fundamentals of futures and options markets, 7th ed, global edition. ch 13, copyright © john c....

Fundamentals of Futures and Options Markets, 7th Ed, Global Edition.Ch 13, Copyright © John C. Hull 2010

Valuing Stock Options

Chapter 12+13

1

2

Sub-Topics

Binomial model of options pricing Black-Scholes-Merton (BSM) model of

options pricing Pricing options on individual stocks and

indices Pricing options on currencies Pricing options on interest rates

Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

3

Introduction

Two methods for pricing options Binomial model: a discrete-time option pricing

model Black-Scholes-Merton model: a continuous

time option pricing model


4

Binomial model of options pricing

One-step binomial model

The binomial model limits the price moves of the underlying asset to one of only two possible new prices

A one-period model limits the time over which the price move occurs to one period, at the end of which the underlying asset moves to one of two possible prices and simultaneously the option expires

We assume that arbitrage profits are arbitraged away to reveal an arbitrage-free price


5



You have a long position in a stock and a short position in a call option on the stock. The current price of the stock is $20. In 3 months it will either be $22 or $18. The 3-month call option has a strike price of $21.

What is the value of the call option at expiry if the stock price is $22?


What volume of stock makes the portfolio riskless? What is the future value of the portfolio?


6



Stock Price = $22

Stock Price = $18

Stock price = $20


7





Stock Price = $22Option Price = $1

Stock Price = $18Option Price = $0

Stock price = $20Option Price=?


8



What volume of stock makes the portfolio riskless?

What is the future value of the portfolio?

The portfolio is riskless so we would expect it to have the same value in either scenario.

25.0

18122

5.425.018

5.4125.022


9



You have a long position in a stock and a short position in a call option on the stock. The current price of the stock is $20. In 3 months it will either be $22 or $18. The 3-month call option has a strike price of $21. The risk-free rate of interest is 12% pa, continuously compounded.

What is the current value of the portfolio? What is the current value of the call option?


10



What is the current value of the portfolio? Riskless portfolios earn the risk-free rate of return, hence

the present value of the portfolio equals the future value discounted at the risk-free rate of return.

What is the current value of the call option? The current value of the portfolio also equals the value of

the stock plus the value of the option, hence

367.45.4 12312.0 e

633.0367.425.020

367.425.020

f

f


11


Generalised one-step binomial model

S0 =stock price f = price of option S0u =stock price moves up S0d =stock price moves down fu = price of option if stock price moves up fd= price of option if stock price moves down


12



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

Su ƒu

Sd ƒd

Sƒ


13



Consider the portfolio that is long shares and short 1 derivative

The portfolio is riskless when Su– ƒu = Sd – ƒd or

SdSufdu

ƒ

Su– ƒu

Sd– ƒd


14



Value of the portfolio at time T is Su– ƒu = Sd– ƒd

Value of the portfolio today is (Su – ƒu )e–rT

Another expression for the portfolio value today is S0– f

Hence ƒ = S0– (Su – ƒu )e–rT

Substituting for we obtain ƒ = [ p ƒu + (1 – p )ƒd ]e–rT

where dude

prT


15




What is the current value of the call option?


16



What is the current value of the call option? The probability of an up movement:

The value of the option:

6523.09.01.1

9.025.12.0

edudep

rT

633.006523.0116523.0

125.012.0

ef

fpfpef durT


17


Illustrate how to arbitrage an anomaly


How would you profit from an arbitrage if the option was quoted at $1.00?


18




If the option is selling at $1.00 and it should be selling at $0.633, it is overpriced.

Sell the option and buy the stock. The number of units of stock bought per option sold:

25.0182201

SS

ff

du

du


19




If we sell 1,000 calls and buy 250 shares, this would require borrowing, at the risk-free rate, funds equal to:

ie borrow $4,000 At expiry the portfolio will equal:

The return on the investment will equal:

000,400.20$25000.1$000,1

500,40$000,118$2501$000,122$250

paor %38,095.0412.0

1000,4500,4


20


Risk-neutral valuation

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

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

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


21




What is the current value of the call option?


22



What is the current value of the call option? In a risk-neutral world the expected return on a stock

must equal the risk-free rate

At the end of three months, the call option has a 0.6523 probability of being worth 1 and a 0.3477 probability of being worth zero. Its expected future value therefore is:

6523.0

18204

201182225.012.0

25.012.0

pep

epp

6523.003477.016523.0 Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

23



What is the current value of the call option? In a risk-neutral world the expected future value should

be discounted at the risk-free rate to get the present value

633.06523.0 25.012.0 e


24


Two-step binomial model: Call option

You have a long position in a stock and a short position in a call option on the stock. The current price of the stock is $20. In consecutive 3-month periods there is an equal chance it will either rise by 10% or fall by 10%. The 3-month call option has a strike price of $21. The risk-free rate of interest is 12% pa continuously compounding.

What is the value of the option at nodes B and C? What is the value of the option at node A?


25



20 A

B

C

D

F

E


26



The value of the stock at nodes D, E and F:

20 A

22

18

24.2

19.8

16.2

2161001100120

8191001100120

2241001100120

...S

...S

...S

F

E

D


27



The value of the option at nodes D, E and F:

20 A

22

18

24.23.2

19.80.0

16.20.0

0212.16,0,0

0218.19,0,0

2.3212.24,0,0

MaxKSMaxC

MaxKSMaxC

MaxKSMaxC

FF

EE

DD


28



The value of the option at nodes B and C:

20 A

222.0257

180.0

24.23.2

19.80.0

16.20.0

6523.0

10.0110.0110.01

0257.2012.325.012.0

25.012.0

epwhere

eppCB


29



The value of the option at node A:

201.2823

222.0257

180.0

24.23.2

19.80.0

16.20.0

6523.0

10.0110.0110.01

2823.1010257.225.012.0

25.012.0

epwhere

eppCB


30


Generalised two-step binomial model

f

fu

fd

fuu

fdd

fud

p

1-p

1-p

1-p

p

p

fpp

fpp

fpp

fpp

ef

dd

ud

ud

uu

tr

11

1

12


31


Generalised two-step binomial model

The value of an option using the generalised two-step binomial model can be calculated

dude

pwhere

fpfppfpefrT

dduduutr

112 222


32


Two-step binomial model: Put option

A two-year European put has a strike of $52 on a stock whose current price is $50. There are two time steps of one year, in each the stock price either moves up by 20% or down by 20%. The risk-free rate of interest is 5% pa continuously compounding.

What is the value of the option?


33




6282.0

20.0120.0120.010.105.0

edude

prT

50

601.4147

409.4636

720

484

3220

0.6282

1-0.6282

0.6282

0.6282

1-0.6282

1-0.6282


34




1923.4

206282.016282.01

46282.016282.02

06282.06282.0205.0

ef

1923.4

4636.96282.01

4147.16282.0105.0

ef


35


American options

In valuing American options The value of the option at the final nodes remains the

same as for European options The value of the option at earlier nodes is the greater

of: The expected payoff discounted at the risk-free rate The payoff from early exercise:

SKfpfpeMaxf

KSfpfpeMaxf

Tdutr

p

Tdutr

c

,1

,1


36


Two-step binomial model: American

A two-year American put has a strike of $52 on a stock whose current price is $50. There are two time steps of one year, in each the stock price either moves up by 20% or down by 20%. The risk-free rate of interest is 5% pa continuously compounding.



37




50

601.4147

4012

720

484

3220

0.6282

1-0.6282

0.6282

0.6282

1-0.6282

1-0.6282

124052,4636.9

4147.16052,4147.1

Maxf

Maxf

C

B


38




0894.5

5052,126282.01

4147.16282.0205.0

efMax

505.0894

601.4147

4012

720

484

3220

0.6282

1-0.6282

0.6282

0.6282

1-0.6282

1-0.6282


39


Delta

Delta () is the ratio of the change in the price of a stock option to the change in the price of the underlying stock

In a multi-step binomial tree the value of varies from node to node

SdSufdu

ƒ


40


Determining u and d In practice u and d are determined from the stock price

volatility:

where is the volatility andt is the length of the time step

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

t

t

eud

eu

1


41


Options on various assets The price on options on various assets, calculated using

the binomial model, is similar except for the calculation of p:

where a equals ert for a non dividend paying stock or bond e(r-q)t for a dividend paying stock or index e(r-rf)t for a currency 1 for a futures contract

duda

p


42

Black-Scholes-Merton model of options pricing

Explain the assumptions of the model The returns of the underlying asset are continuously compounding

and are normally distributed, ie they are log-normally distributed There are no riskless arbitrage opportunities Investors can borrow and lend at the risk-free rate, which in the

short term is constant The volatility of the underlying is known and constant There are no taxes or transaction costs There are no cashflows on the underlying The options are European



Introduction

We look at the standard approach to pricing options where we focus on European options which can only be exercised at a specific time.

A call option gives the buyer the right to buy the asset at time T for the strike price K so at time T

Value of a Call = max(ST - K, 0)

A put option gives the buyer the right to sell the asset at time T for the strike price K so at time T

Value of a Put = max(K - ST , 0)

What should these values be at earlier times?43


Introduction

The value of a call option c has 3 parts

The intrinsic value is the value if the option was exercised at time t which is (St - K)

The time value of money on the strike price is the difference the strike price and its present value which shows how much we save by paying K at time T not now (K - Ke - rT)

The insurance I shows how much investors are willing to pay to limit future losses

So c = (St - K) + (K - Ke- rT ) + I = St - Ke- rT + I .

44


Introduction

In the formula for the value of a call option

c = St - Ke- rT + I

we know K, r and T but we do not know what the share price St will be at any future date.

The best we can do is to make assumptions about how share prices change over time and what this tells us about the probability distribution of possible St values i.e. what type of distribution and what

mean and variance the St values have .

We use these assumed values in our formula for c45


How Share Prices Move

Many studies have shown share prices Si have a

skewed probability distribution like the lognormal distribution shown in Fig 13.1 p 290.

For values with a lognormal distribution, the logs of these values ln(Si) have the normal distribution shown

in Fig 13.2 p 291.

If a share does not have dividends then its continuous rate of return ui is defined as the log of the ratio of the

current price & the previous price

ui = ln (Si / Si-1) = ln (Si) - ln (Si-1)

46



As both ln (Si) and ln (Si-1) are normally distributed so too is their difference ui. Using this result in the Black-

Scholes model it is assumed that

- Returns on a share (S / S) over short time periods are normally distributed

- Returns in different periods are independent

- In 1 period the returns have mean and standard deviation .

- In t periods the returns have mean t and variance of 2t

47



If ST is the share price at time T and it has a

lognormal distribution then it will have

Mean E(ST) = S0e T

Variance Var(ST) = S02e2 T

See next slide

For the long term continuous returns ln (ST / S0)

Mean - 2/2

Variance 2

See Ex 13.2: Confidence Limits for Stock Returns

48

1)( T2e


The Lognormal Distribution

E S S e

S S e e

TT

TT T

( )

( ) ( )

0

02 2 2

1

var

49


The Expected Return p 293

From the CAPM we know the expected return that investors require depends upon the riskiness of an asset & The level of interest rates like rThe value of an option is not affected by but there is an issue you need to be aware of.While the return in a short period t is t the return with continuous compounding over long periods R has a different mean from namely E(R) = –

50


The Expected Return

To see why suppose the t are 1 day periods with 250 trading days in a year then t = 1/250 If the mean daily return is (1/250) the mean yearly return should be … but it is not!! The yearly return over a period of T years with continuous compounding R is given by

For this R value we find E( R) = –

51

0S

S ln

1 R T

T


The Expected Return

This difference reflects the difference between arithmetic and geometric meansGeometric means are always lower because they are less affected by extreme valuesThis is illustrated in the next snapshot

52


53

Mutual Fund Returns (See Business Snapshot 13.1 on page 294)

If Returns are 15%, 20%, 30%, -20% and 25%

Their arithmetic mean of these returns is 14%

(15 + 20 + 30 - 20 + 25) / 5 = 14

The actual value of $100 after 5 yrs is

100 x 1.15 x 1.2 x 1.3 x 0.8 x 1.25 = $179.40

With 14% returns we should have

100 x 1.145 = 192.54

The actual return is the geometric mean 12.4%

100 x 1.1245 = 179.40


The Volatility

The value of the insurance component I of c depends upon the riskiness of the call option which depends upon the volatility.

The volatility which is the standard deviation of the continuously compounded rate of return is in 1 year & in period t

If a stock price is $50 and its volatility is 25% per year what is the standard deviation of the price change in one day?

t

54


Nature of Volatility

Volatility is usually much greater when the market is open (i.e. the asset is trading) than when it is closed

For this reason time is usually measured in “trading days” not calendar days when options are valued where there are 252 trading days in one year and 1 day is a period of t = (1/252)

If = 25% p.a. the volatility for 1 day is

= 25 x 0.063 = 1.575%

55

t


Estimating Volatility from Historical Data (page 295-298)

1. Take observations S0, S1, . . . , Sn on the variable at end of each trading day

2. Define the continuously compounded daily return as:

3. Calculate the standard deviation, s , of the ui ´s (This is for daily returns)

4. The historical volatility per yearly estimate is:

uS

Sii

i

ln1

56

252s

Estimating Volatility from Historical Data (Calculating )

To find the mean for ui we use the formula

To find the variance for ui we use the formula

To find the standard deviation we find the square root of the variance


57

n

iiuu

1

n

1

222 n - 1-n

1 ) -

1-n

1 uuuu

n

i

n

iii

11

2(

Estimating Volatility from Historical Data (If there are Dividends)

Dividends are usually paid twice a year. In those periods where there are no dividends we use the same formula for daily returns ui which is

ui = ln (Si / Si-1)

When dividends D are paid the formula changes to

ui = ln ([Si + D]/ Si-1)

The formulae for the mean and variance are the same as when there are no dividends


58


The Concepts Underlying Black-Scholes Key Assumptions

Share prices have a lognormal distribution with mean and standard deviation

All assets are perfectly divisible and have zero trading costs

There are no dividends in the time to maturity There are no riskless arbitrage opportunities Security trading is continuous Investors can borrow or lend at a constant

risk-free rate r.

59


The Concepts Underlying Black-Scholes

The option price and the stock price depend on the same underlying source of uncertainty

We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty (see p 298)

The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate

60



To obtain this formula we set up a portfolio containing shares and options

The option price & the stock price depend on the same underlying source of uncertainty and move in a well defined way as

c rises when S rises & p falls when S rises We can form a portfolio consisting of the

stock and the option which eliminates this source of uncertainty as it gives a fixed return

61



The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate

In the example on p 298-299 and Fig 13.3 we see that c and S change in the following way

c = 0.4 S Here the riskless portfolio contains

A long position in 40 shares

A short position in 100 call options

N.B. If this relationship changes however we would have to rebalance

62


The Black-Scholes Formulas(See page 299-300)

TdT

TrKSd

T

TrKSd

dNSdNeKp

dNeKdNScrT

rT

10

2

01

102

210

)2/2()/ln(

)2/2()/ln(

)()(

)()(

where

63


The N(x) Function

The other terms have all been used before but in addition to the d terms there is a new termN(d) is the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than d as shown in Fig 13.4The tables for N(d) at the end of the book and at the back of the Formula sheet

The use of the Black-Scholes formula is demonstrated in Ex 13.4 p 301

64


Properties of Black-Scholes Formula

As S0 becomes very large both d1 and d2 also

become large and both N(d1) and N(d2) are both now

close to 1, the area under the Normal curve. With N(d1) and N(d2) values close to 1 we find from our option value formulae that

c tends to S0 – Ke-rT and

p tends to zeroAs S0 becomes very small now

c tends to zero and

p tends to Ke-rT – S0 65


The BSM model Example 10.9

The stock price six months from the expiration of a European option is $42, the exercise price is $40, the risk-free interest rate is 10% per annum, and the volatility is 20% per annum.

What is the value of the option if it is a call? What is the value of the option if it is a put?

66Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010



What is the value of the option if it is a call?

Using tables: N(0.7693) = 0.7791, N(0.6278) = 0.7349, N(-0.7693) = 0.2209 and N(-0.6278) = 0.2651

The price of the European call option is $4.76.

627805020

502220104042

769305020

502220104042

62780407693042

2

1

5010

...

.)/..()/ln(d

...

.)/..()/ln(d where

).(Ne ).(N c ..




What is the value of the option if it is a put?

Using tables: N(0.7693) = 0.7791, N(0.6278) = 0.7349, N(-0.7693) = 0.2209 and N(-0.6278) = 0.2651

The price of the European put option is $0.81.

627805020

502220104042

769305020

502220104042

76930426278040

2

1

5010

...

.)/..()/ln(d

...

.)/..()/ln(d where

).(N ).(N e p ..



Risk-Neutral Valuation

A key result in the pricing of derivatives is the risk-neutral valuation principle which says

Any security dependent on other traded securities can be valued on the assumption that investors are risk neutral

While investors may not actually be risk-neutral we can assume they are when we derive the prices of derivatives. This means

All expected returns are equal to r

We can use r as our discount rate everywhere

69


Applying Risk-Neutral Valuation

Derivatives can be valued with the following procedure

1.Assume that the expected return from an asset is the risk-free rate

2.Calculate the expected payoff from the derivative

3.Discount at the risk-free rate

70


Valuing a Forward Contract with Risk-Neutral Valuation

In the case of forward contracts if we have a long contract then Payoff at expiry is ST – KExpected payoff in a risk-neutral world is

S0erT – KThe value of the forward contract f is given by the present value of the expected payoff

f = e-rT[S0erT – K] = S0 – Ke-rT .

(which is the same as equation 5.5).

71


Implied Volatility

The implied volatility of an option is the

volatility for which the Black-Scholes price equals the market priceThe is a one-to-one correspondence between prices and implied volatilitiesTraders and brokers often quote implied volatilities rather than dollar prices

The CBOE publishes the SPX VIX which shows the implied volatility for a range of 30-day put and call options on the S&P 500

72


Implied Volatility

An index value of 15 means an implied volatility of about 15%

In the futures contracts on the VIX one contract is 1000 times the index

How futures contracts on the VIX work is shown in Ex 13.5 p 304

In the graph in Fig 13.5 on p 305 we can see how the VIX is usually somewhere between 10 and 20 but during the GFC it got up to 80.

73

The VIX Index of S&P 500 Implied Volatility; Jan. 2004 to Sept. 2009


74


Dividends

European options on dividend-paying stocks are valued by substituting the stock price less the present value of dividends into the Black-Scholes-Merton formula

Only dividends with ex-dividend dates during life of option should be included

The “dividend” should be the expected reduction in the stock price on the ex-dividend date

Look at Ex 13.6 p 306

75


American Calls

An American call on a non-dividend-paying stock should never be exercised early

An American call on a dividend-paying stock should only ever be exercised immediately prior to an ex-dividend date

76


Black’s Approximation for Dealing withDividends in American Call Options

This procedure is illustrated in Ex 13.7 p 307

Here we set the American price equal to the maximum of two European prices:

1.The 1st European price is for an option maturing at the same time as the American option

2.The 2nd European price is for an option maturing just before the final ex-dividend date

77


Valuing Employee Stock Options p 306-309

Please read Ch 13.11 very carefully as this material is examinable.

78

fundamentals of futures and options markets, 7th ed, global edition. ch 13, copyright © john c....

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