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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