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Chapter 15 Options on Stock Indices and Currencies.ppt

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Chapter 15 Options on Stock Indices and Currencies.ppt Powered By Docstoc
					Options on stock indices,
currencies, and futures
Options On Stock Indices

       Contracts are on 100 times index; they are
        settled in cash
       On exercise of the option ,the holder of a call
        option receives (S-K)*100 in cash and the writer
        of the option pays this amount in cash ;the
        holder of a put option receives
       (K-S)*100 in cash and the writer of the option
        pays this amount in cash
       S :the value of the index
       K :the strike price
The most popular underlying indices in
the U.S.
        The Dow Jones Index (DJX)
        The Nasdaq 100 Index (NDX)
        The Russell 2000 Index (RUT)
        The S&P 100 Index (OEX)
        The S&P 500 Index (SPX)
LEAPS

   Leaps are options on stock indices that last up to 3
    years
   They have December expiration dates
   Leaps also trade on some individual stocks ,they have
    January expiration dates
Portfolio Insurance

   Consider a manager in charge of a well-diversified
    portfolio whose b is 1.0
   The dividend yield from the portfolio is the same as the
    dividend yield from the index
   The percentage changes in the value of the portfolio
    can be expected to be approximately the same as the
    percentage changes in the value of the index
Portfolio Insurance
Example
   Portfolio has a b 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?
Portfolio Insurance
Example
    Buy 5 three-month put option contracts on the index
     with a strike price of $900
    The index drops to 880 in three months
    the portfolio is worth about
     5*880*100 = $440,000
    the payoff from the options
     5*(900-880)*100 = $10,000
    total value of the portfolio
     440,000+10,000 = $450,000
When the portfolio beta is not 1.0
Example
   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?
When the portfolio beta is not 1.0
Example
   Value of index in three months               1040
   Return from change in index                 40/1000=4%
   Dividends from index                        0.25*4=1%
   Total return from index                     4+1=5%
   Risk-free interest rate                     0.25*12=3%
   Excess return from index                     5-3=2%
   Expected excess return from portfolio         2*2=4%
   Expected return from portfolio               3+4=7%
   Dividends from portfolio                    0.25*4=1%
   Expected increase in value of portfolio       7-1=6%
   Expected value of portfolio       $ 500,000*1.06= $530,000
Relationship between value of index and
value of portfolio for beta = 2.0
      Value of Index in 3    Expected Portfolio Value
           months                in 3 months ($)
            1,080                    570,000
            1,040                    530,000
            1,000                    490,000
              960                    450,000
              920                    410,000
              880                    370,000

   The correct strike price for the 10 put option
   contracts that are purchased is 960
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
Currency options
Example
   An example of a European call option

    Buy $1,000,000 euros with USD at an exchange rate of
    1.2000 USD per euro ,if the exchange rate at the
    maturity of the option is 1.2500 ,the payoff is

    1,000,000*(1.2500-1.2000) = $50,000
Currency options
Example
       An example of a European put option

        Sell $10,000,000 Australian for USD at an
        exchange rate of 0.7000 USD per Australian ,if
        the exchange rate at the maturity of the option
        is 0.6700 ,the payoff is

        10,000,000*(0.7000-0.6700) = $300,000
Range forwards
Short range-forward contract
   Buy a European put option with a strike price of K1 and
    sell a European call option with a strike price of K2
Range forwards
Long range-forward contract
   Sell a European put option with a strike price of K1 and
    buy a European call option with a strike price of K2
Options On Stocks Paying Known
Dividend Yields
   Dividends cause stock prices to reduce on the ex-dividend date
    by the amount of the dividend payment
   The payment of a dividend yield at rate q therefore cause the
    growth rate in the stock price to be less than it would otherwise
    be by an amount q
   With a dividend yield of q ,the stock price grow from S0 today to
    ST
   Without dividends it would grow from S0 today to
    STeqT
   Alternatively ,without dividends it would grow from S0e–qT today to
    ST
European Options on Stocks
Providing a Dividend Yield
        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 at
           rate q
    2.     The stock starts at price S0e–qT and pays no dividends
        We can value European options by reducing the
         stock price to S0e–qT and then behaving as though
         there is no dividend
Put-call parity

   Put-call parity for an option on a stock paying a
    dividend yield at rate q

    c  Ke  rT  p  S 0 e  qT
   For American options, the put-call parity relationship
    is

     S 0 e  qT  K  C  P  S 0  Ke  rT
Pricing Formulas
   By replacing S0 by S0e–qT in Black-Sholes formulas ,we obtain that


              c  S 0 e  qT N ( d1 )  Ke  rT N ( d 2 )
           p  Ke  rT N (  d 2 )  S 0 e  qT N (  d1 )


                    S 0 e  qT      S0
           since ln             ln     qT
                        K           K
                        ln( S 0 / K )  ( r  q   2 / 2)T
           w here d1 
                                        T
                      ln( S 0 / K )  ( r  q   2 / 2)T
                 d2 
                                       T
Risk-neutral valuation

   In a risk-neutral world ,the total return must be r ,the
    dividends provide a return of q ,the expected growth
    rate in the stock price must be r – q
   The risk-neutral process for the stock price



     dS  r  q Sdt  Sdz
Risk-neutral valuation continued
   The expected growth rate in the stock price is
    r –q ,the expected stock price at time T is S0e(r-q)T
   Expected payoff for a call option in a risk-neutral world as




     e ( r q )TdS 0 N (d1 )  KN (d 2 )as above
    Where and d are defined
             1       2
   Discounting at rate r for the T



    c  S 0 e  qT N (d1 )  Ke  rT N (d 2 )
Valuation of European Stock Index
Options
   We can use the formula for an option on a stock
    paying a dividend yield

    c  S0e  qT N (d1 )  Ke  rT N (d 2 )
    p  Ke rT N (d 2 )  S0e qT N (d1 )
   S0 : the value of index
    q : average dividend yield
      : the volatility of the index
    
Example

   A European call option on the S&P 500 that is two
    months from maturity
   S0 = 930, K = 900, r = 0.08
    = 0.2, T = 2/12
   Dividend yields of 0.2% and 0.3% are expected in
    the first month and the second month
Example continued

           The total dividend yield per annum is
            q = (0.2% + 0.3%)*6 = 3%
      ln(930 / 900)  (0.08  0.03  0.2 2 / 2)  2 / 12
d1                                                        0.5444
                         0.2 2 / 12
       ln(930 / 900)  (0.08  0.03  0.2 2 / 2)  2 / 12
d2                                                        0.4628
                         0.2 2 / 12
N (d1 )  0.7069
N (d 2 )  0.6782
c  930  0.7069e 0.032 /12  900  0.6782e 0.082 /12  51.83


         one contract would cost $5,183
Forward price
   Define F0 as the forward price of the index
   F0  S 0 e r  q T

     c  F0 e  rT N ( d1 )  Ke  rT N ( d 2 )
    p  Ke  rT N (  d 2 )  F0 e  rT N (  d1 )


                ln( F0 / K )  ( 2 / 2)T
    w here d1 
                           T
                ln( F0 / K )  ( 2 / 2)T
           d2 
                           T
Implied dividend yields
   By   c  Ke  rT  p  S 0 e  qT

 S 0 e  qT  c  p  Ke  rT
             c  p  Ke  rT
 e  qT   
                   S0
            c  p  Ke  rT
  qT  ln
                  S0
       1   c  p  Ke  rT
 q   ln
       T         S0
Valuation of European Currency Options

   A foreign currency is analogous to a stock paying a known dividend
    yield
   The owner of foreign receives a yield equal to the risk-free interest rate,
    rf
   With q replaced by rf , we can get call price, c, and put price, p

                       rf T
             c  S0e           N ( d1 )  Ke  rT N ( d 2 )
                                                 rf T
             p  Ke  rT N (  d 2 )  S 0 e             N (  d1 )


                                 ln( S 0 / K )  ( r  r f   2 / 2)T
             w here d1 
                                                   T
                                 ln( S 0 / K )  ( r  r f   2 / 2)T
                       d2 
                                                   T
Using forward exchange rates

   Define F0 as the forward foreign exchange rate
            r rf T
   F0  S0e

           c  e  rT F0 N (d1 )  KN (d 2 )
           p  e  rT KN ( d 2 )  F0 N ( d1 )


                      ln( F0 / K )  ( 2 / 2)T
           where d1 
                                 T
                      ln( F0 / K )  ( 2 / 2)T
                 d2 
                                 T
American Options
   The parameter determining the size of up movements, u, the
    parameter determining the size of down movements, d
   The probability of an up movement is


                   ad
                 
    In the casepof options on an index

                   ud

      a  e r  q t
   In the case of options on a currency


              r  r f t
      ae
Nature of Futures Options
 A call futures is the right to enter into a long futures
  contracts at a certain price.
 A put futures is the right to enter into a short futures
  contracts at a certain price.
 Most are American; Be exercised any time during the
  life .



                                                     30
Nature of Futures Options

 When a call futures option is exercised the holder acquires :
1. A long position in the futures
2. A cash amount equal to the excess of
    the futures price over the strike price


 If the futures position is closed out immediately:
  Payoff from call = F0 – K
  where F0 is futures price at time of exercise

                                                         31
Example
   Today is 8/15, One September futures call option on
    copper,K=240(cents/pound), One contract is on 25,000 pounds of copper.
   Futures price for delivery in Sep is currently 251cents
   8/14 (the last settlement) futures price is 250

IF option exercised, investor receive cash:
 25,000X(250-240)cents=$2,500

Plus a long futures, if it closed out immediately:
25,000X(251-250)cents=$250

If the futures position is closed out immediately
 25,000X (251-240)cents=$2,750
                                                                   32
Nature of Futures Options
 When a put futures option is exercised the holder acquires :
 1. A short position in the futures
 2. A cash amount equal to the excess of
    the strike price over the futures price



 If the futures position is closed out immediately:
  Payoff from put =K–F0
  where F0 is futures price at time of exercise

                                                         33
Reasons for the popularity on futures
option

  Liquid and easier to trade
  From futures exchange, price is known immediately.
  Normally settled in cash
  Futures and futures options are traded in the same exchange.
  Lower cost than spot options




                                                        34
       European spot and futures options

   Payoff from call option with strike price K on spot price
    of an asset:

           max(ST  K ,0)
   Payoff from call option with strike price K on futures
    price of an asset:

            max(FT  K ,0)
   When futures contracts matures at the same time as the
    option:
             FT  ST
                                                        35
Put-Call Parity for futures options
 Consider the following two portfolios:
 A. European call plus Ke-rT of cash
 B. European put plus long futures plus cash equal to F0e-rT


 They must be worth the same at time T so that

          c+Ke-rT=p+F0 e-rT



                                                         36
Example
European call option on spot silver for delivery in six month
F0  $8, c  $0.56 , K  $8.50 ,
Rf  10 %
Use equation

    c+Ke-rT=p+F0 e-rT

   →P=c+Ke-rT-F0 e-rT
     P=0.56+8.50e-0.1X0.5-8 e-0.1X0.5
     =1.04
                                                                37
Bounds for futures options
 By
             c+Ke-rT=p+F0 e-rT
         c  Ke  rT  F0 e  rT
 or     c  ( F0  K )e  rT
 Similarly
             Ke  rT  F0 e  rT  P
                                rT
 or          P  ( K  F0 )e

                                       38
Bounds for futures options

Because American futures can be exercised at any time,
 we must have
                   c  F0  K

                    P  K  F0


                                                         39
Valuation by Binomial trees
A 1-month call option on futures has a strike price of 29.



                                         Futures Price = $33
                                         Option Price = $4
 Futures price = $30
 Option Price=?
                                         Futures Price = $28
                                         Option Price = $0



                                                      40
Valuation by Binomial trees
 • Consider the Portfolio: long  futures
                           short 1 call option


                                   3 – 4


                                   -2
 • Portfolio is riskless when 3 – 4 = -2 or
    = 0.8



                                                 41
Valuation by Binomial trees

• The riskless portfolio is:
  long 0.8 futures
  short 1 call option
• The value of the portfolio in 1 month
   is -1.6
• The value of the portfolio today is
  -1.6e – 0.06/12 = -1.592


                                          42
A Generalization
 • A derivative lasts for time T and
  is dependent on a futures price

                            F0u
                            ƒu
           F0
           ƒ                F0d
                            ƒd

                                       43
A Generalization
• Consider the portfolio that is long Δ futures
  and short 1 derivative
                                   F0u   F0  – ƒu


                                F0d  F0 – ƒd


• The portfolio is riskless when
                         fu  f d
                    
                       F0u  F0 d
                                                       44
A Generalization

   • Value of the portfolio at time T is
        F0u Δ –F0 Δ – ƒu

   • Value of portfolio today is – ƒ

   • Hence
       ƒ = – [F0u Δ –F0 Δ – ƒu]e-rT



                                           45
A Generalization

  • Substituting for Δ we obtain
  ƒ = [ p ƒu + (1 – p )ƒd ]e–rT


  where
                    1 d
                 p
                    ud


                                   46
Drift of a futures price in a risk-neutral
word
Valuing European Futures Options
 We can use the formula for an option on a stock
  paying a dividend yield
     Set S0 = current futures price (F0)
     Set q = domestic risk-free rate (r )
 Setting q = r ensures that the expected growth of F
  in a risk-neutral world is zero




                                                   47
      Growth Rates For Futures Prices


• A futures contract requires no initial investment
• In a risk-neutral world the expected return
  should be zero
• The expected growth rate of the futures price is
  therefore zero
• The futures price can therefore be treated like a
  stock paying a dividend yield of r




                                               48
                      Results


  c  S 0 e  qT N ( d1 )  Ke  rT N (d 2 )
    p  Ke  rT N (  d 2 )  S 0 e  qT N (  d1 )
           ln( S 0 / K )  ( r  q   2 / 2)T
where d1 
                           T
           ln( S 0 / K )  (r  q   2 / 2)T
      d2 
                           T




                                                      49
Black’s Formula
• The formulas for European options on futures are
  known as Black’s formulas
  c  e  rT F0 N (d1 )  K N (d 2 )
  p  e rT K N (d 2 )  F0 N (d1 )
              ln( F0 / K )   2T / 2
  where d1 
                        T
            ln( F0 / K )   2T / 2
       d2                            d1   T
                      T
                                                  50
Example

   European put futures option on crude oil,…
  Fo=20,K=20, r=0.09,T=4/12,σ=0.25,ln(F0=K)=0,
  So that
         T
 d1           0.07216
          2
           T
 d2            0.07216
            2
 N ( d1 )  0.4712
 N ( d 2 )  0.5288
  p  e 0.09 X 4 /12 20  0.5288  20  0.4712  $1.12

                                                            51
American futures options vs spot options

• If futures prices are higher than spot prices
  (normal market), an American call on futures is
  worth more than a similar American call on spot.
  An American put on futures is worth less than a
  similar American put on spot
• When futures prices are lower than spot prices
  (inverted market) the reverse is true




                                               52

				
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