Binomial Option Binomial Option Pricing ModelPricing ModelFinance (Derivative Securities) 312

Tuesday, 3 October 2006

Readings: Chapter 11 & 16

Simple Binomial ModelSimple Binomial Model

Suppose that:• Stock price is currently $20• In three months it will be either $22 or $18• 3-month call option has strike price of 21

Stock Price = $22Option Price = $1

Stock Price = $18Option Price = $0

Stock price = $20Option Price = ?

Option PricingOption Pricing

Consider a portfolio:• long shares, short 1 call option

Portfolio is riskless when 22– 1 = 18 = 0.25

22– 1

18

Option PricingOption Pricing

Riskless portfolio:• long 0.25 shares, short 1 call option

Value of the portfolio in three months:• 22 x0.25 – 1 = 4.50

Value of portfolio today (r = 12%):• 4.5e–0.120.25) = 4.3670

Value of shares: 0.25 20 = 5Value of option: 5 – 4.367 = 0.633

GeneralisationGeneralisation

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

Su ƒu

Sd ƒd

Sƒ

GeneralisationGeneralisation

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

ƒu df

Su Sd

GeneralisationGeneralisation

Value of portfolio at time T:• Su– ƒu

Value of portfolio today:• (Su – ƒu)e–rT

Cost of portfolio today: • S– f

Hence ƒ = S– (Su– ƒu )e–rT

GeneralisationGeneralisation

Substituting for we obtain

ƒ = [ pƒu + (1 – p)ƒd ]e–rT

wherep

e d

u d

rT

Risk-Neutral ValuationRisk-Neutral Valuation

Variables p and (1 – p) can be interpreted as the risk-neutral probabilities of up and down movements

Value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate

Expected stock price: pS0 u + (1 – p)S0 d

• Substitute for p, gives S0erT

Risk-Neutral ValuationRisk-Neutral Valuation

Since p is a risk-neutral probability

20e0.12(0.25) = 22p + 18(1 – p) p = 0.6523Alternatively, using the formula:

Su = 22 ƒu = 1

Sd = 18 ƒd = 0

S ƒ

p

(1– p )

6523.09.01.1

9.00.250.12

e

du

dep

rT

Risk-Neutral ValuationRisk-Neutral Valuation

Su = 22 ƒu = 1

Sd = 18 ƒd = 0

Sƒ

0.6523

0.3477

Value of option:• e–0.12(0.25) (0.6523 x 1 + 0.3477 x 0) = 0.633

Two-Step TreeTwo-Step Tree

Value at node B • e–0.12(0.25)(0.6523 x 3.2 + 0.3477 x0) = 2.0257

Value at node A • e–0.12(0.25)(0.6523 x2.0257 + 0.3477 x0) =

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

Valuing a Put OptionValuing a Put Option

504.1923

60

40

720

484

3220

1.4147

9.4636

A

B

C

D

E

F

Valuing American Valuing American OptionsOptions

505.0894

60

40

720

484

3220

1.4147

12.0

A

B

C

D

E

F

DeltaDelta

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

Determining Determining uu and and dd

Determined from stock volatility t

t

eud

eu

1

contract futures afor 1

rate free-risk

foreign theis herecurrency w afor

index on the yield

dividend theis eindex wherstock afor

stock paying dnondividen afor

)(

)(

a

rea

qea

ea

du

dap

ftrr

tqr

tr

f

Tree ParametersTree Parameters

Conditions:

ert = pu + (1 – p)d

2t = pu2 + (1 – p)d 2 – [pu + (1 – p)d ]2

u = 1/ d Where t is small:

tr

t

t

ea

du

dap

ed

eu

Complete TreeComplete Tree

S0

S0u

S0d S0 S0

S0u2

S0d 2

S0u3

S0u

S0d

S0d 3

Example: Put OptionExample: Put Option

Parameters• S0 = 50; K = 50; r = 10%; = 40%;

• T = 5 months = 0.4167; t = 1 month = 0.0833

Implying that:

u = 1.1224; d = 0.8909;

a = 1.0084; p = 0.5076

Example: Put OptionExample: Put Option89.070.00

79.350.00

70.70 70.700.00 0.00

62.99 62.990.64 0.00

56.12 56.12 56.122.16 1.30 0.00

50.00 50.00 50.004.49 3.77 2.66

44.55 44.55 44.556.96 6.38 5.45

39.69 39.6910.36 10.31

35.36 35.3614.64 14.64

31.5018.50

28.0721.93

Effect of DividendsEffect of Dividends

For known dividend yield:• All nodes ex-dividend for stocks multiplied by

(1 – δ), where δ is dividend yield

For known dollar dividend:• Deduct PV of dividend from initial node• Construct tree• Add PV of dividend to each node before ex-

dividend date