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					          Call Option Example
• Suppose that at the February expiry date we
  hold one April $6 ABC Inc. call (the exercise
  price is $6). If the price of a share in ABC Inc.
  is S = $7, would the call be exercised?
      A: Yes (buy @ $6 and sell @ $7)

  Would it be exercised if S = $5.50?
    A: No (buy @ $6 ?? No way)
                                                      E.5
         Basic Call Option Pricing
               Relationships
• At expiry, an American call option is worth the same as a
  European option with the same characteristics.
• If the call is in-the-money, it is worth ST - X.
• If the call is out-of-the-money, it is worthless.
  For a call, the intrinsic value is given by:
  CaT = CeT = Max[ST - X, 0]
• Where
       ST is the value of the stock at expiry (time T)
       X is the exercise price.
       CaT is the value of an American call at expiry
       CeT is the value of a European call at expiry

                                                              E.6
                              Call Option Payoffs
                     6

                     4                                          Buy a call
Option payoffs ($)




                     2

                      0
                          0    1     2      3   4   5   6       7    8       9   10

                     -2       Stock price ($)
                                                            Write a call
                     -4

                     -6

                              Exercise price = $6
                                                                                      E.7
                               Call Option Profits
                     6
                                                         Breakeven = Exercise + Premium

                     4                                                 Buy a call
Option profits ($)




                     2

                      0
                          0    1      2      3   4   5       6     7      8     9   10

                     -2        Stock price ($)
                                                                 Write a call
                     -4

                     -6

                              Exercise price = $6; option premium = $1
                                                                                          E.8
   Basic Put Option Pricing Relationships

• At expiry, an American put option is worth
  the same as a European option with the
  same characteristics.
• If the put is in-the-money, it is worth X - ST.
• If the put is out-of-the-money, it is
  worthless. For a put, the intrinsic value is given
  by:
           PaT = PeT = Max[X - ST, 0]

                                                       E.9
                               Put Option Payoffs
                     6
                                          Buy a put
                     4
Option payoffs ($)




                     2

                      0
                          0    1     2      3     4      5   6   7   8   9   10

                     -2       Stock price ($)


                     -4                    Write a put


                     -6

                              Exercise price = $6
                                                                                  E.10
Option profits ($)               Put Option Profits
                     6
                                                 Breakeven = Exercise + Premium
                     4

                      2                                             Write a put
                      1
                       0
                           0    1     2      3      4     5     6    7   8     9   10
                     -1                                              Buy a put
                     -2        Stock price ($)


                     -4

                     -6

                               Exercise price = $6; option premium = $1
                                                                                        E.11
      More Options Relationships &
              Definitions
• Option premium: this is the value of the option
• Speculative Value OR Time Value
  – The difference between the option premium and the
    intrinsic value of the option. This is the part of the
    option’s value that derives the possibility of future
    favorable movements in the stock price.

     Option             Intrinsic      Time Value
    Premium       =      Value
                                     +
                                                             E.12
                  Put-Call Parity
c0  p0  S 0  PV ( X )
where c0    = value of call
       p0      = value of put
       S0      = value of share
       PV(X) = exercise price discounted at
              risk-free rate
• Note This relationship ONLY holds for European options but
  American options it becomes as follows:
     S 0  X  c 0  p 0  S 0  PV ( X )
                                                               E.13
Valuing Options
• The last section concerned itself with the
  value of an option at expiry.
• This section considers the value of an
  option prior to the expiration date.
• A much more interesting question.


                                               E.14
        Option Value Determinants
                                         Call   Put
1.   Stock price                          +     –
2.   Exercise price                       –     +
3.   Interest rate                        +     –
4.   Volatility in the stock price        +     +
5.   Expiration date                      +     +

        The value of a call option C0 must fall within
                 max (S0 – E, 0) < C0 < S0.

      The precise position will depend on these factors.
                                                           E.15
        Binomial Pricing Models
• Dynamic tracking strategy:
  – used for derivatives whose future payoff is not a
    linear function of payoff of the underlying asset
• Two types:
  – one-period binomial trees
     • tracking-portfolio method
     • risk-neutral valuation method
  – multiperiod binomial trees

                                                        E.16
   Binomial Process (One Period)
• At each point (node), there are only two
  possible values
  – up
  – down                   £620 (up)


    £570

                           £480 (down)
• Good approximation
                                             E.17
   The Tracking-Portfolio Method
• Identify the tracking portfolio
  –  units (shares) of the underlying asset
  –  £’s in the risk-free bond
• Find the current value of the option




                                               E.18
Identifying the Tracking Portfolio

             Su      S =    value of
                             underlying today
S                    Su =   value of underlying in
                             up state
             Sd      Sd =   value of underlying in
                             down state
            (1+rf)
                      rf =   risk-free rate


            (1+rf)
                                                E.19
     Identifying the Tracking Portfolio

                                          t=1
                  t=0           Up                Down
Tracking         S+       Su+(1+rf)         Sd+(1+rf)
portfolio




                                                     =
                                 =
Derivative         ?            Vu                  Vd



System of two equations with two unknowns

                                                              E.20
    Identifying the Tracking Portfolio
Su   (1  rf )  Vu     (1)
S d   (1  rf )  Vd    (2)
Subtracting (2) from (1)
 Su  S d  Vu  Vd
    Vu  Vd spread option prices
                                (3)
    Su  S d   spread share prices
    Vu  S u
               (4)
     1  rf                              E.21
Finding the Current Value of the Option

• If arbitrage doesn’t exist:

  V  S               (5)

Example:
  – what is the current value of a call option with a
    strike price (X) of 90?



                                                        E.22
             Valuing a Call Option I
                          Su=120
                          Vu=30 = 120 - 90
 S=100                                 rf=0.10
 V=?
                          Sd=80
                          Vd=0
                                          t=1
                  t=0             Up             Down
Tracking         100+       120+1.1         80 +1.1
portfolio
Derivative        V?              30               0
                                                            E.23
        Valuing a Call Option I
    Vu  Vd   30  0   3
                  
    Su  S d 120  80 4
                    3
               30  120
    Vu  Su
                 4     54 .55
     1  rf        1.1
Under no arbitrage:
               3
 V  S    100  (54.55)  20.45
               4
                                       E.24
                 Two Remarks
• The valuation process does not refer to the
  probabilities of the up and down states
  – this information is already reflected in the current
    price of the underlying
• The process does not refer to the risk
  preferences of the investor



                                                           E.25
  The Risk-Neutral Valuation Method

Two-step procedure for the valuation of
derivatives:
  1. determine the risk neutral probabilities given that
     investors are risk neutral
  2. multiply each future value of the option by the
     corresponding risk-neutral probability and discount
     the sum of these values by the risk-free rate


                                                           E.26
         Valuing a Call Option II
                           Su=120
                           Vu=30
 S=100                              rf=0.10
 V=?
                           Sd=80
                           Vd=0

1. Determine the risk neutral probabilities
         120     80
   1.1           (1   )
         100    100
      0.75

                                              E.27
         Valuing a Call Option II
2. Multiply the future values of the option by their
   probabilities and discount at the risk-free rate
  – we determine the expected future value of the
    option
     0.75  30  (1  0.75)  0  22.50

  – we determine the present value of the option
    22.50
           20.45
     1.1
                                                       E.28
  The Risk-Neutral Valuation Method

General formula:
  – the risk-neutral probability  is the probability that
    makes the future expected return of an asset equal to
    the risk-free rate
                                          (1  r f )  d
    u  (1   )d  1  r f   
                                              ud
    where
     u= 1 + return of the underlying in the up state
     d= 1 + return of the underlying in the down state
                                                             E.29
       Multiperiod Binomial Trees
• In real life it is more likely to have more than two prices
  for the security over a long period say of 3 months,
  therefore, it is better to apply to model over shorter
  periods of time.
• Then, if we extend the model for n intervals, we can then
  work out the possible movements of the security in the
  up-state and down-state as follows:
            / n
     u e                   AND             d = 1/u
  where n is the number of intervals in a year,  is the S.D of
  the annualized stock return
                                                                  E.30
         The Black-Scholes Model
• Discrete valuation models:
   – finite number of future outcomes for the share price
   – finite number of periods
• Continuous-time models:
   – infinite number of future outcomes for the share price
   – infinite number of periods
• A binomial process with a large number of short
  periods can approximate the continuous-time process.

                                                              E.31
       The Black-Scholes Model
                   C0  S  N(d1 )  Ee rT  N(d 2 )
 Where
 C0 = the value of a European option at time t = 0
 r = the risk-free interest rate.
                         σ2          N(d) = Probability that a
       ln( S / E )  (r  )T         standardized, normally
  d1                    2
                 T                  distributed, random variable
                                     will be less than or equal to
  d 2  d1   T                     d.
The Black-Scholes Model allows us to value options in the real
  world just as we have done in the 2-state world.
                                                                     E.32
        The Black-Scholes Model
Find the value of a six-month call option on the
  Microsoft with an exercise price of $150
The current value of a share of Microsoft is $160
The interest rate available in the U.S. is r = 5%.
The option maturity is 6 months (half of a year).
The volatility of the underlying asset is 30% per annum.
Before we start, note that the intrinsic value of the option
  is $10—our answer must be at least that amount.

                                                               E.33
        The Black-Scholes Model

 First calculate d1 and d2
       ln( S / E )  (r  .5σ 2 )T
  d1 
                   T
         ln( 160 / 150)  (.05  .5(0.30) 2 ).5
   d1                                           0.5282
                        0.30 .5

Then,
  d 2  d1   T  0.52815  0.30 .5  0.31602
                                                           E.34
   The Black-Scholes Model
            C0  S  N(d1 )  Ee rT  N(d 2 )
d1  0.5282          N(d1) = N(0.52815) = 0.7013
d 2  0.31602        N(d2) = N(0.31602) = 0.62401

  C0  $160  0.7013  150e .05.5  0.62401
  C0  $20.92



                                                    E.35
    Stocks and Bonds as Options
• Levered Equity is a Call Option.
   – The underlying asset comprise the assets of the firm.
   – The strike price is the payoff of the bond.
• If at the maturity of their debt, the assets of the
  firm are greater in value than the debt, the
  shareholders have an in-the-money call, they will
  pay the bondholders and “call in” the assets of
  the firm.
• If at the maturity of the debt the shareholders
  have an out-of-the-money call, they will not pay
  the bondholders (i.e. the shareholders will
  declare bankruptcy) and let the call expire.        E.37
    Stocks and Bonds as Options
• Levered Equity is a Put Option.
   – The underlying asset comprise the assets of the firm.
   – The strike price is the payoff of the bond.
• If at the maturity of their debt, the assets of the
  firm are less in value than the debt, shareholders
  have an in-the-money put.
• They will put the firm to the bondholders.
• If at the maturity of the debt the shareholders
  have an out-of-the-money put, they will not
  exercise the option (i.e. NOT declare
  bankruptcy) and let the put expire.                 E.38

				
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