Chap 10 Slides Introduction to Binomial Trees Updated2

8/2/2019 Chap 10 Slides Introduction to Binomial Trees Updated2

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FIN 480 (Instructor- Saif Rahman)

Derivative Securities

Chapter 10

Introduction to Binomial Trees


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FIN 480 (Instructor- Saif Rahman)1

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 = $18Stock price = $20


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Stock Price = $22Option Price = $1

Stock Price = $18Option Price = $0

Stock price = $20Option Price=?

A Call Option (Figure 11.1, page 248)

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


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Consider the Portfolio: long sharesshort 1 call option

Portfolio is riskless when 22 1 = 18 or

= 0.25

22 1

18

Setting Up a Riskless Portfolio


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

The riskless portfolio is:

long 0.25 shares

short 1 call option

The value of the portfolio in 3 months is

22 0.251 = 4.50

The value of the portfolio today is

4.5e 0.12 0.25 = 4.3670


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Generalization (Figure 11.2, page 249)

A derivative lasts for time Tand is dependent on a

stock

Su

u

Sdd

S


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Generalization (continued)

Consider the portfolio that is long shares and short 1 derivative

The portfolio is riskless when Su u = Sd d or

SdSu

f du

Su u

Sd d


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Value of the portfolio at time Tis Su u

Value of the portfolio today is (Su u )erT

Another expression for the portfolio value today is

S f

Hence = S (Su u )erT


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Substituting for we obtain

= [p u + (1p )d]erT

where

p e du d

rT


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Risk-Neutral Valuation

= [p u + (1p )d]e-rT

The variablesp and (1p ) 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

u

Sd

d

S


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Original Example Revisited

Sincep is a risk-neutral probability --

20e0.12 0.25 = 22p + 18(1p );p = 0.6523

Alternatively, we can use the formula

6523.09.01.1

9.00.250.12

e

du

dep

rT

Su = 22u = 1

Sd= 18

d= 0

S


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Valuing the Option

The value of the option is

e0.12 0.25 [0.6523 1 + 0.3477 0]

= 0.633

Su = 22u = 1

Sd= 18d= 0

S


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Valuing a Call OptionFigure 11.4, page 253

Value at node B= e0.12 0.25(0.6523 3.2 + 0.3477 0) = 2.0257

Value at node A= e0.12 0.25(0.6523 2.0257 + 0.3477 0)

= 1.2823

Each time step is 3 months

K=21, r=12%

201.2823

22

18

24.23.2

19.80.0

16.20.0

2.0257

0.0

A

B

C

D

E

F


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A Put Option Example;K=52Figure 11.7, page 256

K= 52, t= 1yr

r= 5%

504.1923

60

40

720

484

3220

1.4147

9.4636

A

B

C

D

E

F


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What Happens When an Option is American(Figure 11.8, page 257)

505.0894

60

40

720

484

32

20

1.4147

12.0

A

B

C

D

E

F


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Delta

= Delta = Change in price of the

option/Change in price of the stock = Number of

units of the stock that are held for each call

option shorted to create a risk free hedge

The value of varies from node to node


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Choosing u andd

One way of matching the volatility is to set

where is the volatility and tis the length of the time

step. This is the approach used by Cox, Ross, and

Rubinstein

t

t

eud

eu

1


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The Probability of an Up Move

contractfuturesafor

ratefree-risk

foreigntheisherecurrency wafor

indextheonyield

dividendtheiseindex wherstockafor

stockpayingdnondividenafor

1

)(

)(

a

rea

qea

ea

du

dap

f

trr

tqr

tr

f

Chap 10 Slides Introduction to Binomial Trees Updated2

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