1. An Introduction to Binomial Trees Chapter 11

2. A Simple Binomial Model
A stock price is currently $20
In three months it will be either $22 or $18
Stock Price = $22
Stock price = $20
Stock Price = $18

3. A Call Option
A 3-month call option on the stock has a strike price of 21.
Figure (p. 244)
Stock Price = $22
Option Price = $1
Stock price = $20
Option Price=?
Stock Price = $18
Option Price = $0

4. Setting Up a Riskless Portfolio
Consider the Portfolio: long  shares short 1 call option
Portfolio is riskless when 22– 1 = 18 or  = 0.25
22– 1x1
18

5. Valuing the Portfolio ( Risk-Free Rate is 12% )
The riskless portfolio is:
long shares short 1 call option
The value of the portfolio in 3 months is 220.25 – 1 = 4.50
The value of the portfolio today is e – 0.120.25 =

6. Valuing the Option The portfolio that is
long shares short 1 option
is worth 4.367
The value of the shares is (= 0.2520 )
The value of the option is therefore (= – )

7. Generalization Stock goes up by u-1 percent or down by d-1 percent.
Figure (p. 245) Su ƒu Sd ƒd S ƒ

8. Generalization (continued)
Consider the portfolio that is long  shares and short 1 derivative
The portfolio is riskless when Su – ƒu = Sd – ƒd or
Su – ƒu
Sd – ƒd

9. Generalization (continued)
Value of the portfolio at time T is Su – ƒu
Value of the portfolio today is (Su – ƒu )e–rT
Another expression for the portfolio value today is S – ƒ
Hence ƒ = S – (Su – ƒu )e–rT

10. Generalization (continued)
Substituting for  we obtain
ƒ = [ p ƒu + (1 – p )ƒd ]e–rT Eqn (11.2) p. 246
where

11. Generalization – Continued
Note: u > 1 and d < 1
Note: d < erT < u (why?)
So 0 < p < 1

12. Risk-Neutral Valuation
ƒ = [ p ƒu + (1– p )ƒd ]e-rT Eqn (11.2)
The variables p & (1– p ) can be interpreted as the risk-neutral probabilities of up and down movements
The value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate Su ƒu Sd ƒd S ƒ p
(1– p )

13. Risk-Neutral Valuation
In risk-neutral world (RNW) the expected return on all assets equal the risk-free rate
Determine risk-neutral probabilities: p and 1-p
Then c = e-rT E[max{0, S-X}] = e-rT [p(S-X) + (1-p)(0)], where expectation is wrt to risk- neutral probabilities

14. Risk Neutral Valuation
Suppose u = 1.1, d = .95, and erT = 1.05
Let S = $1 (it does not matter). Then in RNW E[ST] = S erT: 1.1p + .95(1-p) = 1.05
Or p = ( )/( ) = 2/3 same as eq (10.3)
Suppose X = 100, S = 100, and payoff = max{0, ST – 100}
Then c = [(2/3)( ) + (1/3)(0)](1/1.05) = $6.35

15. Risk Neutral Valuation: Basic Idea
Option value based upon absence of arbitrage profit
Arbitrage profit desirable regardless of investors’ risk preferences
All investors regardless of risk preferences assign same value to options
So in a world of risk neutral investors options would have the same value as in risk-averse world
Equal to the present value of expected payoff

16. Irrelevance of Stock’s Expected Return
When we are valuing an option in terms of the underlying stock the expected return on the stock is irrelevant
Why? Because in a risk-neutral world options have the same value as in our risk-averse world. But all assets have an expected return equal to the risk free rate. So expected return does not matter.

17. Interesting Question
Suppose stock A has an expected return of 10% and a risk of 20%
Stock B has an expected return of 15% and a risk of 20%
How can the call option values be the same?
Wouldn’t you rather own option on stock B?
In a risk-neutral world both would have an expected return equal to the risk-free rate and a risk of 20%.
Therefore, both would have the same value.

18. Original Example Revisited
Su = 22
ƒu = 1
Since p is a risk-neutral probability 20e0.12 0.25 = 22p + 18(1– p ); p =
Alternatively, we can use the formula: p S ƒ
Sd = 18
ƒd = 0
(1– p )

19. Valuing the Option
Su = 22
ƒu = 1
Sd = 18
ƒd = 0 S ƒ
0.6523
0.3477
The value of the option is e–0.120.25 [0.6523 0]
= 0.633

20. Delta
Delta () is the ratio of the change in the price of a stock option to the change in the price of the underlying stock
The value of  varies from node to node