The document discusses various pricing models for options including: 1. Upper and lower bounds for call option value based on the stock price. 2. Factors that determine option value such as exercise price, expiration date, stock price, stock price variability, and interest rate. 3. The binomial model and risk-neutral valuation method for pricing options using a single period binomial tree framework. 4. The Black-Scholes model for pricing European options using a continuous time framework based on the stock's lognormal distribution. 5. Assumptions of the Black-Scholes model and adjustments needed for dividends, short-term options, and long-term options.

1. OPTIONS
Pricing Models

2. Option Value: Bounds
UPPER AND LOWER BOUNDS
FOR THE VALUE OF CALL OPTION
VALUE OF UPPER LOWER
CALL OPTION BOUND (S0) BOUND ( S0 – E)
STOCK PRICE
0 E

3. Factors Determining The Option Value
• EXERCISE PRICE
• EXPIRATION DATE
• STOCK PRICE
• STOCK PRICE VARIABILITY
• INTEREST RATE
C0 = f [S0 , E, 2, t , rf ]
+ - + + +

4. Binomial Model
Option Equivalent Method - 1
A SINGLE PERIOD BINOMIAL (OR 2 - STATE) MODEL
• S CAN TAKE TWO POSSIBLE VALUES NEXT YEAR, uS OR
dS (uS > dS)
• B CAN BE BORROWED .. OR LENT AT A RATE OF r, THE
RISK-FREE RATE .. (1 + r) = R
• d < R > u
• E IS THE EXERCISE PRICE
Cu = MAX (u S - E, 0)
Cd = MAX (dS - E, 0)

5. Binomial Model
Option Equivalent Method - 1
PORTFOLIO
 SHARES OF THE STOCK AND B RUPEES OF BORROWING
STOCK PRICE RISES :  uS - RB = Cu
STOCK PRICE FALLS :  dS - RB = Cd
Cu - Cd SPREAD OF POSSIBLE OPTION PRICE
 = =
S (u - d) SPREAD OF POSSIBLE SHARE PRICES
dCu - uCd
B =
(u - d) R
SINCE THE PORTFOLIO (CONSISTING OF  SHARES AND B
DEBT) HAS THE SAME PAYOFFAS THAT OFA CALL OPTION,
THE VALUE OF THE CALL OPTION IS
C =  S - B

6. Illustration
S = 200, u = 1.4, d = 0.9
E = 220, r = 0.10, R = 1.10
Cu = MAX (u S - E, 0) = MAX (280 - 220, 0) = 60
Cd = MAX (dS - E, 0) = MAX (180 - 220, 0) = 0
Cu - Cd 60
 = = = 0.6
(u - d) S 0.5 (200)
dCu - uCd 0.9 (60)
B = = = 98.18
(u - d) R 0.5 (1.10)
0.6 OF A SHARE + 98.18 BORROWING … 98.18 (1.10) = 108 REPAYT
PORTFOLIO CALL OPTION
WHEN u OCCURS 1.4 x 200 x 0.6 - 108 = 60 Cu = 60
WHEN d OCCURS 0.9 x 200 x 0.6 - 108 = 0 Cd = 0
C =  S - B = 0.6 x 200 - 98.18 = 21.82

7. Binomial Model Risk-Neutral Method
WE ESTABLISHED THE EQUILIBRUIM PRICE OF THE
CALL OPTION WITHOUT KNOWING ANYTHING
ABOUT THE ATTITUDE OF INVESTORS TOWARD
RISK. THIS SUGGESTS … ALTERNATIVE METHOD …
RISK-NEUTRAL VALUATION METHOD
1. CALCULATE THE PROBABILITY OF RISE IN A
RISK NEUTRAL WORLD
2. CALCULATE THE EXPECTED FUTURE VALUE ..
OPTION
3. CONVERT .. IT INTO ITS PRESENT VALUE USING
THE RISK-FREE RATE

8. Pioneer Stock
1. PROBABILITY OF RISE IN A RISK-NEUTRAL WORLD
RISE 40% TO 280
FALL 10% TO 180
EXPECTED
RETURN = [PROB OF RISE x 40%] + [(1 - PROB OF RISE) x - 10%]
= 10% p = 0.4
2. EXPECTED FUTURE VALUE OF THE OPTION
STOCK PRICE Cu = RS. 60
STOCK PRICE Cd = RS. 0
0.4 x RS. 60 + 0.6 x RS. 0 = RS. 24
3. PRESENT VALUE OF THE OPTION
RS. 24
= RS. 21.82
1.10

9. Black-Scholes Model
E
C0 = S0 N (d1) - N (d2)
ert
N (d) = VALUE OF THE CUMULATIVE
NORMAL DENSITY FUNCTION
S0 1
ln E + r + 2 2 t
d1 =
  t
d2 = d1 -   t
r = CONTINUOUSLY COMPOUNDED RISK - FREE
ANNUAL INTEREST RATE
 = STANDARD DEVIATION OF THE CONTINUOUSLY
COMPOUNDED ANNUAL RATE OF RETURN ON
THE STOCK

10. Black-Scholes Model
Illustration
S0 = RS.60 E = RS.56  = 0.30
t = 0.5 r = 0.14
STEP 1 : CALCULATE d1 AND d2
S0 2
ln E + r + 2 t
d1 =
  t
.068 993 + 0.0925
= = 0.7614
0.2121
d2 = d1 -   t
= 0.7614 - 0.2121 = 0.5493
STEP 2 : N (d1) = N (0.7614) = 0.7768
N (d2) = N (0.5493) = 0.7086
STEP 3 : E 56
= = RS. 52.21
ert e0.14 x 0.5
STEP 4 : C0 = RS. 60 x 0.7768 - RS. 52.21 x 0.7086
= 46.61 - 37.00 = 9.61

11. Assumptions
• THE CALL OPTION IS THE EUROPEAN OPTION
• THE STOCK PRICE IS CONTINUOUS AND IS
DISTRIBUTED LOGNORMALLY
• THERE ARE NO TRANSACTION COSTS AND
TAXES
• THERE ARE NO RESTRICTIONS ON OR
PENALTIES FOR SHORT SELLING
• THE STOCK PAYS NO DIVIDEND
• THE RISK-FREE INTEREST RATE IS KNOWN AND
CONSTANT

12. Adjustment For Dividends Short-Term Options
Divt
ADJUSTED STOCK PRICE = S = S - 
(1 + r)t
E
VALUE OF CALL = S N (d1) - N (d2)
ert
S 2
ln E + r + 2 t
d1 =
  t

13. Adjustment For Dividends – 2 Long-Term Options
C = S e -yt N (d1) - E e -rt N (d2)
S 2
ln E + r - y + 2 t
d1 =
  t
d2 = d1 -   t y - dividend yield
THE ADJUSTMENT
• DISCOUNTS THE VALUE OF THE STOCK TO THE PRESENT
AT THE DIVIDEND YIELD TO REFLECT THE EXPECTED
DROP IN VALUE ON ACCOUNT OF THE DIVIDEND PAYMENTS
• OFFSETS THE INTEREST RATE BY THE DIVIDEND YIELD TO
REFLECT THE LOWER COST OF CARRYING THE STOCK

14. Put – Call Parity - Revisited
JUST BEFORE EXPIRATION
C1 = S1 + P1 - E
IF THERE IS SOME TIME LEFT
C0 = S0 + P0 - E e -rt
THE ABOVE EQUATION CAN BE USED TO
ESTABLISH THE PRICE OF A PUT OPTION &
DETERMINE WHETHER THE PUT - CALL PARITY IS
WORKING