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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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