Outline

A One-Step Binomial Model Risk-Neutral Valuation Two-Step Binomial Model American Options Delta Matching Volatility With u and d Options On Other Assets

Binomial Trees Binomial Tree representing different possible

paths that might be followed by the stock price over the life of an option

In each time step, it has a certain probability of moving up by a certain percentage amount and a certain probability of moving down by a certain percentage amount

A one-step binomial model

A Simple Binomial Model A 3-month call option on the stock has a

strike price of 21. A stock price is currently $20 In three months it will be either $22 or $18Stock Price = $22

Option Price = $1

Stock price = $20Stock Price = $18Option Price = $0

Setting Up a Riskless Portfolio

Consider the Portfolio:long shares short 1 call option

Portfolio is riskless when 22– 1 = 18 =

0.25

A riskless portfolio is therefore=> Long : 025 shares Short : 1 call option

22– 1

18

Valuing the Portfolio

The riskless portfolio is:

long 0.25 shares

short 1 call option The value of the portfolio in 3 months is

22 ×0.25 – 1 = 4.5 or 18 ×0.25=4.5 The value of the portfolio today is (if Rf=12%)

4.5e – 0.12×0.25 = 4.367

Valuing the Option

Stock price today = $20 Suppose the option price = f

the portfolio today is 0.25 20 – f = 5 – f

It follows that

5 – f =4.367

So

f=0.633 ---- the current value

of option

Generalization

S0 = stock price

u= percentage increase in

the stock price

d= percentage decrease in

the stock price

ƒ= option on stock price whose current price

ƒu = payoff from the option(when price moves up)

ƒd= payoff from the option(when price moves down)

T= the duration of the option

S0u ƒu

S0d ƒd

S0

ƒ

Generalization (continued)

Consider the portfolio that is long shares and short 1 call option

The portfolio is riskless when S0u– ƒu = S0d– ƒd or

dSuS

fdu

00

ƒ

S0u– ƒu

S0d– ƒd

Generalization (continued)

Value of the portfolio at time T is (S0u – ƒu)e–rT

The cost of setting up the portfolio is S0– f

Hence S0– ƒ = (S0u– ƒu )e–rT

ƒ = S0– (S0u– ƒu )e–rT

Substituting for we obtain

ƒ = [ pƒu + (1 – p)ƒd ]e–rT ---(11.2) where

du

dep

rT

dSuS

fdu

00

ƒ

Generalization (continued)

Ex. (see Figure11.1)

u=1.1, d=0.9,r=0.12,T=0.25,fu=1, ƒd=0

ƒ = [ pƒu + (1 – p)ƒd ]e–rT

= [ 0.6523×1 + 0.3477×0 ]e–0.12×0.25

= 0.633

6523.09.01.1

9.012/312.0

e

du

dep

rT

The option pricing formula in equation(11.2) does not involve the probabilities of stock price moving up or down.

The key reason is that we are not valuing the option in absolute terms. We are calculating its value in terms of the price of the underlying stock. The probabilities of future up or down movements are already incorporated into the stock price.

Risk-Neutral Valuation We assume p and 1-p as probabilities of up and down

movements.

Expected option payoff = p × ƒu + (1 – p ) × ƒd

The expected stock price at time T is

E(ST) = pS0u + (1-p) S0d = pS0 (u-d) + S0d

substituting => E(ST)=S0erT

From this equation, we can see that the stock price grows on average

at the risk-free rate. Because setting the probability of the up

movement equal to p is therefore equivalent to assuming that the

return on the stock equals the risk-free rate.

du

dep

rT

Risk-Neutral Valuation (continued)

In a risk-neutral world all individuals are indifferent to risk. In such a world , investors require no compensation for risk, and the expected return on all securities is the risk-free interest rate.

Risk-neutral valuation states that we can with complete impunity assume the world is risk neutral when pricing options.

Original Example Revisited* European 3-month call option

*Rf=12%

Since p is the probability that gives a return on the stock equal to the risk-free rate. We can find it from

E(ST)=S0erT => 22p + 18(1 – p ) = 20e0.12 0.25 => p = 0.6523At the end of the three months, the call option has a 0.6523 probability of being worth 1 and a 0.3477 probability of being worth zero. So the expect value is

Expected option payoff = p × ƒu + (1 – p ) × ƒd

0.6523×1 + 0.3477×0 = 0.6523In a risk-neutral world this should be discounted at the risk-free rate. The value of the option today is 0.6523e–0.12×0.25= 0.633

S0u = 22 ƒu = 1S0d = 18 ƒd = 0

S0=20 ƒ

p

(1– p )

Real world compare with

Risk-Neutral world

It is not easy to know the correct discount rate to apply to the expected payoff in the real world.

Using risk-neutral valuation can solve this problem because we know that in a risk-neutral world the expected return on all assets is the risk-free rate.

Two-Step Binomial Model

Stock price=$20 , u=10% , d=10% Each time step is 3 months r=12%, K=21

(Figure 11.3 Stock prices in a two-step tree)

20

22

18

24.2

19.8

16.2

Valuing a Call Option(Figure 11.4)

p=

Value at node B = e–0.120.25(0.65233.2 + 0.34770) = 2.0257

Value at node A = e–0.120.25(0.65232.0257 + 0.34770)

= 1.2823

*ƒ = [ pƒu + (1 – p)ƒd ]e–rT ---(11.2) where

201.2823

22

18

24.2

19.8

16.2

2.0257A

B

C

D

E

F

6523.09.01.1

9.012/312.0

e

du

dep

rT

3.2=max{24.2-21,0}

0=max{19.8-21,0}

0=max{16.2-21,0}

0

Generalization

S0u ƒu

S0d ƒd

S0

ƒ

S0d2

ƒdd

S0u2

ƒuu

S0ud ƒud

Figure11.6 Stock and option prices in general two-step tree

Generalization (continued)

ƒ = e–r t[ pƒu + (1 – p)ƒd ]--------------(1) (11.2)

(11.3)

ƒu = e–r t[ pƒuu + (1 – p)ƒud ]-----------(2)

ƒd= e–r t[ pƒud + (1 – p)ƒdd ]------------(3)

ƒ = e–2rt[ p2ƒuu +2p (1 – p)ƒud + (1 – p)2ƒdd ]

du

dep

tr

*The length of time step ist years

A Put Example (Figure 11.7)

K = 52, duration = 2yr, current price = $50

u=20%, d=20%, r = 5%

504.1923

60

40

72

48

32

1.4147

9.4636

A

B

C

D

E

F

6282.08.02.1

8.0105.0

e

p

ƒ = e–2rt[ p2ƒuu +2p (1 – p)ƒud + (1 – p)2ƒdd ] = e–2*0.05*1 [ 0.62822 + 2 0.6282(1 – 0.6282)+ (1 –0.6282)2 20] = 4.1923

0=max{52-72,0}

4=max{52-48,0}

20=max{52-32,0}

American Options

American options can be valued using a binomial tree

The procedure is to work back through the tree from the end to the beginning, testing at each node to see whether early exercise is optimal

American Options(Figure 11.8)

American Put option

K = 52, duration = 2yr, current price = $50,u=20%, d=20%, r = 5%

505.0894

60

40

72

48

32

1.4147

12.0

A

B

C

D

E

F

0=max{52-72,0}

4=max{52-48,0}

20=max{52-32,0}

max{1.4147,52-60}

max{9.4636,52-40}

max{5.0894,52-50}

Value at node B = e–-0.051(0.62820 +0.37184)=1.4147Value at node C = e–-0.051(0.62824+ 0.371820)=9.4636 Value at node A = e–-0.051(0.62821.4147 +0.371812)=5.0894

6282.08.02.1

8.0105.0

e

p

Delta

Delta () is an important parameter in the pricing and hedging of option.

The delta ( of stock option

stock underlying theof price in the change the

optionstock a of price in the change the

Delta

(Figure 11.7) Delta At the end of the first time step is

At the end of the second time step is either

The two-step examples show that delta changes over time

13248

204or 1667.0

4872

40

4024.04060

4636.94147.1

Matching Volatility With u and dIn practice, when constructing a binomial tree to represent the movements in a stock price. We choose the parameters u and d to match the volatility of the stock price.

= volatility t = the length of the time step

This is the approach used by Cox, Ross, and Rubinstein

t

t

eud

eu

1

Options On Other AssetsOption on stocks paying a continuous dividend

yield Dividend yield at rate = q Total return from dividends and capital gains in a

risk-neutral world = r.

=> Capital gains return = r-q The stock expected value after one time step of

length t is S0e(r-q) t

pS0u+(1-p)S0d=S0e(r-q) t =>

( )r q te dp

u d

udeu t /1;

Options On Other Assets

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

Options On Other AssetsOption on stock indices ( a= e(r-q) t ) European 6-month call option on an index level

when index level is 810,K=800, rf=5%, σ=20%,q=2%

5126.0)9048.01052.1(

)9048.00075.1(

0075.1

,9048.0/1

,1052.1

,25.0

25.0)02.005.0(

25.02.0

p

ea

ud

eeu

tt

81053.39

895.19100.66

732.925.06

989.34189.34

810.0010

663.170.00

Node time:

0 0.25 0.5

Node time:

0 0.25 0.5

Node time:

0 0.25 0.5

du

dap

Options On Other Assets

Option on currencies ( a= e(r-rf) t ) Three-step tree:American 3-month call.when the value of

the currency is 0.61,K=0.6,rf=5%, σ=20%,foreign rf=7%

471.0)9439.00594.1(

)9439.09983.0(

9983.0

,9439.0/1

,0594.1

,0833.0

0833.0)07.005.0(

0833.02.0

p

ea

ud

eu

t

0.610.019

0.6320.033

0.5890.007

0.6540.054

0.610.015

0.5690.00

0.5890.00

0.6320.032

0.6770.077

0.5500.00

0 0.0833 0.1667 0.25

du

dap

Options On Other Assets

Option on futures ( a= 1 ) Three-step tree: American 9-month put. when the futures

price is 31,K=30,rf=5%, σ=30%

4626.0)8607.01618.1(

)8607.01(

1

,8607.0/1

,1618.1

,25.025.03.0

p

a

ud

eu

t

312.84

36.020.93

26.684.54

41.850

311.76

22.977.03

26.683.32

36.020

48.620

19.7710.23

0 0.25 0.5 0.75

du

dap