1. 21 Valuing options McGraw-Hill/Irwin
Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.

2. 21-1 simple option-valuation model
Apple call options have exercise price of $400
Case 1
Stock price falls to $320.00
Option value = $0
Case 2
Stock price rises to $500.00
Option value = $100.00
In the binomial model there are only two outcomes at each stage; up or down. The value of Apple stock in down state and up state is given. The corresponding option value on the expiration date is shown.

3. 21-1 simple option-valuation model
Buy .556 Apple shares and borrow $ from bank at 1.5%
Value of call
= 400 x .556 –
= $44.62
Stock Price
$320
$500
.556 shares
$177.78
$277.78
Repayment of loan + interest
Total payoff $0
$100
Assume that you own .556 shares of Apple stock and also borrow the present value of .556 of the down-state value.

4. 21-1 simple option-valuation model
Apple call option is equal to leveraged position in .556 shares
Option delta:
Option delta = (spread of possible option prices)/(spread of possible share prices)

5. 21-1 simple option-valuation model
When risk-neutral, expected returns on Apple call options is 1.73%
Risk-neutral approach:
Expected return = [probability of rise × 25] + [(1 – probability of rise) × (–20)] = 1.73%

6. 21-1 simple option-valuation model
Value of Apple Option
Risk-neutral approach:
Option value = [(probability of rise) × 100] + [(1 – probability of rise) × (0)]/(1.0173)
= $47.48

7. 21-1 simple option-valuation model
Valuation of Apple Put Option
Case 1
Stock price falls to $320.0.
Option value = $80
Case 2
Stock price rises to $500
Option value = $0
The put option can also be valued using the same approach. The first slide in the example shows the possible outcomes.

8. 21-1 simple option-valuation model
Apple put option is equal to leveraged position in .444 shares
Option delta:
The next step is to calculate the option delta.

9. 21-1 simple option-valuation model
Sell .444 of an Apple share and lend 1.5%
Value of put
-(.444) x /1.0173= $40.65
Stock Price
$320
$500
Sale of .444 shares
-$142.22
$222.22
Repayment of loan + interest
Total payoff
$80.00 $0
The method now requires the sale of .444 of Apple shares and loaning the proceeds.

10. Figure 21.2 present and possible future stock prices, apple
Assume price will rise by 17.09% or fall by % in each three-month period
This slide shows the present and possible future prices of Apple stock.

11. 21-2 binomial method for valuing options
Probability stock will rise, given expected return of .86% per quarter
Expected value in month six
This slide shows the math to calculate the option value.

12. Figure 21.3 present and possible future stock prices, apple
Figures in parentheses show the corresponding values of a six-month call option with an exercise price of $400.

13. 21-2 binomial method for valuing options
Binomial Model
Probability up = p = (a – d)/(u – d)
Probability down = 1 – p
a = e(r) × (h ); u = e(σ) × (√h) ; d = e–(σ) × (√h ); h = ∆t = time interval as % of year

14. 21-2 binomial method for valuing options
Example
Price = 36
Strike = 40
s = .40
t = 90/365
Dt = 30/365
r = 10%
a =
u =
d = .8917
Pu = .5075
Pd = .4925
Price = 36; σ = t = 90/365; ∆t = 30/365; Strike (exercise) = 40; t = 10%
a = e(r) × (h) = ; u = e(σ) × (√ h) = ; d = e– (σ) × (√ h) = ; Pu = ; Pd =

15. 21-2 binomial method for valuing options
40.37
32.10 36
Up price = 36 × = 40.37; Down price = 36 × =

16. 21-2 binomial method for valuing options
40.37
32.10 36
Up price = P0 × U = PU1 = 36 × = 40.37; Down price = P0 × D = PD1 = 36 × = 32.10

17. 21-2 binomial method for valuing options
50.78 = price
40.37
32.10
25.52
45.28 36
28.62
The tree is developed for the three periods.

18. 21-2 binomial method for valuing options
50.78 = price
10.78 = intrinsic value
40.37
.37
32.10
25.52
45.28 36
28.62
Option price on expiration = stock price – strike price = – 40 = 10.78, etc.

19. 21-2 binomial method for valuing options
50.78 = price
10.78 = intrinsic value
40.37
.37
32.10
25.52
45.28
5.60 36
28.62
The greater of
The value at the node in this slide is the higher of the intrinsic value or the present value of the two branches.
5.60 = [(10.78)(0.5075) + (0.37)(0.4925)]/1.0083
0.19 = [(0.37)(0.5075) + 0]/1.0083
The values are worked backwards from the day of expiration.

20. 21.2 binomial method for valuing options
50.78 = price
10.78 = intrinsic value
40.37
.37
32.10
25.52
45.28
5.60 36
.19
28.62
2.91
.10
1.51
5.60 = [(10.78)(0.5075) + (0.37)(0.4925)]/1.0083
0.19 = [(0.37)(0.5075) + 0]/1.0083
The values are worked backwards from the day of expiration.
2.91 = [(5.60)(0.5075) + (0.19)(0.4925)]/1.0083
0.1 = [(0.19)(0.5075) + 0]/1.0083
The values are worked backwards.
1.51 = [(2.91)(5075) + (0.1)(0.4925)]/1.0083
The call option price today is $1.51

21. Figure 21.1a possible six-month price changes, apple
As we reduce the time interval, the binomial model converges to Black-Scholes model. By shortening the time interval, we can get more steps and more possible price changes. When a limited number of steps are used, binomial outcomes differ from Black-Scholes value. This is illustrated using Apple numerical example.

22. Figure 21.1b possible six-month price changes, apple
As we reduce the time interval, the binomial model converges to Black-Scholes model. By shortening the time interval, we can get more steps and more possible price changes. When a limited number of steps are used, binomial outcomes differ from Black-Scholes value. This is illustrated using Apple numerical example.

23. Figure 21.1c possible six-month price changes, apple
As we reduce the time interval, the binomial model converges to Black-Scholes model. By shortening the time interval, we can get more steps and more possible price changes. When a limited number of steps are used, binomial outcomes differ from Black-Scholes value. This is illustrated using Apple numerical example.

24. 21-3 the black-scholes formula
Components of Option Price
Underlying stock price
Strike or exercise price
Volatility of stock returns (standard deviation of annual returns)
Time-to-option expiration
Time value of money (discount rate)
These are the components of the option price for the Black-Scholes model. Option price depends on these five variables:
Call Put
Underlying stock price –
Exercise price – +
Volatility of stock return (standard deviation of annual returns)
Time to expiration
Discount rate (time value of money) –

25. 21-3 the black-scholes formula
Black-Scholes Option Pricing Model
OC: Call option price
P: Stock price
N(d1): Cumulative normal probability density function of (d1)
PV(EX): Present value of strike or exercise price
N(d2): Cumulative normal probability density function of (d2)
r: Discount rate (90-day commercial paper rate or risk-free rate)
t: Time to maturity of option (as % of year)
v: Volatility-annualized standard deviation of daily returns
Value of a call = (delta)(price) – (bank loan) = N(d1)×(P) – N(d2) ×[PV(EX)]
N(d1) and N(d2) are obtained from cumulative normal distribution tables or from Excel function NORMSDIST(d).

26. Figure 21.4 life of option
d1 should be calculated using the above formula. This is the first step.

27. 21-3 the black-scholes formula
Calculating d2 from d1 is the second step.

28. 21-3 the black-scholes formula
Example
Determine call option price given following values
P = 430
r = 3%
v = .4068
EX = 430
t = 108 days/365
Apple example: t should be in years (180/365) and r (0.05) should be in decimal to plug-in the formula.
d1 and N(d1) are calculated. This is the first step.

29. 21-3 the black-scholes formula
Example
Determine call option price given following values
P = 430
r = 3%
v = .4068
EX = 430
t = 108 days/ 365
d2 and N(d2) are calculated. This is the second step.
The option price is calculated in the final step. The option price is only a fraction of the stock price.

30. Figure 21.5 value of apple call option versus value of apple stock
This is the same time decay chart presented in the previous chapter. It shows the decreasing value of the Apple option as the time to expiration declines.

31. 21-3 the black-scholes formula
Example
Determine call-option price given following values
P = 36
r = 10%
v = .40
EX = 40
t = 90 days/365
Another example is provided for practice.

32. 21-3 the black-scholes formula
Example
Determine call-option price given following values
P = 36
r = 10%
v = .40
EX = 40
t = 90 days/365
Another example is provided for practice.

33. 21-3 the black-scholes formula
Value of a call = (delta)([rice) – (bank loan) = N(d1)×(P) – N(d2) ×[PV(EX)]
PV(EX) = (EX) (e– (r) × (t ) ) is calculated using the continuous time discounting formula.

34. 21-4 black-scholes in action
Establishment Industries and Digital Organics stocks are compared as an executive stock-option decision.

35. Figure 21.6 standard deviations of market returns
Implied volatility is explained. Volatility is an unobservable variable. Volatility can be calculated using observed option prices.

36. 21-4 black-scholes in action
Carrying cost = r × EX × t
Put – Call Parity can be used to calculate the put price given the call price.
 Put price = OC + EX – P – carrying cost + div.
Where: Carrying cost = r × EX × t

37. 21-5 Option values at a glance
Valuation Variations
American calls with no dividends
European puts with no dividends
American puts with no dividends
European calls and puts on dividend-paying stocks
American calls on dividend-paying stocks
The formulae presented can be modified to incorporate various types of options. This slide lists various option alternatives.

38. Table 21.1 binomial versus black-scholes
Expand binomial model to allow more possible price changes
As the number of steps is increased, the binomial model converges to the Black–Scholes model. This is illustrated using a numerical example.

39. 21-6 option menagerie Example
Determine call-option price given the following
P = 36
r = 10%
v = .40
EX = 40
t = 90 days/365
Binomial price = $1.51
Black-Scholes price = $1.70
When only a limited number of steps are used, binomial outcome will approach, but not necessarily equal, the Black–Scholes price.

40. 21-6 option menagerie
Estimated call price in relation to number of binomial steps
No. of Steps Estimated Value
Black-Scholes
Binomial model and Black–Scholes model convergence is illustrated.

41. 21-6 option menagerie Dilution
Exercise of a warrant increases the number of shares outstanding. The dilution factor reflects that fact.