Pricing Derivatives

7/30/2019 Pricing Derivatives

1/55

A Simple Binomial Model

A stock price is currently $20

In three months it will be either $22 or $18

Stock Price = $22

Stock Price = $18Stock price = $20


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Stock Price = $22Option Price = $1

Stock Price = $18Option Price = $0

Stock price = $20Option Price=?

A Call Option

A 3-month call option on the stock has a strike price of 21.


3/55

Consider the Portfolio: long sharesshort 1 call option

Portfolio is riskless when 22 1 = 18 or = 0.25

22 1

18

Setting Up a Riskless Portfolio


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Valuing the Portfolio

(Risk-Free Rate is 12%)

The riskless portfolio is:

long 0.25 shares short 1

call option

The value of the portfolio in 3 months is

220.25 1 = 4.50

The value of the portfolio today is

4.5e 0.120.25 = 4.3670


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Valuing the Option

The portfolio that is

long 0.25 shares short 1

option

is worth 4.367

The value of the shares is

5.000 (= 0.2520 )

The value of the option is therefore

0.633 (= 5.000 4.367 )


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Generalization

(continued) Consider the portfolio that is long shares and short 1

derivative

The portfolio is riskless when Su u = Sd d or

=

u df

Su Sd

Su u

Sd d


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Generalization

(continued)

Substituting for we obtain

= [p u + (1 p )d]erT

where

p e du d

rT

=


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Risk-Neutral Valuation

= [p u + (1 p )d]e-rT

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

The value of a derivative is its expected payoff in a risk-

neutral world discounted at the risk-free rate

Su

u

Sd

d

S

p

(1p)


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Irrelevance of Stocks Expected

Return

When we are valuing an option in terms of

the underlying stock the expected return onthe stock is irrelevant


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Original Example Revisited

Sincep is a risk-neutral probability

20e0.12 0.25 = 22p + 18(1 p );p = 0.6523 Alternatively, we can use the formula

6523.09.01.1

9.00.250.12

=

=

=

e

du

dep

rT

Su = 22u = 1

Sd= 18

d= 0

S

p

(1p)


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Valuing the Option

The value of the option is

e0.120.25 [0.65231 + 0.34770]

= 0.633

Su = 22u = 1

Sd= 18d= 0

S

0.6523

0.3477

A T S E l


14/55

A Two-Step Example

Each time step is 3 months

20

22

18

24.2

19.8

16.2


15/55

Valuing a Call Option

Value at node B = e0.120.25(0.65233.2 + 0.34770) = 2.0257

Value at node A = e0.12

0.25(0.65232.0257 + 0.34770)

= 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


16/55

A Put Option Example;X=52

50

60

40

720

48

32

A

B

C

D

E

F


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A Put Option Example;X=52

504.1923

60

40

720

484

3220

1.4147

9.4636

A

B

C

D

E

F


18/55

What Happens When an

Option is American

50

60

40

720

48

32

A

B

C

D

E

F


19/55

What Happens When an

Option is American

505.0894

60

40

720

484

32

20

1.4147

12.0

AB

C

D

E

F


20/55

Delta

Delta () is the ratio of the changein the price of a stock option to the

change in the price of the underlyingstock

The value of varies from node to

node


21/55

Choosing u and d

One way of matching the volatility is to set

where is the volatility and tis thelength of the time step. This is the approach

used by Cox, Ross, and Rubinstein

u e

d e

t

t

=

=


22/55

The Black-Scholes Random Walk

Assumption Consider a stock whose price is S

In a short period of time of length tthe

change in the stock price is assumed to benormal with mean Stand standarddeviation

is expected return and is volatilitytS


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The Lognormal Property These assumptions imply ln STis normally

distributed with mean:

and standard deviation:

Because the logarithm ofSTis normal, STislognormally distributed

TS )2/(ln 20 +

T


24/55

The Lognormal Property

continued

where [m,s] is a normal distribution withmean m and standard deviations

[ ]

[ ]TTS

S

TTSS

T

T

=

+

,)2(ln

or

,)2(lnln

2

0

2

0


25/55

The Lognormal Distribution

E S S e

S S e e

T

T

T

T T

( )

( ) ( )

=

= 0

0

2 2 2 1

var


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The Expected Return

The expected value of the stock price isS0eT

The expected return on the stock withcontinuous compounding is 2/2

The arithmetic mean of the returns overshort periods of length t is

The geometric mean of these returns is 2/2


27/55

The Volatility

The volatility is the standard deviation ofthe continuously compounded rate ofreturn in 1 year

The standard deviation of the return intime tis

If a stock price is $50 and its volatility is

25% per year what is the standarddeviation of the price change in one day?

t


28/55

Estimating Volatility from

Historical Data

1. Take observations S0, S1, . . . , Sn at intervals of years

2. Define the continuously compounded return as:

3. Calculate the standard deviation,s , of the ui s

4. The historical volatility estimate is:

uS

Si

i

i

=

ln1

=s


29/55

Categorization of Stochastic

Processes Discrete time; discrete variable

Discrete time; continuous variable Continuous time; discrete variable

Continuous time; continuous variable


30/55

Modeling Stock Prices

We can use any of the four types of

stochastic processes to model stock prices

The continuous time, continuous variableprocess proves to be the most useful for the

purposes of valuing derivative securities


31/55

Markov Processes (See pages 218-9)

In a Markov process future movements in

a variable depend only on where we are,

not the history of how we got where weare

We will assume that stock prices followMarkov processes


32/55

Weak-Form Market Efficiency

The assertion is that it is impossible to

produce consistently superior returns with a

trading rule based on the past history of stockprices. In other words technical analysis does

not work.

A Markov process for stock prices is clearlyconsistent with weak-form market efficiency


33/55

Example of a Discrete Time

Continuous Variable Model

A stock price is currently at $40

At the end of 1 year it is consideredthat it will have a probability

distribution of (40,10) where

(,) is a normal distribution withmean and standard deviation .


34/55

Questions

What is the probability distribution of thestock price at the end of 2 years?

years?

years?

tyears?

Taking limits we have defined a continuous

variable, continuous time process


35/55

Variances & Standard

Deviations

In Markov processes changes in

successive periods of time are independent This means that variances are additive

Standard deviations are not additive


36/55

Variances & Standard Deviations

(continued)

In our example it is correct to say that

the variance is 100 per year. It is strictly speaking not correct to say

that the standard deviation is 10 per

year.


37/55

A Wiener Process

We consider a variablez whose value changes

continuously

The change in a small interval of time t is z The variable follows a Wiener process if

1.

2. The values ofz for any 2 different (non-overlapping) periods of time are independent

z t= (0,1)where is a random drawing from


38/55

Ito Process

In an Ito process the drift rate and the

variance rate are functions of time

dx=a(x,t)dt+b(x,t)dz The discrete time equivalent

is only true in the limit as ttends to

zero

x a x t t b x t t= +( , ) ( , )


39/55

Itos Lemma

If we know the stochastic process

followed byx, Itos lemma tells us the

stochastic process followed by somefunction G (x, t)

Since a derivative security is a function

of the price of the underlying & time,Itos lemma plays an important part in

the analysis of derivative securities


40/55

From stock price process to derivative process

Itos Lemma

Taking limits

Substituting

We obtain

This is Ito's Lemma

dG Gx

dx Gt

dt Gx

b dt

dx a dt b dz

dG G

xa G

t

G

xb dt G

xb dz

= + +

= +

= + +

+

2

2

2

2

2

2

Stock Price Process dx=a(x,t)dt+b(x,t)dz


41/55

The Concepts Underlying Black-

Scholes The option price & the stock price depend on

the same underlying source of uncertainty

We can form a portfolio consisting of the stockand the option which eliminates this source ofuncertainty

The portfolio is instantaneously riskless andmust instantaneously earn the risk-free rate

This leads to the Black-Scholes differentialequation


42/55

1 of 3: The Derivation of the

Black-Scholes Differential Equation

S S t S z

S S t S S t S S z

S

= +

= + +

+

We set up a portfolio consisting of

: derivative

+

: shares

2

2

2 2

1


43/55

The value of the portfolio is given by

The change in its value in time is given by

= +

= +

SS

t

SS




44/55



The return on the portfolio must be the risk - free rate. Hence

We substitute for and in these equations to get the

Black - Scholes differential equation:

=

+ + =

r t

S

trS

SS

Sr

2 22

2


45/55

The Black-Scholes Formulas

c S N d X e N d

p X e N d S N d

dS X r T

T

d S X r T

Td T

rT

rT

=

=

=+ +

= + =

0 1 2

2 0 1

10

20

1

2 2

2 2

where

( ) ( )

( ) ( )

ln( / ) ( / )

ln( / ) ( / )


46/55

TheN(x) Function

N(x) is the probability that a normally

distributed variable with a mean of zero and

a standard deviation of 1 is less thanx See Normal distribution tables

P i f Bl k S h l


47/55

Properties of Black-Scholes

Formula

As S0 becomes very large c tends to

S Xe-rTandp tends to zero

As S0 becomes very small c tends to zero

andp tends toXe-rT S


48/55


The variable does not appear in the Black-Scholes equation

The equation is independent of all variables

affected by risk preference

This is consistent with the risk-neutral valuation

principle


49/55

Applying Risk-Neutral Valuation

1. Assume that the expected

return from an asset is the risk-

free rate2. Calculate the expected payoff

from the derivative

3. Discount at the risk-free rate


50/55

Valuing a Forward Contract with


Payoff is ST K

Expected payoff in a risk-neutral world isSerT K

Present value of expected payoff is

e-rT[SerT K]=S Ke-rT


51/55

Implied Volatility

The implied volatility of an option is the

volatility for which the Black-Scholes price

equals the market price There is a one-to-one correspondence

between prices and implied volatilities

Traders and brokers often quote impliedvolatilities rather than dollar prices


52/55

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


53/55

Dividends

European options on dividend-payingstocks are valued by substituting the stock

price less the present value of dividends

into the Black-Scholes formula Only dividends with ex-dividend dates

during life of option should be included

The dividend should be the expectedreduction in the stock price expected


54/55

American Calls

An American call on a non-dividend-paying stock

should never be exercised early

An American call on a dividend-paying stockshould only ever be exercised immediately

prior to an ex-dividend date


55/55

Blacks Approach to Dealing with

Dividends in American Call Options

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 optionmaturing just before the final ex-dividend date

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