1. Binomial Option Pricing
Professor P. A. Spindt

2. A simple example A stock is currently priced at $40 per share.
In 1 month, the stock price may
go up by 25%, or
go down by 12.5%.

3. A simple example Stock price dynamics: t = now t = now + 1 month
up state
$40x(1+.25) = $50
$40
$40x(1-.125) = $35
down state

4. Call option A call option on this stock has a strike price of $45 t=0
Stock Price=$50;
Call Value=$5
Stock Price=$40;
Call Value=$c
Stock Price=$35;
Call Value=$0

5. A replicating portfolio
Consider a portfolio containing D shares of the stock and $B invested in risk-free bonds.
The present value (price) of this portfolio is DS + B = $40 D + B

6. Portfolio value t=0 t=1 up state down state $50 D + (1+r/12)B
$40 D + B
down state

7. A replicating portfolio
This portfolio will replicate the option if we can find a D and a B such that
Up state
$50 D + (1+r/12) B = $5
and
Down state
$35 D + (1+r/12) B = $0
Portfolio payoff =
Option payoff

8. The replicating portfolio
Solution:
D = 1/3
B = -35/(3(1+r/12)).
Eg, if r = 5%, then the portfolio contains
1/3 share of stock (current value $40/3 = $13.33)
partially financed by borrowing $35/(3x ) = $11.62

9. The replicating portfolio
Payoffs at maturity

10. The replicating portfolio
Since the the replicating portfolio has the same payoff in all states as the call, the two must also have the same price.
The present value (price) of the replicating portfolio is $ $11.62 = $1.71.
Therefore, c = $1.71

11. A general (1-period) formula

12. An observation about D
As the time interval shrinks toward zero, delta becomes the derivative.

13. Put option What about a put option with a strike price of $45 t=0 t=1
Stock Price=$50;
Put Value=$0
Stock Price=$40;
Put Value=$p
Stock Price=$35;
Put Value=$10

14. A replicating portfolio
up state
$50 D + (1+r/12)B
$35 D + (1+r/12)B
$40 D + B
down state

15. A replicating portfolio
This portfolio will replicate the put if we can find a D and a B such that
Up state
$50 D + (1+r/12) B = $0
and
Down state
$35 D + (1+r/12) B = $10
Portfolio payoff =
Option payoff

16. The replicating portfolio
Solution:
D = -2/3
B = 100/(3(1+r/12)).
Eg, if r = 5%, then the portfolio contains
short 2/3 share of stock (current value $40x2/3 = $26.66)
lending $100/(3x ) = $33.19.

17. Two Periods
Suppose two price changes are possible during the life of the option
At each change point, the stock may go up by Ru% or down by Rd%

18. Two-Period Stock Price Dynamics
For example, suppose that in each of two periods, a stocks price may rise by 3.25% or fall by 2.5%
The stock is currently trading at $47
At the end of two periods it may be worth as much as $50.10 or as little as $44.68

19. Two-Period Stock Price Dynamics
$50.10
$48.53
$47
$47.31
$45.83
$44.68

20. At expiration, a call with a strike price of $45 will be worth:
Terminal Call Values
At expiration, a call with a strike price of $45 will be worth:
Cuu =$5.10
$Cu
$C0
Cud =$2.31
$Cd
Cdd =$0

21. Two Periods
The two-period Binomial model formula for a European call is

22. Example
TelMex Jul CB 23/16 -5/ ,703

23. Estimating Ru and Rd
According to Rendleman and Barter you can estimate Ru and Rd from the mean and standard deviation of a stock’s returns

24. Estimating Ru and Rd
In these formulas, t is the option’s time to expiration (expressed in years) and n is the number of intervals t is carved into

25. For Example
Consider a call option with 4 months to run (t = .333 yrs) and
n = 2 (the 2-period version of the binomial model)

26. For Example
If the stock’s expected annual return is 14% and its volatility is 23%, then

27. For Example
The price of a call with an exercise price of $105 on a stock priced at $108.25

28. Anders Consulting
Focusing on the Nov and Jan options, how do Black-Scholes prices compare with the market prices listed in case Exhibit 2?
Hints:
The risk-free rate was 7.6% and the expected return on stocks was 14%.
Historical Estimates: sIBM = .24 & sPepsico = .38