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7/30/2019 Pricing Derivatives
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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.
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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
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A Two-Step Example
Each time step is 3 months
20
22
18
24.2
19.8
16.2
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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
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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
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What Happens When an
Option is American
50
60
40
720
48
32
A
B
C
D
E
F
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What Happens When an
Option is American
505.0894
60
40
720
484
32
20
1.4147
12.0
AB
C
D
E
F
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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
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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
=
=
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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
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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
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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
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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
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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
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Categorization of Stochastic
Processes Discrete time; discrete variable
Discrete time; continuous variable Continuous time; discrete variable
Continuous time; continuous variable
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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
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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
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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
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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 .
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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
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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
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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.
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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
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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= +( , ) ( , )
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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
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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
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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
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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
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The value of the portfolio is given by
The change in its value in time is given by
= +
= +
SS
t
SS
2 of 3: The Derivation of the
Black-Scholes Differential Equation
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3 of 3: The Derivation of the
Black-Scholes Differential Equation
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
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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( / ) ( / )
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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
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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
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Risk-Neutral Valuation
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
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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
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Valuing a Forward Contract with
Risk-Neutral Valuation
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
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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
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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
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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
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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
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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