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Binomial Option Pricing

Professor P. A. Spindt

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A simple example

A stock is currently priced at $40 pershare.

In 1 month, the stock price may go up by 25%, or

go down by 12.5%.

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A simple example

Stock price dynamics:

$40

$40x(1+.25) = $50

$40x(1-.125) = $35

t = now t = now + 1 month

up state

down state

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Call option

A call option on this stock has a strikeprice of $45

t=0 t=1

Stock Price=$40;

Call Value=$c

Stock Price=$50;

Call Value=$5

Stock Price=$35;

Call Value=$0

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A replicating portfolio

Consider a portfolio containing Dsharesof the stock and $B invested in risk-free

bonds. The present value (price) of this portfolio isDS + B = $40 D + B

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Portfolio value

t=0 t=1

$50 D + (1+r/12)B

$35 D + (1+r/12)B

$40 D + B

up state

down state

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This portfolio will replicate the option ifwe can find a D and a B such that

$50 D + (1+r/12) B = $5

$35 D + (1+r/12) B = $0

and

Portfolio payoff = Option payoff

Up state

Down state

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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/(3x1.00417) = $11.62

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Payoffs at maturity

up state dow n state

Stock Price 50.00$ 35.00$

1/3 Share 16.67$ 11.67$

Bond Repayment 11.67$ 11.67$

Net portfolio 5.00$ -$

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Since the the replicating portfolio hasthe 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 $13.33 - $11.62 =

$1.71.Therefore, c = $1.71

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A general (1-period) formula

D Cu Cd

Su SdB

SuCd SdCu

1 r Su Sd

p r d

u d

c DS B pCu 1 p Cd

1 r

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An observation aboutD

As the time interval shrinks towardzero, delta becomes the derivative.

D Cu Cd

Su Sd

C

S

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Put option

What about a put option with a strikeprice of $45

t=0 t=1

Stock Price=$40;

Put Value=$p

Stock Price=$50;

Put Value=$0

Stock Price=$35;

Put Value=$10

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t=0 t=1

$50 D + (1+r/12)B

$35 D + (1+r/12)B

$40 D + B

up state

down state

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This portfolio will replicate the put ifwe can find a D and a B such that

$50 D + (1+r/12) B = $0

$35 D + (1+r/12) B = $10

and

Portfolio payoff = Option payoff

Up state

Down state

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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/(3x1.00417) = $33.19.

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Two Periods

Suppose two price changes are possibleduring the life of the option

At each change point, the stock may goup by Ru% or down by Rd%

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Two-Period Stock Price

DynamicsFor 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

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Two-Period Stock Price

Dynamics

$47

$48.53

$45.83

$50.10

$47.31

$44.68

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Terminal Call Values

$C0

$Cu

$Cd

Cuu =$5.10

Cud =$2.31

Cdd =$0

At expiration, a call with a strikeprice of $45 will be worth:

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Two Periods

The two-period Binomial model formulafor a European call is

Cp2CUU 2p(1 p)CUD (1 p)

2CDD

1 r 2

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ExampleTelMex Jul 45 143 CB 23/16 -5/16 47 2,703

Two Per iod Binomial Model

Call Option Price Calculator

Stock Price $47.00

Exercise Price $45.00Years to Maturity 0.08

Risk-free Rate (per annum) 5.00%

Ru 3.25%

Rd -2.50%

p 47.10%

Stock Value in Up Up State 50.10$

Call Value in Up Up State 5.10$

Stock Value in Down Up State 47.31$

Call Value in Down Up State 2.31$

Stock Value in Down Down State 44.68$

Call Value in Down Down State -$

Call Value 2.28$

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Estimating Ruand Rd

According to Rendleman and Barter you canestimate Ru and Rd from the mean and

standard deviation of a stocks returns

Ru exptn

tn 1

Rd exptn

tn 1

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Estimating Ruand Rd

In these formulas, t is the options time to expiration(expressed in years) and n is the number of intervalst is carved into

Ru exptn

tn 1

Rd exptn

tn 1

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For Example

Consider a call option with 4 months torun (t = .333 yrs) and

n = 2 (the 2-period version of thebinomial model)

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For Example

The price of a call with an exercise price of $105 on a stockpriced at $108.25

Two Period Binomial Model

Call Option Price Calculator

Stock Price $108.25

Exercise Price $105.00

Years to Maturi ty 0.33

Risk-free Rate (per annum) 7.00%

Ru 12.36%

Rd -6.79%

p 41.49%

Stock Value in Up Up State 136.66$Call Value in Up Up State 31.66$

Stock Value in Down Up State 113.37$

Call Value in Down Up State 8.37$

Stock Value in Down Down State 94.05$

Call Value in Down Down State -$

Call Value 9.30$

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Anders Consulting

Focusing on the Nov and Jan options,how do Black-Scholes prices compare

with the market prices listed in caseExhibit 2?

Hints:

The risk-free rate was 7.6% and the expectedreturn on stocks was 14%.

Historical Estimates: IBM = .24 & Pepsico = .38

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