Chapter 20 - Options by LK2bg5

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									                                                                             Chapter 21: Option Valuation-1




Chapter 21 Supplement: Steps and explanations in some of Chapter 21’s equations

   A. Deriving Equation 21.5 from Equation 21.4

       Su + (1+rf)B = Cu                                                                     (21.4a)
       Sd + (1+rf)B = Cd                                                                     (21.4b)

       1) Take difference between 21.4a and 21.4b

           => Su – Sd      =     u   – Cd

       2) Solve for B in equation 21.4b

                Cd  S d 
           B
                 1 rf

   B. Deriving Equation (21.9) from (20.3)

       P = C – S + PV(K)                                                        (20.3 solving for P)
            S  N d1   PV K   N d 2   S  PV ( K )                  (Substituting in 21.7)
            PV ( K )  PV K   N d 2   S  S  N d1 
            PV K 1  N d 2   S 1  N d1 

   C. Deriving Equation 21.20 and 21.21

       C = S + B                                                                              (21.6)
               S
       C          S                                                                        (21.17)
             S  B

       In equations 21.17 and 21.6, C is a call on stock, S is the stock, and B is the investment in
           risk-free bonds to create the replicating portfolio

       Note: recall that equity is essentially a call on the firm’s assets

           => let E = equity of firm, D = debt of firm, A = assets of firm, U = beta of unlevered
              equity = beta of firm’s assets. All are market values.

           => substitute E for C and A (or U) for S in equations 21.6 and 21.17

       E = A + B




                                                                                         Corporate Finance
                                                       Chapter 21: Option Valuation-2


      A
E       U
       E
       E  D 
                U
          E
       E D
       U
       E E
                D
=>  E   1   U                                                    (21.19)
                E

        E        D
U        E      D                                                   (14.9)
       ED      ED

              E      D
   => U        E  D
               A     A
        A        E
   =>      U   E   D Note: multiply both sides by A/D
        D        D
               A     E
   =>    D  U   E
              D      D
               A     E    D
   =>    D  U  1   U Note: substituting 21.19 for E
              D      D    E
               A E  D 
   =>    D    1   U
              D D      E 
            A   E   
   =>  D      1 U
            D    D 
            A    E D 
   =>  D        U
            D    D D 
              A        A 
   =>  D       U
              D        D 
                       A 
   =>  D   1     U
                       D 
                       D  E 
   =>  D   1            U
                       D 
                      E 
=>  D   1   1   U                                           (21.20)
                   D 

          E
U                Note: solving for U in (21.19)                      (21.21)
            D
        1  
            E



                                                                   Corporate Finance

								
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