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Valuing Stock Options
Chapter 12+13
Fundamentals of Futures and Options Markets, 7th Ed, Global Edition.
Ch 13, Copyright © John C. Hull 2010
1
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
2
Introduction

Two methods for pricing options


Binomial model: a discrete-time option pricing
model
Black-Scholes-Merton model: a continuous
time option pricing model
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
3
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
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
4
Binomial model of options pricing
One-step binomial model





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 is the value of the call option at expiry if the stock
price is $18?
What volume of stock makes the portfolio riskless?
What is the future value of the portfolio?
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
5
Binomial model of options pricing
One-step binomial model

Stock Price = $22
Stock price = $20
Stock Price = $18
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
6
Binomial model of options pricing
One-step binomial model


What is the value of the call option at expiry if
the stock price is $22?
What is the value of the call option at expiry if
the stock price is $18?
Stock Price = $22
Option Price = $1
Stock price = $20
Option Price=?
Stock Price = $18
Option Price = $0
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
7
Binomial model of options pricing
One-step binomial model

What volume of stock makes the portfolio riskless?

22  1  18
   0.25
What is the future value of the portfolio?
22  0.25  1  4.5

18  0.25  4.5
The portfolio is riskless so we would expect it to have the
same value in either scenario.
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
8
Binomial model of options pricing
One-step binomial model



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?
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
9
Binomial model of options pricing
One-step binomial model




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.
4.5 e0.123 12  4.367
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
20  0.25  f  4.367
 f  20  0.25  4.367  0.633
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
10
Binomial model of options pricing
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
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
11
Binomial model of options pricing
Generalised one-step binomial model

A derivative lasts for time T and is dependent
on a stock
S
ƒ
Su
ƒu
Sd
ƒd
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
12
Binomial model of options pricing
Generalised one-step binomial
model

Consider the portfolio that is long  shares and short 1
derivative
Su – ƒu
Sd – ƒd

The portfolio is riskless when Su – ƒu = Sd  – ƒd or
ƒu  f d

Su  Sd
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
13
Binomial model of options pricing
Generalised one-step binomial
model





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
e rT  d
p
ud
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
14
Binomial model of options pricing
One-step binomial model


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 call option?
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
15
Binomial model of options pricing
One-step binomial model


What is the current value of the call option?
The probability of an up movement:
rT
 d e0.12.25  0.9
e
p

 0.6523
ud
1.1  0.9

The value of the option:
f  erT  p f u  1  p  f d 
f  e0.120.25 0.6523 1  1  0.6523  0  0.633
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
16
Binomial model of options pricing
Illustrate how to arbitrage an anomaly


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.
How would you profit from an arbitrage if the option was
quoted at $1.00?
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
17
Binomial model of options pricing
Illustrate how to arbitrage an anomaly




How would you profit from an arbitrage if the option was
quoted at $1.00?
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:
1 0
f  fd
 u

 0.25
S u  S d 22  18
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
18
Binomial model of options pricing
Illustrate how to arbitrage an
anomaly





How would you profit from an arbitrage if the option was
quoted at $1.00?
If we sell 1,000 calls and buy 250 shares, this would
require borrowing, at the risk-free rate, funds equal to:
1,000  $1.00  250  $20.00  4,000 ie borrow $4,000
At expiry the portfolio will equal:
250  $22  1,000  $1  250  $18  1,000  $0  4,500
The return on the investment will equal:
 4,500  0.12
 1 
 0.095, or 38% pa

4
 4,000 
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
19
Binomial model of options pricing
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 riskfree interest rate.
The value of a derivative is its expected payoff in a riskneutral world discounted at the risk-free rate
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
20
Binomial model of options pricing
One-step binomial model


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 call option?
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
21
Binomial model of options pricing
One-step binomial model


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
22 p  181  p   20 e0.120.25
4 p  20e0.120.25  18
p  0.6523

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:
0.6523 1  0.3477  0  0.6523
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
22
Binomial model of options pricing
One-step binomial model


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
0.6523 e0.120.25  0.633
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
23
Binomial model of options pricing
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?
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
24
Binomial model of options pricing
Two-step binomial model: Call option
D
B
20
A
E
C
F
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
25
Binomial model of options pricing
Two-step binomial model: Call
option

The value of the stock at nodes D, E and F:
S D  20  1  0.10  1  0.10  24.2
S E  20  1  0.10  1  0.10  19.8
S F  20  1  0.10  1  0.10  16.2
20
24.2
22
19.8
A
18
16.2
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
26
Binomial model of options pricing
Two-step binomial model: Call option

The value of the option at nodes D, E and F:
C D  Max0, S D  K   Max0, 24.2  21  3.2
C E  Max0, S E  K   Max0,19.8  21  0
C F  Max0, S F  K   Max0,16.2  21  0
24.2
3.2
22
20
19.8
0.0
A
18
16.2
0.0
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
27
Binomial model of options pricing
Two-step binomial model: Call
option

The value of the option at nodes B and C:
0.120.25

 2.0257



p

3
.
2




1

p


0

e
CB
0.120.25
 1  0.10
e
where p 
 0.6523
1  0.10  1  0.10
24.2
3.2
22
2.0257
20
19.8
0.0
A
18
0.0
16.2
0.0
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
28
Binomial model of options pricing
Two-step binomial model: Call
option

The value of the option at node A:
 1.2823
C   p2.0257  1 p 0e
 1  0.10
where p  e
 0.6523
1  0.10  1  0.10
0.120.25
B
0.120.25
24.2
3.2
22
2.0257
20
1.2823
19.8
0.0
18
0.0
16.2
0.0
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
29
Binomial model of options pricing
Generalised two-step binomial model
 p  p   f uu

  p  1  p   f

ud

f  e2 rt 
 1  p   p   f ud



 1  p   1  p   f dd 
p
p
fu
fuu
1-p
f
fud
1-p
fd
p
1-p
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
fdd
30
Binomial model of options pricing
Generalised two-step binomial model

The value of an option using the generalised
two-step binomial model can be calculated
f e
 2 rt
p
2
f uu  2 p1  p  f ud  1 p  f dd 
2
e rT  d
where p 
ud
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
31
Binomial model of options pricing
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?
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
32
Binomial model of options pricing
Two-step binomial model: Put option

What is the value of the option?
e rT  d
e 0.051.0  1  0.20
p

 0.6282
u  d 1  0.20  1  0.20
0.6282
60
1.4147
0.6282
1-0.6282
50
40
1-0.6282 9.4636
72
0
0.6282
1-0.6282
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
48
4
32
20
33
Binomial model of options pricing
Two-step binomial model: Put option

What is the value of the option?
f e
0.051
0.6282 1.4147

 1  0.6282  9.4636


 4.1923
0.6282  0.6282  0

f  e0.052  2  0.6282  1  0.6282  4 


 1  0.6282  1  0.6282  20
 4.1923
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
34
Binomial model of options pricing
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:
f c  Maxert  p f u  1  p  f d , S T  K 
f p  Maxert  p f u  1  p  f d , K  S T 
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
35
Binomial model of options pricing
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.
What is the value of the option?
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
36
Binomial model of options pricing
Two-step binomial model: American

What is the value of the option?
f B  Max1.4147, 52  60  1.4147
f C  Max9.4636, 52  40  12
0.6282
0.6282
60
1.4147
1-0.6282
50
40
1-0.6282 12
72
0
0.6282
1-0.6282
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
48
4
32
20
37
Binomial model of options pricing
Two-step binomial model: American

What is the value of the option?
0.6282 1.4147 


0.052 
Max f  e
 1  0.6282 12, 52  50 




 5.0894
0.6282
50
5.0894
60
1.4147
40
1-0.6282 12
0.6282
72
0
1-0.6282
0.6282
1-0.6282
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
48
4
32
20
38
Binomial model of options pricing
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
ƒu  f d

Su  Sd
In a multi-step binomial tree the value of  varies from
node to node
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
39
Binomial model of options pricing
Determining u and d

In practice u and d are determined from the stock price
volatility:
u  es
t
d  1 u  e s

t
where s is the volatility and t is the length of the time
step
This is the approach used by Cox, Ross, and Rubinstein
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
40
Binomial model of options pricing
Options on various assets

The price on options on various assets, calculated using
the binomial model, is similar except for the calculation
of p:
ad
p
ud
where a equals




ert 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
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
41
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
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
42
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?
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
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
.
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
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 c
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
45
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)
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
46
How Share Prices Move
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 m and
standard deviation s.
- In t periods the returns have mean m t and
variance of s2 t
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
47
How Share Prices Move
If ST is the share price at time T and it has a
lognormal distribution then it will have
Mean
E(ST) = S0e m T
Variance Var(ST) = S02e2 m T (es T  1)
See next slide
For the long term continuous returns ln (ST / S0)
Mean
m - s2/2
2
Variance
s2
See Ex 13.2: Confidence Limits for Stock Returns
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
48
The Lognormal Distribution
E ( ST )  S0 e mT
2 2 mT
var ( ST )  S0 e
(e
s2T
 1)
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
49
The Expected Return
p 293
From the CAPM we know the expected return
m that investors require depends upon
the riskiness of an asset &
The level of interest rates like r
The value of an option is not affected by m but
there is an issue you need to be aware of.
While the return in a short period t is mt the
return with continuous compounding over long
periods R has a different mean from m namely
E(R) = m – s2/2
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
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 m (1/250) the mean
yearly return should be m … but it is not!!
The yearly return over a period of T years with
continuous compounding R is given by
1 ST
R  ln
T S0
For this R value we find E( R) = m – s2/2
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
51
The Expected Return
This difference reflects the difference between
arithmetic and geometric means
Geometric means are always lower because
they are less affected by extreme values
This is illustrated in the next snapshot
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
52
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
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
53
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
s in 1 year & s t 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?
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
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 s = 25% p.a. the volatility for 1 day is
s t = 25 x 0.063 = 1.575%
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
55
Estimating Volatility from Historical
Data (page 295-298)
1.
2.
Take observations S0, S1, . . . , Sn on the
variable at end of each trading day
Define the continuously compounded daily
return as:
 Si 

ui  ln
 Si 1 
3.
4.
Calculate the standard deviation, s , of the
ui ´s (This is for daily returns)
The historical volatility per yearly estimate
is:
s  252
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
56
Estimating Volatility from Historical Data
(Calculating s)
To find the mean for ui we use the formula
1 n
u   ui
n i 1
To find the variance for ui we use the formula
1 n
1 n 2
2
2
s 
(ui - u ) 
ui - n u 



n - 1 i 1
n - 1  i 1

2
To find the standard deviation s we find the square
root of the variance
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
57
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
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
58
The Concepts Underlying Black-Scholes
Key Assumptions






Share prices have a lognormal distribution
with mean m and standard deviation s
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.
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
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
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
60
The Concepts Underlying Black-Scholes



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
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
61
The Concepts Underlying Black-Scholes
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

Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
62
The Black-Scholes Formulas
(See page 299-300)
c  S 0 N (d1 )  K e
 rT
N (d 2 )
p  K e  rT N (d 2 )  S 0 N (d1 )
2
ln( S 0 / K )  (r  s / 2)T
where d1 
s T
ln( S 0 / K )  (r  s 2 / 2)T
d2 
 d1  s T
s T
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
63
The N(x) Function
The other terms have all been used before but in
addition to the d terms there is a new term
 N(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 13.4
 The 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
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
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 zero
As S0 becomes very small now
c tends to zero and
p tends to Ke-rT – S0
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
65
Black-Scholes-Merton model of options pricing
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?
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
66
Black-Scholes-Merton model of options pricing
The BSM model
Example 10.9



What is the value of the option if it is a call?
c  42 N ( 0.7693 )  40 e 0.10.5 N ( 0.6278 )
ln( 42 / 40 )  ( 0.1  0.22 / 2 )  0.5
where
d1 
 0.7693
0.2 0.5
ln( 42 / 40 )  ( 0.1  0.22 / 2 )  0.5
d2 
 0.6278
0.2 0.5
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.
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
67
Black-Scholes-Merton model of options pricing
The BSM model
Example 10.9



What is the value of the option if it is a put?
p  40 e 0.10.5 N ( 0.6278 )  42 N ( 0.7693 )
ln( 42 / 40 )  ( 0.1  0.22 / 2 )  0.5
where
d1 
 0.7693
0.2 0.5
ln( 42 / 40 )  ( 0.1  0.22 / 2 )  0.5
d2 
 0.6278
0.2 0.5
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.
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
68
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
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
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
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
70
Valuing a Forward Contract with RiskNeutral Valuation
In the case of forward contracts if we have a long
contract then
Payoff at expiry is ST – K
Expected payoff in a risk-neutral world is
S0erT – K
The 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).
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
71
Implied Volatility
The implied volatility of an option is the
volatility for which the Black-Scholes price equals
the market price
The is a one-to-one correspondence between
prices and implied volatilities
Traders 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
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
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.
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
73
The VIX Index of S&P 500 Implied
Volatility; Jan. 2004 to Sept. 2009
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
74
Dividends




European options on dividend-paying stocks
are valued by substituting the stock price less
the present value of dividends into the BlackScholes-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 exdividend date
Look at Ex 13.6 p 306
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
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
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
76
Black’s Approximation for Dealing with
Dividends 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
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
77
Valuing Employee Stock Options
p 306-309
Please read Ch 13.11 very carefully as this
material is examinable.
Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010
78