd ]
12.7
The Binomial Model
continued
• In a risk-neutral world the stock price
grows at r-q rather than at r when there
is a dividend yield at rate q
• The probability, p, of an up movement
must therefore satisfy
pS0u+(1 – p)S0d=S0e (r-q)T
so that
( r q ) T
e           d
p
u d
12.8
Index Options
• The most popular underlying indices in the
U.S. are
–   The Dow Jones Index times 0.01 (DJX)
–   The Nasdaq 100 Index (NDX)
–   The Russell 2000 Index (RUT)
–   The S&P 100 Index (OEX)
–   The S&P 500 Index (SPX)
• Contracts are on 100 times index; they are
settled in cash; OEX is American and the
rest are European.
12.9

LEAPS
• Leaps are options on stock indices that
last up to 3 years
• They have December expiration dates
• They are on 10 times the index
• Leaps also trade on some individual
stocks
12.10

Index Option Example

• Consider a call option on an
index with a strike price of 560
• Suppose 1 contract is exercised
when the index level is 580
• What is the payoff?
12.11
Using Index Options for
Portfolio Insurance
• Suppose the value of the index is S0 and the strike
price is X
• If a portfolio has a b of 1.0, the portfolio insurance
is obtained by buying 1 put option contract on the
index for each 100S0 dollars held
• If the b is not 1.0, the portfolio manager buys b put
options for each 100S0 dollars held
• In both cases, X is chosen to give the appropriate
insurance level
12.12

Example 1 (Table 12.2, page 262)
•   Portfolio has a beta of 1.0
•   It is currently worth \$500,000
•   The index currently stands at 1000
•   What trade is necessary to provide
insurance against the portfolio value
falling below \$450,000?
12.13

Example 2 (Table 12.5, page 264)
• Portfolio has a beta of 2.0
• It is currently worth \$500,000 and index
stands at 1000
• The risk-free rate is 12% per annum
• The dividend yield on both the portfolio
and the index is 4%
• How many put option contracts should
be purchased for portfolio insurance?
12.14
Calculating Relation Between Index
Level and Portfolio Value in 3 months

• If index rises to 1040, it provides a
40/1000 or 4% return in 3 months
• Total return (incl. dividends)=5%
• Excess return over risk-free rate=2%
• Excess return for portfolio=4%
• Increase in Portfolio Value=4+3–1=6%
• Portfolio value=\$530,000
12.15
Determining the Strike Price
(Table 12.4, page 263)
Value of Index in 3      Expected Portfolio Value
months                  in 3 months (\$)
1,080                     570,000
1,040                     530,000
1,000                     490,000
960                     450,000
920                     410,000
880                     370,000
An option with a strike price of 960 will provide protection
against a 10% decline in the portfolio value
12.16
Valuing European Index
Options
We can use the formula for an option
on a stock paying a continuous dividend
yield
Set S0 = current index level
Set q = average dividend yield expected
during the life of the option
12.17

Currency Options
• Currency options trade on the Philadelphia
Exchange (PHLX)
• There also exists an active over-the-counter
(OTC) market
• Currency options are used by corporations
to buy insurance when they have an FX
exposure
12.18
The Foreign Interest Rate
• We denote the foreign interest rate by rf
• When a U.S. company buys one unit of
the foreign currency it has an
investment of S0 dollars
• The return from investing at the foreign
rate is rf S0 dollars
• This shows that the foreign currency
provides a “dividend yield” at rate rf
12.19
Valuing European Currency
Options
• A foreign currency is an asset that
provides a continuous “dividend yield”
equal to rf
• We can use the formula for an option
on a stock paying a continuous
dividend yield :
Set S0 = current exchange rate
Set q = rƒ
12.20
Formulas for European
Currency Options
(Equations 12.9 and 12.10, page 266)
 rf T
c  S0 e            N (d1 )  Xe  rT N ( d 2 )
 rT                           rf T
p  Xe              N (  d 2 )  S0 e            N (  d1 )
ln( S0 / X )  (r  r                  2 / 2) T
f
where d1 
 T
ln( S0 / X )  ( r  r               2 / 2) T
f
d2 
 T
12.21
Alternative Formulas
(Equations 12.11 and 12.12, page 267)
( r rf ) T
Using        F0  S0 e

c  e  rT [ F0 N (d1 )  XN ( d 2 )]
p  e  rT [ XN (  d 2 )  F0 N (  d1 )]
ln( F0 / X )   T / 2
2
d1 
 T
d 2  d1   T

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 views: 6 posted: 5/8/2012 language: pages: 21