risk and retun

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.

10-1

Finance 457

10

Chapter Ten

Introduction to Binomial Trees

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Finance 457

Chapter Outline10.1 A one-step binomial model10.2 Risk Neutral Valuation10.3 Two-step binomial trees10.4 A put example10.5 American Options10.6 Delta10.7 Matching volatilities with u and d10.8 Binomial Trees in Practice


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Prospectus:

• The last chapter concerned itself with the value of an option at expiry.

• This section considers the value of an option prior to the expiration date.

• A much more interesting question.


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An Option‑Pricing Formula

• We will start with a binomial option pricing formula to build our intuition.

• Then we will graduate to the normal approximation to the binomial for some real-world option valuation.


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Binomial Option Pricing Model

Suppose a stock is worth $25 today and in one period will either be worth $28.75 or $21.25.

The risk-free rate is 5%. What is the value of an at-the-money call option?

$25

$21.25

$28.75S1S0


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1. A call option on this stock with exercise price of $25 will have the following payoffs.

2. We can replicate the payoffs of the call option. With a levered position in the stock.

$25

$21.25

$28.75S1S0 c1

$3.75

$0


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Binomial Option Pricing ModelBorrow the present value of $21.25 today and buy 1 share. The net payoff for this levered equity portfolio in one period is

either $7.50 or $0. The levered equity portfolio has twice the option’s payoff so

the portfolio is worth twice the call option value.

$25

$21.25

$28.75S1S0 debt

- $21.25portfolio$7.50

$0

( - ) ==

=

c1$3.75

$0- $21.25


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The levered equity portfolio value today is today’s value of one share less the present value of a $21.25 debt:

fre 25.21$25$

$25

$21.25

$28.75S1S0 debt


$0

( - ) ==

=

c1$3.75

$0- $21.25


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Finance 457


We can value the option today as half of the value of the levered

equity portfolio: freC 25.21$25$21

0

$25

$21.25

$28.75S1S0 debt


$0

( - ) ==

=

c1$3.75

$0- $21.25


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Finance 457

If the interest rate is 5%, the call is worth:

The Binomial Option Pricing Model

$25

$21.25

$28.75S1S0 debt


$0

( - ) ==

=

c1$3.75

$0- $21.25

freC 25.21$25$21

0 39.2$

$2.39

c0


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Finance 457


the replicating portfolio intuition.the replicating portfolio intuition.

Many derivative securities can be valued by valuing portfolios of primitive securities when those portfolios have the same payoffs as the derivative securities.

The most important lesson (so far) from the binomial option pricing model is:


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Finance 457

Delta and the Hedge Ratio

• In the example just previous, we replicated the payoffs of the call option with a levered equity portfolio. This has everything to do with anything for the rest of the semester, so let’s take a minute to wrap our brains around it now rather than later.

• The delta of a stock option is the ratio of change in the price of the option to the change in the price of the underlying asset:

• The delta is the number of units of stock we should hold for each option shorted in order to create a riskless hedge.

dSuSff du

00


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Delta and the Hedge Ratio

• This practice of the construction of a riskless hedge is called delta hedging.

• The delta of a call option is positive.– Recall from the example:

dSuSff du

00

• The delta of a put option is negative. • Deltas change through time.

-This is a feature of options that we will return to in chapter 14

21

5.7$75.3$

25.21$75.28$075.3$


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Finance 457

The Risk-Neutral Approach to Valuation

We could value f as the value of the replicating portfolio. An equivalent method is risk-neutral valuation

S0

f

p

1- p

S0u

fu

S0d

fd

])1([ durT fpfpef


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Finance 457


S0 is the value of the underlying asset today.

S0u and S0d are the values of the asset in the next period following an up move and a down move, respectively.

fu and fd are the values of the derivative asset in the next period following an up move and a down move, respectively.

p is the risk-neutral probability of an “up” move.

S0

f

p

1- p

S0u

fu

S0d

fd


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Finance 457


• The key to finding p is to note that it is already impounded into an observable security price: the value of S0:

])1([ 000 dSpuSpeS Tr f

A minor bit of algebra yields:

S0

f

p

1- p

S0u

fu

S0d

fd

])1([ durT fpfpef

dudep

Tr f


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Finance 457

Example of the Risk-Neutral Valuation of a Call:

Suppose a stock is worth $25 today and in one period will either be worth 15% more or 15% less. (u = 1.15; d = 0.85)

The risk-free rate is 5%. What is the value of an at-the-money call option?

The binomial tree would look like this:

$21.25

Cd

p

1- p

$25.00

c0

$28.75

Cu


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Finance 457


The next step would be to compute the risk neutral probabilities

dudep

Tr f

85.015.185.005.

ep

32

$21.25

Cd

$25.00

c0

$28.75

Cu32

31


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Finance 457


After that, find the value of the call in the up state and down state.

$21.25

$0

$25.00

c0

$28.75

$3.7532

31

0$

3175.3$

32

0Tr fec

39.2$0 c


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Finance 457

This risk-neutral result is consistent with valuing the call using a replicating portfolio.

Risk-Neutral Valuation and the Replicating Portfolio

39.2$0$3175.3$

32

0

Tr fec

39.2$25.21$25$21

0 frec


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Finance 457

More on the Binomial Model

• The binomial option pricing model is an alternative to the Black-Scholes option pricing model—especially given the computational efficiency of spreadsheets such as Excel.

• In some situations, it is a superior alternative.• For example if you have path dependency in your

option payoff, you must use the binomial option pricing model.– Path dependency occurs when how you arrive at a price (the path

you follow) for the underlying asset is important.– One example of a path dependent security is a “no regret” call option

where the exercise price is the lowest price of the stock during the option life.


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Finance 457

3 Period Binomial Option Pricing Example

• There is no reason to stop with just two periods.

• Find the value of a three-period at-the-money call option written on a $25 stock that can go up or down 15 percent each period when the risk-free rate is 5 percent.


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Finance 457

Three Period Binomial Process: Stock Prices

$25

28.75

21.25

2/3

1/3

)15.1(00.25$

2)15.1(00.25$

)15.1)(15.1(00.25$

2)15.1(00.25$

)15.1(00.25$

3)15.1(00.25$

)15.1()15.1(00.25$ 2

2)15.1()15.1(00.25$

3)15.1(00.25$

33.06

24.44

2/3

1/3

18.06

2/3

1/3

15.35

2/3

1/3

38.02

2/3

1/3

20.77

2/3

1/3

28.10


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Finance 457

2/3 ]0$3110.3$

32[05.

,,

e

CC UDDU

$25

28.75

21.25

2/3

1/3

33.06

24.44

2/3

1/3

18.06

2/3

1/3

15.35

2/3

1/3

38.02

2/3

1/3

20.771/3

28.10

]0,25$02.38max[$,, UUUC

13.02

]0,25$10.28max[$,,,,,,

DUUUDUUUD CCC

3.10

]0,25$77.20max[$,,,,,,

UDDDUDDDU CCC

0

]0,25$35.15max[$,,

DDDC

0

Three Period Binomial Process: Call Option Prices

10.3$3102.13$

3205.

,

e

C UU

9.28

1.98

0

]0310

32[05.

,

e

C DD

]97.1$3125.9$

32[05.

e

CU

6.54

]0$3197.1$

32[05.

e

CD

1.26

]25.1$3150.6$

32[05.

0 eC

4.57


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Finance 457

Valuation of a Lookback Option

• When the stock price falls due to the stock market as a whole falling, the board of directors tends to reset the exercise price of executive stock options.

• To see how this reset provision adds value, let’s price that same three-period call option (exercise price initially $25) with a reset provision.

• Notice that the exercise price of the call will be the smallest value of the stock price depending upon the path followed by the stock price to get there.


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Finance 457

Three Period Binomial Process: Lookback Call Option Prices

$25

28.75

21.25

33.06

24.44

18.06

24.44

15.35

20.77

28.10

20.77

20.77

28.10

38.02

28.10


10-27

Finance 457


$25

28.75

21.25

33.06

24.44

18.06

$15.35

$38.02

$20.77

$28.10

28.10

$28.10

24.44

$20.77

$20.77

]0,25$02.38max[$,, UUUC

$13.02

$3.1010.3]0,25$10.28max[$,, DUUC

$6.85

$3.66

0]0,44.24$77.20max[$,, DDUC $0

$0

$2.71

$0]0,06.1836.15max[$,, DDDC

66.3]0,44.24$10.28max[$,, UDUC

85.6]0,25.21$10.28max[$,, UUDC

0]0,25.21$77.20max[$,, DUDC

71.2]0,06.18$77.20max[$ UDDC ,,


10-28

Finance 457


$25

28.75

21.25

33.06

24.44

18.06

24.44

10.3$

3102.13$

3205.

, eC UU9.25

0$

3166.3$

3205.

, eC DU

0$

3385.6$

3205.

, eC UD

2.33

4.35

0$

3171.2$

3205.

, eC DD 1.72 $15.35

$38.02

$20.77

$28.10

28.10

$28.10

$20.77

$20.77

$13.02

$3.10

$6.85

$3.66

$0

$0

$2.71

$0


10-29

Finance 457


$25

28.75

21.25

33.06

24.44

18.06

24.44

$15.35

$0

$38.02

$13.02

$20.77

$0

$28.10 $3.10

$28.10

$3.66

$28.10

$6.85

$20.77

$2.71

$20.77

$0

9.25

2.33

4.35

1.72

33.2$

3125.9$

3205.eCU

6.61

3.31

72.1$

3135.4$

3205.eCD

31.3$

3161.6$

3205.

0 eC

5.25


10-30

Finance 457

10.4 A put example

At the money. Before we start, we expect value less than $5.25

$25

28.75

21.25

2/3

1/3

33.06

24.44

2/3

1/3

18.06

2/3

1/3

15.35

2/3

1/3

38.02

2/3

1/3

20.77

2/3

1/3

28.10


10-31

Finance 457

2/3]23.4$310$

32[05.

,,

e

pp UDDU

$25

28.75

21.25

2/3

1/3

33.06

24.44

2/3

1/3

18.06

2/3

1/3

15.35

2/3

1/3

38.02

2/3

1/3

20.771/3

28.10

0$,, UUUp

0

DUUUDUUUD ppp ,,,,,,

0

UDDDUDDDU ppp ,,,,,,

4.23

DDDp ,,

9.65

10.4 A put example

0$310$

3205.

,

e

p UU

0

1.32

5.72

]65.9$3123.4$

32[05.

,

e

p DD

]32.1$310$

32[05.

e

pU

0.43

]72.5$3132.1$

32[05.

e

pD

2.63

]63.2$3142$.

32[05.

0 ep

1.09


10-32

Finance 457

10.4 A put example

• We can check our work with put-call parity:

25$09.1$25$57.4$ 305.

000

e

SpKec rT


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Finance 457

10.5 American Options

• At each node prior to expiry, compare immediate exercise with the option’s value.

• If the proceeds of immediate exercise are higher than the value of the option, exercise.

• Use the exercise value at that node to work backward through the tree to find the value of an American option at time 0.


10-34

Finance 457

Optimal Early Exercise: American Put

2/3

$25

28.75

21.25

2/3

1/3

33.06

24.44

2/3

1/3

18.06

2/3

1/3

15.35

2/3

1/3

38.02

2/3

1/3

20.771/3

28.10

0

0

4.23

9.65

0

1.32

5.72

0.43

2.63

1.09

6.94

3.75

1.21

]75.3$3143$.

32[05.

0 ep

]945.6$3132.1$

32[02.3$ 05. e

3.02


10-35

Finance 457

Optimal Exercise of American Calls

• There are two cases to consider:– A stock paying a known dividend yield– The dollar amount of the dividend is known.


10-36

Finance 457

Known Dividend Yield

Ex-dividend date

S0

S0 u

S0 d

S0 u2

S0

S0 d2

S0 u3(1-)

S0 u(1-)

S0 d(1-)

S0 d3(1-)

ud 1

Ex-dividend date

S0 u2(1-)

S0 (1-)

S0 d2(1-)


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Finance 457

Known Dollar Dividend

ud 1

S0

S0 u

S0 d

S0 u2

S0

S0 d2

Ex-dividend date

S0 u2– D

S0 – D

S0 d2 – D

(S0 u2– D) u

(S0 – D) u

(S0 d2 – D)u

(S0 d2 – D)d

(S0 u2– D) d

(S0 – D) d


10-38

Finance 457

10.7 Matching Volatility with u and d

• In practice, we choose the parameters u and d to match the volatility of the stock price.

t

t

ed

euδ

δ


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Finance 457

10.8 Binomial Trees in PracticeThe BOPM is easily incorporated into Excel spreadsheetsAfter 30 or so steps, the results are excellent.

14% s 28.75$ 1 Maturity 25.00$ 1 n 3.75$ 1 D t q

25.00$ S 0

25.00$ X Stock Price 25.00$ 5% r f Exercise Price 25.00$

1.1500 u Ordinary Call 2.38$ 0.8500 d1.0500 a 1- q

66.67% Risk Neutral Prob 21.25$ 33.33% 1- R.N. Prob 25.00$

-$

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