11.1

Binomial Trees

Chapter 11

11.2

A Simple Binomial Model

A stock price is currently $20In three months it will be either $22 or $18

Stock Price = $22

Stock Price = $18

Stock price = $20

11.3

Stock Price = $22Option Price = $1

Stock Price = $18Option Price = $0

Stock price = $20Option Price=?

A Call Option (Figure 11.1, page 242)

A 3-month call option on the stock has a strike price of 21.

11.4

Consider the Portfolio: long Δ sharesshort 1 call option

Portfolio is riskless when 22Δ – 1 = 18Δ or Δ = 0.25

22Δ – 1

18Δ

Setting Up a Riskless Portfolio

11.5

Valuing the Portfolio(Risk-Free Rate is 12%)

The riskless portfolio is: long 0.25 sharesshort 1 call option

The value of the portfolio in 3 months is 22 x 0.25 – 1 = 4.50

The value of the portfolio today is 4.5e – 0.12x0.25 = 4.3670

11.6

Valuing the OptionThe portfolio that is

long 0.25 sharesshort 1 option

is worth 4.367The value of the shares is

5.000 (= 0.25 x 20 )The value of the option is therefore

0.633 (= 5.000 – 4.367 )

11.7

Generalization (Figure 11.2, page 243)

A derivative lasts for time T and is dependent on a stock

S0uƒu

S0dƒd

S0ƒ

11.8

Generalization(continued)

Consider the portfolio that is long Δ shares and short 1 derivative

The portfolio is riskless when S0uΔ – ƒu = S0dΔ – ƒd or

dSuSfdu

00 −−

=Δƒ

S0uΔ – ƒu

S0dΔ – ƒd

11.9

Generalization(continued)

Value of the portfolio at time T is S0uΔ – ƒu

Value of the portfolio today is (S0uΔ – ƒu)e–rT

Another expression for the portfolio value today is S0Δ – fHence

ƒ = S0Δ – (S0uΔ – ƒu )e–rT

11.10

Generalization(continued)

Substituting for Δ we obtainƒ = [ pƒu + (1 – p)ƒd ]e–rT

where

p e du d

rT

=−

−

11.11

p as a Probability

It is natural to interpret p and 1-p as probabilities of up and down movementsThe value of a derivative is then its expected payoff in a risk-neutral world discounted at the risk-free rate

S0uƒu

S0dƒd

S0ƒ

p

(1 – p )

11.12

Irrelevance of Stock’s Expected Return

When we are valuing an option in terms of the the price of the underlying asset, the probability of up and down movements in the real world are irrelevantThis is an example of a more general result stating that the expected return on the underlying asset in the real world is irrelevant

11.13

Risk-neutral Valuation

When the probability of an up and down movements are p and 1-p the expected stock price at time T is S0erT

This shows that the stock price earns the risk-free rateBinomial trees illustrate the general result that to value a derivative we can assume that the expected return on the underlying asset is the risk-free rate and discount at the risk-free rate (ƒ = [ p ƒu + (1 – p )ƒd ]e-rT )This is known as using risk-neutral valuation

11.14

Original Example Revisited

Since p is the probability that gives a return on the stock equal to the risk-free rate. We can find it from20e0.12 x 0.25 = 22p + 18(1 – p )which gives p = 0.6523Alternatively, we can use the formula

6523.09.01.1

9.00.250.12

=−

−=

−−

=×e

dudep

rT

S0u = 22ƒu = 1

S0d = 18ƒd = 0

S0ƒ

p

(1 – p )

11.15

Valuing the Option Using Risk-Neutral Valuation

The value of the option is e–0.12 x 0.25 (0.6523 x 1 + 0.3477 x 0)

= 0.633

S0u = 22ƒu = 1

S0d = 18ƒd = 0

S0ƒ

0.6523

0.3477

11.16

A Two-Step ExampleFigure 11.3, page 246

Each time step is 3 monthsK=21, r=12%

20

22

18

24.2

19.8

16.2

11.17

Valuing a Call OptionFigure 11.4, page 247

Value at node B = e–0.12 x 0.25(0.6523 x 3.2 + 0.3477 x 0) = 2.0257

Value at node A = e–0.12 x 0.25(0.6523 x 2.0257 + 0.3477 x 0)= 1.2823

201.2823

22

18

24.23.2

19.80.0

16.20.0

2.0257

0.0

A

B

C

D

E

F

11.18

A Put Option Example; K=52Figure 11.7, page 250

K = 52, time step = 1yrr = 5%

504.1923

60

40

720

484

3220

1.4147

9.4636

A

B

C

D

E

F

11.19

What Happens When an Option is American (Figure 11.8, page 251)

505.0894

60

40

720

484

3220

1.4147

12.0

A

B

C

D

E

F

11.20

Delta

Delta (Δ) is the ratio of the change in the price of a stock option to the change in the price of the underlying stockThe value of Δ varies from node to node

11.21

Choosing u and d

One way of matching the volatility is to set

where σ is the volatility and Δt is the length of the time step. This is the approach used by Cox, Ross, and Rubinstein

t

t

eud

euΔσ−

Δσ

==

=

1