PERTEMUAN 14
Dengan mengambil materi
penilaian opsi
DERIVATIVES
by
Prof. ROY SEMBEL, PhD
2007
OPENING QUIZ
Siapa pemenang Hadiah Nobel Ekonomi 1997
dan apa karya mereka ?
DERIVATIVE SECURITIES
SECURITIES WHOSE VALUES DEPEND
ON OTHER MORE ELEMENTARY ASSET.
USES OF DERIVATIVE:
SPECULATION
ARBITRAGE
HEDGING
TYPES OF DERIVATIVES
FORWARDS / FUTURES
OPTIONS
SWAPS
DERIVATIVE DEBACLES
Importance of
BARINGS
Good governance
ORANGE COUNTY
Well defined strategy
LTCM
Risk management
ENRON
PRICING DERIVATIVES:
Binomial Trees
Prof. Roy Sembel, PhD
Smart_WISDOM@yahoogroups.com
Pricing Derivatives 7
Mengapa
mereka memenangkan hadiah Nobel ?
Apa hebatnya karya mereka ?
Artikel “Nobel untuk Opsi yang Sexy”
Pricing Derivatives 8
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 = $20
Stock Price = $18
Pricing Derivatives 9
A Call Option
A 3-month call option on the stock has a strike price of 21.
Stock Price = $22
Option Price = $1
Stock price = $20
Option Price=?
Stock Price = $18
Option Price = $0
Pricing Derivatives 10
Setting Up a Riskless Portfolio
• Consider the Portfolio:
long D shares
short 1 call option
22D – 1
18D
• Portfolio is riskless when 22D – 1 = 18D or
D = 0.25
Pricing Derivatives 11
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
220.25 – 1 = 4.50
• The value of the portfolio today is
4.5e – 0.120.25 = 4.3670
Pricing Derivatives 12
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 (= 0.2520 )
• The value of the option is therefore
0.633 (= 5.000 – 4.367 )
Pricing Derivatives 13
Generalization
• A derivative lasts for time T and is
dependent on a stock
S
ƒ
Su
ƒu
Sd
ƒd
Pricing Derivatives 14
Generalization
(continued)
• Consider the portfolio that is long D shares and short 1
derivative
SuD – ƒu
SdD – ƒd
• The portfolio is riskless when SuD – ƒu = Sd D – ƒd or
ƒu  f d
D
Su  Sd
Pricing Derivatives 15
Generalization
(continued)
• Value of the portfolio at time T is
Su D – ƒu
• Value of the portfolio today is
(Su D – ƒu )e–rT
• Another expression for the
portfolio value today is S D – f
• Hence
ƒ = S D – (Su D – ƒu )e–rT
Pricing Derivatives 16
Generalization
(continued)
• Substituting for D we obtain
ƒ = [ p ƒu + (1 – p )ƒd ]e–rT
where
e d
p
ud
rT
Pricing Derivatives 17
Risk-Neutral Valuation
• ƒ = [ p ƒu + (1 – p )ƒd ]e-rT
• The variables p and (1 – p ) can be interpreted as the riskneutral probabilities of up and down movements
• The value of a derivative is its expected payoff in a riskneutral world discounted at the risk-free rate
S
ƒ
Su
ƒu
Sd
ƒd
Pricing Derivatives 18
Irrelevance of Stock’s Expected
Return
When we are valuing an option in terms of
the underlying stock the expected return on
the stock is irrelevant
Pricing Derivatives 19
Original Example Revisited
Su = 22
ƒu = 1
S
ƒ
Sd = 18
ƒd = 0
• Since p is a risk-neutral probability
• 20e0.12 0.25 = 22p + 18(1 – p ); p = 0.6523
• Alternatively, we can use the formula
e rT  d e 0.120.25  0.9
p

 0.6523
ud
1.1  0.9
Pricing Derivatives 20
Valuing the Option
Su = 22
ƒu = 1
S
ƒ
Sd = 18
ƒd = 0
The value of the option is
e–0.120.25 [0.65231 + 0.34770]
= 0.633
Pricing Derivatives 21
A Two-Step Example
24.2
22
19.8
20
18
16.2
• Each time step is 3 months
Pricing Derivatives 22
Valuing a Call Option
D
22
20
1.2823
A
B
2.0257
18
24.2
3.2
E
19.8
0.0
C
0.0
F
16.2
0.0
• Value at node B
= e–0.120.25(0.65233.2 + 0.34770) = 2.0257
• Value at node A
= e–0.120.25(0.65232.0257 + 0.34770)
= 1.2823
Pricing Derivatives 23
A Put Option Example; X=52
D
60
50
B
A
48
E
40
72
0
C
32
F
Pricing Derivatives 24
A Put Option Example; X=52
D
60
50
4.1923
A
B
1.4147
40
72
0
48
4
E
C
9.4636
F
32
20
Pricing Derivatives 25
What Happens When an
Option is American
D
60
50
B
A
48
E
40
72
0
C
32
F
Pricing Derivatives 26
What Happens When an
Option is American
D
60
50
5.0894
A
B
1.4147
40
72
0
48
4
E
C
12.0
F
32
20
Pricing Derivatives 27
Delta
• Delta (D) 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 D varies from node to
node
Pricing Derivatives 28
Choosing u and d
One way of matching the volatility is to set
ue
s
Dt
d  e  s Dt
where s is the volatility and Dt is the length
of the time step. This is the approach used
by Cox, Ross, and Rubinstein
PRICING DERIVATIVES:
Black-Scholes
Formula
Prof. Roy Sembel, PhD
Smart_WISDOM@yahoogroups.com
Pricing Derivatives 30
The Black-Scholes Random
Walk Assumption
• Consider a stock whose price is S
• In a short period of time of length dt the
change in the stock price is assumed to be
normal with mean mSdt and standard
deviation
sS dt
 m is expected return and s is volatility
Pricing Derivatives 31
The Lognormal Property
• These assumptions imply ln ST is normally
distributed with mean:
ln S 0  (m  s 2 / 2)T
and standard deviation:
s T
• Because the logarithm of ST is normal, ST is
lognormally distributed
Pricing Derivatives 32
The Lognormal Property
continued

ln S T   ln S 0  (m  s 2 2)T , s T

or

ST
2
ln
  (m  s 2)T , s T
S0

where  m,s] is a normal distribution with
mean m and standard deviation s
Pricing Derivatives 33
The Lognormal Distribution
E ( ST )  S0 e mT
2 2 mT
var ( ST )  S0 e
(e
s2T
 1)
Pricing Derivatives 34
The Expected Return
• The expected value of the stock price is
S0emT
• The expected return on the stock with
continuous compounding is m – s2/2
• The arithmetic mean of the returns over
short periods of length dt is m
• The geometric mean of these returns is m –
s2/2
Pricing Derivatives 35
The Volatility
• The volatility is the standard deviation of
the continuously compounded rate of
return in 1 year
• The standard deviation of the return in time
dt is s dt
• 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?
Pricing Derivatives 36
Estimating Volatility from
Historical Data
1. Take observations S0, S1, . . . , Sn at
intervals of t years
2. Define the continuously compounded
return as:
 Si 

ui  ln
 Si 1 
3. Calculate the standard deviation, s , of the
ui ´s
s
4. The historical volatility estimate is: sˆ 
t
Categorization of Stochastic
Processes
•
•
•
•
Discrete time; discrete variable
Discrete time; continuous variable
Continuous time; discrete variable
Continuous time; continuous variable
Modeling Stock Prices
• We can use any of the four types of
stochastic processes to model stock prices
• The continuous time, continuous variable
process proves to be the most useful for the
purposes of valuing derivative securities
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
we are
• We will assume that stock prices follow
Markov processes
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 stock prices. In other words
technical analysis does not work.
• A Markov process for stock prices is clearly
consistent
with weak-form market
efficiency
Example of a Discrete Time
Continuous Variable Model
• A stock price is currently at $40
• At the end of 1 year it is considered
that it will have a probability
distribution of
(40,10) where
(m,s) is a normal distribution with
mean m and standard deviation s.
Questions
• What is the probability distribution of the
stock price at the end of
2 years?
• ½ years?
• ¼ years?
• Dt years?
Taking limits we have defined a continuous
variable, continuous time process
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
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.
A Wiener Process
• We consider a variable z whose value changes
continuously
• The change in a small interval of time Dt is Dz
• The variable follows a Wiener process if
1. Dz   Dt where  is a random drawing from (0,1)
2. The values of Dz for any 2 different (nonoverlapping) periods of time are independent
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
Dx  a( x, t )Dt  b( x, t ) Dt
is only true in the limit as Dt tends to
zero
Ito’s Lemma
• If we know the stochastic process
followed by x, Ito’s lemma tells us the
stochastic process followed by some
function G (x, t )
• Since a derivative security is a function
of the price of the underlying & time,
Ito’s lemma plays an important part in
the analysis of derivative securities
Ito’s Lemma
From stock price process to derivative
2 process
Taking limits
G
G
G 2
dG 
dx 
dt  ½ 2 b dt
x
t
x
Substituting
Stock
Price Process dx  a dt  b dz
We obtain
 G
G
 2G 2 
G
dG  
a
 ½ 2 b  dt 
b dz
t
x
x
 x

This is Ito's Lemma
The Concepts Underlying BlackScholes
• The option price & 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
• The portfolio is instantaneously riskless and
must instantaneously earn the risk-free rate
• This leads to the Black-Scholes differential
equation
The Derivation of the
Black-Scholes Differential Equation
1 of 3:
DS  mS Dt  sS Dz
 ƒ
ƒ
2 ƒ 2 2 
ƒ
D ƒ   mS 
 ½ 2 s S  D t  sS D z
t
S
S
 S

We set up a portfolio consisting of
 1: derivative
ƒ
+ : shares
S
The Derivation of the
Black-Scholes Differential Equation
2 of 3:
The value of the portfolio  is given by
ƒ
  ƒ 
S
S
The change in its value in time Dt is given by
ƒ
D   D ƒ 
DS
S
The Derivation of the
Black-Scholes Differential Equation
3 of 3:
The return on the portfolio must be the risk - free rate. Hence
D  r Dt
We substitute for D ƒ and DS in these equations to get the
Black - Scholes differential equation:
ƒ
ƒ
2 2  ƒ
 rS  ½ s S
 rƒ
2
t
S
S
2
Pricing Derivatives 53
The Black-Scholes Formulas
c  S 0 N ( d1 )  X e
p Xe
 rT
 rT
N (d 2 )
N (  d 2 )  S 0 N (  d1 )
2
ln( S0 / X )  (r  s / 2)T
where d1 
s T
ln( S0 / X )  (r  s 2 / 2)T
d2 
 d1  s T
s T
Pricing Derivatives 54
The N(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 than x
• See Normal distribution tables
Pricing Derivatives 55
Properties of Black-Scholes
Formula
• As S0 becomes very large c tends to
S – Xe-rT and p tends to zero
• As S0 becomes very small c tends to zero
and p tends to Xe-rT – S
Pricing Derivatives 56
Risk-Neutral Valuation
• The variable m does not appear in the BlackScholes equation
• The equation is independent of all variables
affected by risk preference
• This is consistent with the risk-neutral valuation
principle
Pricing Derivatives 57
Applying Risk-Neutral Valuation
1. Assume that the expected
return from an asset is the riskfree rate
2. Calculate the expected payoff
from the derivative
3. Discount at the risk-free rate
Pricing Derivatives 58
Valuing a Forward Contract with
Risk-Neutral Valuation
• Payoff is ST – K
• Expected payoff in a risk-neutral world is
SerT – K
• Present value of expected payoff is
e-rT[SerT – K]=S – Ke-rT
Pricing Derivatives 59
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 implied
volatilities rather than dollar prices
Pricing Derivatives 60
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
Pricing Derivatives 61
Dividends
• European options on dividend-paying
stocks 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 expected
reduction in the stock price expected
Pricing Derivatives 62
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
Pricing Derivatives 63
Black’s 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 option
maturing just before the final ex-dividend
date
Pricing Derivatives 64
Kegiatan dan Forum SCL
• Discovery Learning:
a. Dosen menjelaskan secara rinci penilaian opsi.
b. Mahasiswa diminta untuk terjun ke dunia riil
memahami secara rinci penilaian opsi.
c. Dosen
memberikan evaluasi, sebagai guide
adalah bahan ajar dalam hybrid learning.