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					Financial Analysis, Planning and
          Forecasting
    Theory and Application
               Chapter 10
 Option Pricing Theory and firm Valuation

                  By
             Alice C. Lee
     San Francisco State University
             John C. Lee
          J.P. Morgan Chase
             Cheng F. Lee
          Rutgers University
                     Outline
 10.1 Introduction
 10.2 Basic concepts of Options
 10.3 Factors affecting option value
 10.4 Determining the value of options
 10.5 Option pricing theory and capital structure
 10.6 Warrants
 10.7 Summary
 Appendix 10A. Application of the Binomial
                 Distribution to evaluate call options
10.2 Basic concepts of Options
 Option   price information
10.2 Basic concepts of Options
Exhibit 10-1 Listed Options Quotations

  Close   Strike           Calls                   Puts
  Price   Price    Sep      Oct     Jan     Sep     Oct    Jan

JNJ

  65.12   45.00    20.40    N/A     N/A     N/A     N/A    N/A

  65.12   50.00    N/A      N/A     N/A     N/A     N/A    N/A

  65.12   55.00    10.30   10.40    11.10   N/A     0.02   0.30

  65.12   60.00    5.30     N/A     6.40    N/A     N/A    0.70

  65.12   65.00    0.10     1.15    2.60    0.05    0.85   1.95

  65.12   70.00    N/A      0.05    0.55    4.80    4.50   5.00
10.2 Basic concepts of Options
Exhibit 10-1 Listed Options Quotations (Cont’d)

   Close   Strike           Calls                    Puts
   Price   Price    Sep      Oct     Jan      Sep     Oct    Jan
   MRK

   51.82   47.50    4.34     4.90    6.04     N/A     N/A    1.30

   51.82   50.00    1.85     2.70    4.40     0.03    0.65   2.10

   51.82   52.50    0.03     1.10    2.85     0.55    1.50   3.10

   51.82   55.00    0.01     0.35    1.70     3.20    3.30   4.60

   51.82   57.50    N/A      N/A     0.95     5.70    N/A    N/A

   51.82   60.00    N/A      N/A     0.50     8.20    N/A    8.40
10.2 Basic concepts of Options
Exhibit 10-1 Listed Options Quotations (Cont’d)

  Close   Strike           Calls                    Puts
  Price   Price    Sep     Oct      Jan     Sep     Oct     Jan

   PG

  69.39   40.00    29.70   29.70   30.10    N/A     N/A     N/A

  69.39   45.00    24.60   24.70   N/A      N/A     N/A     N/A

  69.39   55.00    14.50   N/A     15.20    N/A     N/A     0.15

  69.39   60.00    9.70    9.80    10.70    N/A     0.08    0.42

  69.39   65.00    4.50    4.90    6.19     N/A     0.21    1.00

  69.39   70.00    0.05    0.90    2.75     0.60    1.50    2.65

  69.39   75.00    N/A     0.05    0.70     5.50    5.70    5.70

  69.39   80.00    N/A     N/A     0.15     10.40   10.50   N/A
10.2 Basic concepts of Options



         Vc  MAX (0, P  E )


        V p  MAX (0, E  P)
10.2 Basic concepts of Options
Figure 10-1
Value of $50 Exercise Price Call Option (a) to Holder, (b) to Seller
10.2 Basic concepts of Options
Figure 10-1
Value of $50 Exercise Price Call Option (a) to Holder, (b) to Seller
10.3 Factors affecting option value


    Determining  the value of a call
     option before the expiration date
10.3 Factors affecting option value
Figure 10-2 Value of Call Option
10.3 Factors affecting option value
Figure 10-3 Call Option Value as a Function of Stock Price
10.3 Factors affecting option value
 TABLE 10-1
 Probabilities for Future Prices of Two Stocks

            Less Volatile Stock                More Volatile Stock

 Future Price($)         Probability   Future Price ($)      Probability

       42                    .10             32                  .15
       47                    .20             42                  .20
       52                    .40             52                  .30
       57                    .20             62                  .20
       62                    .10             72                  .15
10.3 Factors affecting option value
Figure 10-4 Call-Option Value as Function of Stock Price for High-,
   Moderate-, and Low-Volatility Stocks
10.3 Factors affecting option value
TABLE 10-2 Data for a Hedging Example

                   Current price per share:       $100
                   Future price per share:        $125 with probability .6
                   $85 with probability .4
                   Exercise price of call option: $100




TABLE 10-3 Possible Expiration-Date Outcomes for Hedging Example

       Expiration-Date                 Value per Share                  Value per Share
         Stock Price                  of Stock Holdings                of Options Written
            $125                            $125                             -$25
            $85                             $85                               $0
10.3 Factors affecting option value

           125H  25  85H

                  25 5
               H   
                  40 8

               P E
            H  U

               P  PL
                U
10.3 Factors affecting option value


          (5)(125)-(8)(25) = $425

          (5) (85)+(8) (0) = $425

          1.08(500  8 Vc )  425

        Vc 
             1.08 500  425  $13.31
                 1.088
10.4 Determining the value of options


    Expected  value estimation
    The Black-Scholes option
    Pricing model
    Taxation of options
    American options
10.4 Determining the value of options

expected value of share = (.6)(125) + (.4)(85) = $109

   expected value of call = (.6)(25) + (.4)(0) = $15

                                  15  13.31
expected rate of return on call              100  12.7%
                                    13.31


               VP  MAX  E  P,0
10.4 Determining the value of options
Figure 10-5 Put-Option Value
10.4 Determining the value of options

      Vc  P[ N (d1 )]  e  rt E[ N (d 2 )]    (10-1)

                   P          2 t
                ln    rt  
           d1     E            2
                                               (10-2A)
                       t

            P          2 t
         ln    rt  
    d2     E            2
                              d1   t        (10-2B)
                t
10.4 Determining the value of options
FIGURE 10-6 Probability Distribution of Stock Prices
10.4 Determining the value of options
   Example 10-1
                                P
            .6             ln    ln .9   .1054
                                E
                                                         P            t
       P            t                               ln    rt   2
    ln    rt   2
                                                  d2   
                                                          E             2
d1   
        E             2
                                                             t
           t
                                                                                  1
                            1                           .1054  .08 .50       .36 .50 
      .1054  .08.50   .36 .50                                          2
                            2                       
                                          .06                       .6 .50
                   .6 .50
                                                     .37
10.4 Determining the value of options
   Example 10-1

             N  d1   N .06  .5239

            N  d2   N  .37  .3557

                    rt        .08.5
               e          e                 .9608

       Vc  PN  d1   e rt EN  d 2 
            90 .5239   .9608 100 .3557   $12.98
10.5 Option pricing theory and capital structure



  Proportion of debt in capital structure
  Riskiness of business operations
10.5 Option pricing theory and capital structure
               V  MAX  0,V f  B        (10-3)
FIGURE 10-7 Option Approach to Capital Structure
10.5 Option pricing theory and capital structure

 Example               10-2
                           P
                        ln( )  ln(1.4)  .3365
                           E


       P            t                                    P            t
    ln    rt   2                                   ln    rt   2
d1                                               d2   
        E             2                                     E             2
           t                                                  t
                              1                                                   1
      .3365  .08  6       .2  6                 .3365  .08  6       .2  6 
                             2             1.91                                 2
                                                                                               1.42
                   .2 6                                                .2 6
10.5 Option pricing theory and capital structure

 Example     10-2
             N  d1   N 1.91  .9719

             N  d2   N 1.42  .9222

                             .08 6 
                 e rt  e                  .6188

       V  PN (d1 )  e rt EN (d 2 )
          (14)(.9719)  (.6188)(10)(.9222)  $7.90 million
10.5 Option pricing theory and capital structure

                     rt
     V  PN (d1 )  e EN (d 2 )
        (14)(.9996)  (.6188)(5)(.9977)
        $10.91 million

   value of debt = 14-10.91 = $3.09 million
10.5 Option pricing theory and capital structure


Table 10-4
Effect of Different Levels of Debt on Debt Value

     Face Value of Debt     Actual Value of Debt   Actual Value per Dollar Debt
        ($ millions)            ($ millions)              Face Value of Debt
             5                     3.09                       $.618

            10                     6.10                       $.610
10.5 Option pricing theory and capital structure


       V  PN (d1 )  e rt EN (d 2 )
          (14)(.9066)  (.6188)(10)(.6331)
          $8.77 million

   value of debt = 14-8.77 = $5.23 million
10.5 Option pricing theory and capital structure

Table 10-5
Effect of Different Levels of Business Risk on the Value of $10
Million Face Value of Debt


     Variance of Rate of       Value of Equity           Value of Debt
            Return               ($ millions)             ($ millions)

             .2                     7.90                     6.10

             .4                     8.77                     5.23
10.6 Warrants

          Vw  MAX  0, NP  E 


old equity = stockholders’ equity + warrants

new equity = old equity + exercise money

H(new equity) = H (old equity) + H (exercise money)
10.6 Warrants

            Nw
        H
           Nw  N

        Nw 
    P             Value of old equity 
        Nw  N 
        N 
    E             exercise money 
        Nw  N 
10.6 Warrants
old equity = 100(lm) + 20(.5m) = $110 million

new equity = $110m + $40m = $150 million

      Nw                     500, 000
  H             or
     Nw  N            500, 000  1, 000, 000

  1          1         1
    150m   110m    40m   50 million
  3          3         3
10.7 Summary
 In Chapter 10, we have discussed the basic
 concepts of call and put options and have
 examined the factors that determine the
 value of an option. One procedure used in
 option valuation is the Black-Scholes model,
 which allows us to estimate option value as
 a function of stock price, option-exercise
 price, time-to-expiration date, and risk-free
 interest rate. The option pricing approach to
 investigating capital structure is also
 discussed, as is the value of warrants.
Appendix 10A: Applications of the Binomial Distribution
              to Evaluate Call Options


   What  is an option?
   The simple binomial option pricing
    model
   The Generalized Binomial Option
    Pricing Model
Appendix 10A: Applications of the Binomial Distribution
              to Evaluate Call Options


          Cu  Max(0, uS  X )                 (10A.1)

          Cd  Max(0, dS  X )                 (10A.2)

           h(uS )  Cu  h(dS )  Cd           (10A.3)

                  Cu  Cd
               h                              (10A.4)
                  (u  d ) S
Appendix 10A: Applications of the Binomial Distribution
              to Evaluate Call Options


  (1  r )(hS  C )  h(uS )  Cu  h(dS )  Cd   (10A.5)

            R  d      u  R 
       C         Cu       Cd  R           (10A.6)
            u  d      ud  
           Rd                   u  R 
        p            so 1  p                 (10A.7)
           ud                   u  d 

             C   pCu  (1  p)C d  R           (10A.8)
Appendix 10A: Applications of the Binomial Distribution
              to Evaluate Call Options
Table 10A.1 Possible Option Value at Maturity
                 Today
     Stock (S)          Option (C)                      Next Period (Maturity)
                                     uS = $110   Cu =       Max (0,uS – X)
                                                 =          Max (0,110 – 100)
                                                 =          Max (0,10)
                                                 =          $ 10
   $100             C
                                     dS = $ 90   Cd =       Max (0,dS – X)
                                                 =          Max (0,90 – 100)
                                                 =          Max (0, –10)
                                                 =          $0
Appendix 10A: Applications of the Binomial Distribution
              to Evaluate Call Options
      CT = Max [0, ST – X]               (10A.9)

      Cu = [pCuu + (1 – p)Cud] / R           (10A.10)

      Cd= [pCdu + (1 – p)Cdd] / R            (10A.11)

                                              
  C  p 2 C uu  2 p(1  p)C ud  (1  p) 2 C dd R 2
                                               (10A.12)
Appendix 10A: Applications of the Binomial Distribution
              to Evaluate Call Options

        n
   1         n!
C n
  R
        k!(n  k )!
       k 0
                     p k (1  p) n  k Max[0, u k d n  k S  X ] (10A.13)


  C1 = Max [0, (1.1)3(.90)0(100) – 100] = 33.10

  C2 = Max [0, (1.1)2(.90) (100) – 100] = 8.90

  C3 = Max [0, (1.1) (.90)2(100) – 100] = 0

  C4 = Max [0, (1.1)0(.90)3(100) – 100] = 0
Appendix 10A: Applications of the Binomial Distribution
              to Evaluate Call Options

       1  3!                         3!
 C                (.85) (.15) X 0 
                        0     3
                                          (.85)1 (.15) 2 X 0
    (1.07)3  0!3!
                                    1!2!
           3!                        3!                      
              (.85) (.15) X 8.90 
                    2     1                   3     0
                                         (.85) (.15) X 33.10 
          2!1!                      3!0!                     
        1          3X 2 X1                         3X 2 X1                         
             00           (.7225)(.15)(8.90)               X (.61413)(1)(33.10) 
      1.225 
                   2 X 1X 1                      3 X 2 X 1X 1                      
        1
           (.32513 X 8.90)  (.61413 X 33.10)]
      1.225
     $18.96
 Appendix 10A: Applications of the Binomial Distribution
               to Evaluate Call Options

       n       n!                     k nk
                                  nk u d     X  n    n!               nk 
C  S                 p (1  p)
                         K
                                               n            p (1  p) 
                                                                 k

      k m k!(n  K )!                 R  R k m k!(n  k )!
                                          n
                                                                             
                                                               (10A.14)
                            k   nk
                           u d
        P (1  p)
           k         nk
                                       p rk (1  p ) n  k
                             Rn


                              X
        C  SB1 (n, p , m)  n B2 (n, p, m) (10A.15)
                             R
Appendix 10A: Applications of the Binomial Distribution
              to Evaluate Call Options



                           n
      B1 (n, p , m)   C p  (1  p )
                                n
                                k
                                     k           nk

                         k m



                           n
      B2 (n, p, m)   C p (1  p)
                                n
                                k
                                    k          nk

                         k m
Appendix 10A: Applications of the Binomial Distribution
              to Evaluate Call Options      190.61
                                                    162.22
Figure 10A.1                               138.06
                                                             137.89

  Price Path of                                     117.35
                                                             137.89
                                  117.50                      99.75
  Underlying
  Stock                                             117.35
                                                             137.89
                                                             99.75
Source:                                    99.88
  R.J.Rendelman,                                             99.75
                                                    84.90
  Jr., and                                                   72.16
                        $100.00
  B.J.Bartter (1979),
                                                             137.89
  “Two-State                                        117.35
                                                             99.75
  Option Pricing,”                         99.88
  Journal of                                                 99.75
                                                    84.90
  Finance 34                                                 72.16
                                  85.00
  (December), 1906.
                                                             99.75
                                                    84.90
                                                             72.16
                                           72.25
                                                             72.16
                                                    61.41
                                                             52.20

                           0         1        2        3      4
Appendix 10A: Applications of the Binomial Distribution
              to Evaluate Call Options

           16
            Pi190.61  137.89  . . .  52.20
       P    
           i 1
          16                16
          $105.09

                                                           12
           (190.61  105.09)  . . .  (52.20  105.09) 
                              2                        2
     P                                                
                              16                        
         $34.39

				
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