Download Model of the Behavior of Stock Prices Chapter 12

Stochastic
Calculus and The
Black-Scholes
Model
What is calculus?
• To obtain long term estimation from short
term information
Why we need stochastic calculus
• Many important results can not be
obtained by simple averaging.
• Some examples
Investment Return
• The initial value of a portfolio is 1000
dollars. Suppose the portfolio either gain
60% with 50% probability or lose 60% with
50% probability. What is its average rate
of return? What is the most likely value of
the portfolio after 10 years?
Solution
• The average rate of return is
0.6*50% + (-0.6)*50% = 0
• The most likely value of the portfolio after
10 years is therefore 1000. Right?
• Let’s calculate
• 1000*(1+0.6)^5*(1-0.6)^5 = 107.4
• This is much less than 1000.
The value distribution for the
first four years
6553.6
4096
2560
1600
1000
1638.4
1024
640
400
409.6
256
160
102.4
64
25.6
Discussion
• The arithmetic mean of value is always
1000 dollars each year. But high end, as
well as low end, values are extremely rare.
• The most likely value decline over time.
Arithmetic mean and geometric
mean
• Suppose a portfolio either change r1 with
probability of p1 or r2 with probability of
p2.
• The arithmetic rate of return: r1*p1+r2*p2
• The geometric rate of return:
(1+r1)^p1*(1+r2)^p2-1
• The geometric rate of return of the last portfolio
• (1+0.6)^0.5*(1-0.6)^0.5-1= -0.2
• The value of portfolio after 10 years, calculated
from geometric rate of return
• 1000*(1-0.2)^10 = 107.4
• The same as earlier calculation
• The value of portfolio after 4 years, calculated
from geometric rate of return
• 1000*(1-0.2)^4 = 409.6
• The same as the middle number in year 4.
Examples
• Amy and Betty are Olympic athletes. Amy got
two silver medals. Betty got one gold and one
bronze. What are their average ranks in two
sports? Who will get more attention from media,
audience and advertisers?
• Rewards will be given to Olympic medalists
according to the formula 1/x^2 million dollars,
where x is the rank of an athlete in an event.
How much rewards Amy and Betty will get?
Solution
• Amy will get
1
2  2  0.5 million
2
• Betty will get
1
1
 1.11 million
2
3
Discussion
• Although the average ranks of Amy and
Betty are the same, the rewards are not
the same.
Mathematical derivatives and
financial derivatives
• Calculus is the most important intellectual
invention. Derivatives on deterministic variables
• Mathematically, financial derivatives are
derivatives on stochastic variables.
• In this course we will show the theory of financial
derivatives, developed by Black-Scholes, will
lead to fundamental changes in the
understanding social and life sciences.
The history of stochastic calculus
and derivative theory
• 1900, Bachelier: A student of Poincare
– His Ph.D. dissertation: The Mathematics of Speculation
– Stock movement as normal processes
– Work never recognized in his life time
• No arbitrage theory
– Harold Hotelling
• Ito Lemma
– Ito developed stochastic calculus in 1940s near the end of WWII,
when Japan was in extreme difficult time
– Ito was awarded the inaugural Gauss Prize in 2006 at
age of 91
The history of stochastic calculus
and derivative theory (continued)
• Feynman (1948)-Kac (1951) formula,
• 1960s, the revival of stochastic theory in
economics
• Thorp, E. O., & Kassouf, S. T. (1967). Beat the
market: a scientific stock market system.
• 1973, Black-Scholes
– Black, F. and Scholes, M. (1973). The Pricing of
Options and Corporate Liabilities
– Fischer Black died in 1995, Scholes and Merton were
awarded Nobel Prize in economics in 1997.
The history of stochastic calculus
and derivative theory (continued)
• Recently, real option theory and an
analytical theory of project investment
inspired by the option theory
• It often took many years for people to
recognize the importance of a new theory
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 and time, Ito’s
lemma plays an important part in the
analysis of derivative securities
• Why it is called a lemma?
The Question
Suppose
dx  a ( x, t )dt  b( x, t )dz
How G(x, t) changes with the change of x and t?
Taylor Series Expansion
• A Taylor’s series expansion of G(x, t)
gives
G
G
 2G 2
G 
x 
t  ½ 2 x
x
t
x
 2G
 2G 2

x t  ½ 2 t  
xt
t
Ignoring Terms of Higher
Order Than t
In ordinary calculus we have
G
G
G 
x 
t
x
t
In stochastic calculus this becomes
G
G
 2G 2
G 
x 
t  ½
x
2
x
t
x
because x has a component which is
of order t
Substituting for x
Suppose
dx  a( x, t )dt  b( x, t )dz
so that
x = a t + b  t
Then ignoring terms of higher order than t
G
G
 2G 2 2
G 
x 
t  ½ 2 b  t
x
t
x
The 2Dt Term
Since   (0,1) E ()  0
E ( 2 )  [ E ()]2  1
E ( )  1
2
It follows that E ( 2 t )  t
The variance of t is proportion al to t and can
2
be ignored. Hence
G
G
1 G 2
G 
x 
t 
b t
2
x
t
2 x
2
Taking Limits
Taking limits
G
G
2G 2
dG 
dx 
dt  ½ 2 b dt
x
t
x
Substituting
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
Differentiation in stochastic and
deterministic calculus
• Ito Lemma can be written in another form
G
G
1  2G 2
dG 
dx 
dt 
b dt
2
x
t
2 x
• In deterministic calculus, the differentiation
is
G
G
dG 
dx 
dt
x
t
The simplest possible model of
stock prices
• Over long term, there is a trend
• Over short term, randomness dominates.
It is very difficult to know what the stock
price tomorrow.
A Process for Stock Prices
dS  mSdt  sSdz
where m is the expected return s is
the volatility.
The discrete time equivalent is
S  mSt  sS t
Application of Ito’s Lemma
to a Stock Price Process
The stock price process is
d S  mS dt  sS d z
For a function G of S and t
 G
G
 2G 2 2 
G
dG  
mS 
 ½ 2 s S dt 
sS dz
t
S
S
 S

Examples
1. The forward price of a stock for a contract
maturing at time T
G  S e r (T  t )
dG  ( m  r )G dt  sG dz
2. G  ln S

s2 
dt  s dz
dG   m 
2 

Arithmetic mean and geometric
mean
• The initial price of a stock is 1000 dollars. The
stock’s return over the past six years are
• 19%, 25%, 37%, -40%, 20%, 15%.
Questions
–
–
–
–
What is the arithmetic return
What is the geometric return
What is the variance
What is mu – 1/2sigma^2? Compare it with the
geometric return.
– What is the final price of the stock?
– What are the final prices of the stock calculated from
arithmetic and geometric rate of returns?
– Which number: arithmetic return or geometric return
is more relevant to investors?
Answer
•
•
•
•
•
Arithmetic mean: 12.67%
Geometric mean: 9.11%
Variance: 7.23%
Arithmetic mean -1/2*variance: 9.05%
Geometric mean is more relevant because
long term wealth growth is determined by
geometric mean.
The BlackScholes
Model
Randomness matters in
nonlinearity
• An call option with strike price of 10.
• Suppose the expected value of a stock at
call option’s maturity is 10.
• If the stock price has 50% chance of
ending at 11 and 50% chance of ending at
9, the expected payoff is 0.5.
• If the stock price has 50% chance of
ending at 12 and 50% chance of ending at
8, the expected payoff is 1.
ds
 rdt  sdz
s
• Applying Ito’s Lemma, we can find
S  S0e
1
( r  s 2 )t
s z (t )
2
e
• Therefore, the geometric rate of return is r0.5sigma^2.
• The arithmetic rate of return is r
The history of option pricing models
• 1900, Bachelier, the purpose, risk
management
• 1950s, the discovery of Bachelier’s work
• 1960s, Samuelson’s formula, which
contains expected return
• Thorp and Kassouf (1967): Beat the
market, long stock and short warrant
• 1973, Black and Scholes
The influence of Beat the Market
• Practical experience is not merely the
ultimate test of ideas; it is also the ultimate
source. At their beginning, most ideas are
dimly perceived. Ideas are most clearly
viewed when presented as abstractions,
hence the common assumption that
academics --- who are proficient at
presenting and discussing abstractions --are the source of most ideas. (p. 6,
Treynor, 1973) (quoted in p. 49)
Why Black and Scholes
• Jack Treynor, developed CAPM theory
• CAPM theory: Risk and return is the same
thing
• Black learned CAPM from Treynor. He
understood return can be dropped from
the formula
Fischer Black (1938 – 1995 )
•
•
•
•
•
Start undergraduate in physics
Transfer to computer science
Finish PhD in mathematics
Looking for something practical
Join ADL, meet Jack Treynor, learn finance and
economics
• Developed Black-Scholes
• Move to academia, in Chicago then to MIT
• Return to industry at Goldman Sachs for the last
11 years of his life, starting from 1984
• Fischer never took a course in either economics or
finance, so he never learned the way you were
supposed to do things. But that lack of training proved to
be an advantage, Treynor suggested, since the
traditional methods in those fields were better at
producing academic careers than new knowledge.
Fischer’s intellectual formation was instead in physics
and mathematics, and his success in finance came from
applying the methods of astrophysics. Lacking the ability
to run controlled experiments on the stars, the
astrophysist relies on careful observation and then
imagination to find the simplicity underlying apparent
complexity. In Fischer’s hands, the same habits of
research turned out to be effective for producing new
knowledge in finance. (p. 6)
• Both CAPM and Black-Scholes are thus much
simpler than the world they seek to illuminate,
but according to Fischer that’s a good thing, not
a bad thing. In a world where nothing is
constant, complex models are inherently fragile,
and are prone to break down when you lean on
them too hard. Simple models are potentially
more robust, and easier to adapt as the world
changes. Fischer embraced simple models as
his anchor in the flux because he thought they
were more likely to survive Darwinian selection
as the system changes. (p. 14)
• John Cox, said it best, ‘Fischer is the only
real genius I’ve ever met in finance. Other
people, like Robert Merton or Stephen
Ross, are just very smart and quick, but
they think like me. Fischer came from
someplace else entirely.” (p. 17)
• Why Black is the only genius?
• No one else can achieve the same level of
understanding?
• Fischer’s research was about developing clever
models ---insightful, elegant models that
changed the way we look at the world. They
have more in common with the models of
physics --- Newton’s laws of motion, or
Maxwell’s equations --- than with the
econometric “models” --- lists of loosely
plausible explanatory variables --- that now
dominate the finance journals. (Treynor, 1996,
Remembering Fischer Black)
The objective of this course
• We will learn Black-Scholes theory.
• Then we will develop an economic theory
of life and social systems from basic
physical and economic principles.
• We will show that the knowledge that
helps Black succeed will help everyone
succeed.
• There is really no mystery.
Effect of Variables on Option
Pricing
Variable
S0
K
T
s
r
D
c
+
–
?
+
+
–
p
–
+?
+
–
+
C
+
–
+
+
+
–
P
–
+
+
+
–
+
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
• The portfolio is instantaneously riskless and must
instantaneously earn the risk-free rate
• This leads to the Black-Scholes differential equation
• Thorp and Kassouf (1967): Beat the market, long
stock and short warrant. This provided the stimulus
for this line of thinking.
The Derivation of the BlackScholes Differential Equation
S  mS t  sS z
 ƒ
ƒ
2 ƒ 2 2 
ƒ
ƒ   mS 
 ½ 2 s S t 
sS z
t
S
S
 S

W e set up a portfolio consisting of
 1 : derivative
ƒ
+
: shares
S
The Derivation of the Black-Scholes
Differential Equation continued
The value of the portfolio  is given by
ƒ
  ƒ 
S
S
The change in its value in time t is given by
ƒ
  ƒ 
S
S
The Derivation of the Black-Scholes
Differential Equation continued
The return on the portfolio must be the risk - free
rate. Hence
  r t
We substitute for ƒ and S in these equations
to get the Black - Scholes differenti al equation :
2
ƒ
ƒ

ƒ
2 2
 rS
½ s S
 rƒ
2
t
S
S
The Differential Equation
• Any security whose price is dependent on the
stock price satisfies the differential equation
• The particular security being valued is determined
by the boundary conditions of the differential
equation
• In a forward contract the boundary condition is
ƒ = S – K when t =T
• The solution to the equation is
ƒ = S – K e–r (T
–t)
The payoff structure
• When the contract matures, the payoff is
C ( S ,0)  max( S  K ,0)
• Solving the equation with the end condition,
we obtain the Black-Scholes formula
The Black-Scholes Formulas
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
How they found the solution
• The equation had been obtained quite
awhile ago. But they could not find a
solution for some time.
• Later they use formulas from others which
contains expected rate of return. They set
the return to be the risk free rate. That was
the formula.
• It can be solved directly from the equation
and the initial condition.
The basic property of BlackSchoels formula
C  S  Ke
 rT
Rearrangement of d1, d2
S
ln(  rT )
1
Ke
d1 
 s T
2
s T
S
ln(  rT )
1
Ke
d2 
 s T
2
s T
Properties of B-S formula
• When S/Ke-rT increases, the chances of
exercising the call option increase, from
the formula, d1 and d2 increase and N(d1)
and N(d2) becomes closer to 1. That
means the uncertainty of not exercising
decreases.
• When σ increase, d1 – d2 increases,
which suggests N(d1) and N(d2) diverge.
This increase the value of the call option.
Similar properties for put options
 rT
Ke
ln(
)
1
S
 d2 
 s T
2
s T
 rT
Ke
ln(
)
1
S
 d1 
 s T
2
s T
Calculating option prices
• The stock price is $42. The strike price for
a European call and put option on the
stock is $40. Both options expire in 6
months. The risk free interest is 6% per
annum and the volatility is 25% per
annum. What are the call and put prices?
Solution
• S = 42, K = 40, r = 6%, σ=25%, T=0.5
ln( S0 / K )  (r  s 2 / 2)T
d1 
s T
• = 0.5341
ln( S0 / K )  (r  s 2 / 2)T
d2 
s T
• = 0.3573
Solution (continued)
c  S 0 N (d1 )  K e
 rT
N (d 2 )
• =4.7144
pKe
 rT
• =1.5322
N (d 2 )  S 0 N (d1 )
The Volatility
• The volatility of an asset is the standard
deviation of the continuously
compounded rate of return in 1 year
• As an approximation it is the standard
deviation of the percentage change in the
asset price in 1 year
Estimating Volatility from
Historical Data
1. Take observations S0, S1, . . . , Sn at
intervals of t years
2. Calculate the continuously compounded
return in each interval as:
 Si 

ui  ln
 Si 1 
3. Calculate the standard deviation, s , of
the ui´s
4. The historical volatility estimate is: sˆ 
s
t
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
Causes 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
Dividends
• European options on dividend-paying
stocks are valued by substituting the stock
price less the present value of dividends
into Black-Scholes
• 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
Calculating option price with
dividends
• Consider a European call option on a
stock when there are ex-dividend dates in
two months and five months. The dividend
on each ex-dividend date is expected to
be $0.50. The current share price is $30,
the exercise price is $30. The stock price
volatility is 25% per annum and the risk
free interest rate is 7%. The time to
maturity is 6 month. What is the value of
the call option?
Solution
• The present value of the dividend is
• 0.5*exp (-2/12*7%)+0.5*exp(-5/12*7%)=0.9798
• S=30-0.9798=29.0202, K =30, r=7%,
σ=25%, T=0.5
• d1=0.0985
• d2=-0.0782
• c= 2.0682
Investment strategies and
outcomes
• With options, we can develop many
different investment strategies that could
generate high rate of return in different
scenarios if we turn out to be right.
• However, we could lose a lot when market
movement differ from our expectation.
Example
• Four investors. Each with 10,000 dollar
initial wealth.
• One traditional investor buys stock.
• One is bullish and buys call option.
• One is bearish and buy put option.
• One believes market will be stable and
sells call and put options to the second
and third investors.
Parameters
S
K
R
T
sigma
d1
d2
c
p
20
20
0.03
0.5
0.3
0.1768
-0.035
1.8299
1.5321
• Number of call options the second investor
buys
10000/ 1.8299 = 5464.84
• Number of put options the second investor
buys
10000/ 1.5321 = 6526.91
Final wealth for four investors with
different levels of final stock price.
Final stock price
20
15
30
10000
7500
15000
Second investor
0
0
54648.4
Third investor
0
32635
0
30000
-2635
-24648.4
First investor
Fourth investor
American Calls
• An American call on a non-dividend-paying
stock should never be exercised early
– Theoretically, what is the relation between an
American call and European call?
– Which one customers prefer? Why?
• An American call on a dividend-paying stock
should only ever be exercised immediately
prior to an ex-dividend date
Put-Call Parity; No Dividends
• Consider the following 2 portfolios:
– Portfolio A: European call on a stock + PV of the
strike price in cash
– Portfolio C: European put on the stock + the stock
• Both are worth MAX(ST , K ) at the maturity of the
options
• They must therefore be worth the same today
– This means that
c + Ke -rT = p + S0
An alternative way to derive PutCall Parity
• From the Black-Scholes formula
C  P  SN (d )  Ke rT N (d )  {Ke rT N (d )  SN (d )}
1
2
2
1
 S  Ke rT
Arbitrage Opportunities
• Suppose that
c =3
S0 = 31
T = 0.25
r = 10%
K =30
D=0
• What are the arbitrage
possibilities when
p = 2.25 ?
p=1?
Application to corporate liabulities
• Black, Fischer; Myron Scholes (1973).
"The Pricing of Options and Corporate
Liabilities
Put-Call parity and capital structure
• Assume a company is financed by equity and a zero
coupon bond mature in year T and with a face value of
K. At the end of year T, the company needs to pay off
debt. If the company value is greater than K at that time,
the company will payoff debt. If the company value is
less than K, the company will default and let the bond
holder to take over the company. Hence the equity
holders are the call option holders on the company’s
asset with strike price of K. The bond holders let equity
holders to have a put option on their asset with the strike
price of K. Hence the value of bond is
• Value of debt = K*exp(-rT) – put
• Asset value is equal to the value of
financing from equity and debt
• Asset = call + K*exp(-rT) – put
• Rearrange the formula in a more familiar
manner
• call + K*exp(-rT) = put + Asset
Example
• A company has 3 million dollar asset, of
which 1 million is financed by equity and 2
million is finance with zero coupon bond
that matures in 5 years. Assume the risk
free rate is 7% and the volatility of the
company asset is 25% per annum. What
should the bond investor require for the
final repayment of the bond? What is the
interest rate on the debt?
equity financing
1million
debt financing
2million
total asset
3million
debt maturity
5years
risk free rate
volatility
7%
25%
S
K
R
T
sigma
c
p
value of debt
debt rate
3
3.253908
0.07
5
0.25
1
0.29299
2
0.097342
Discussion
• From the option framework, the equity
price, as well as debt price, is determined
by the volatility of individual assets. From
CAPM framework, the equity price is
determined by the part of volatility that covary with the market. The inconsistency of
two approaches has not been resolved.
Homework1
• The stock price is $50. The strike price for
a European call and put option on the
stock is $50. Both options expire in 9
months. The risk free interest is 6% per
annum and the volatility is 25% per
annum. If the stock doesn’t distribute
dividend, what are the call and put prices?
Homework2
Three investors are bullish about Canadian stock market.
Each has ten thousand dollars to invest. Current level
of S&P/TSX Composite Index is 12000. The first
investor is a traditional one. She invests all her money
in an index fund. The second investor buys call options
with the strike price at 12000. The third investor is very
aggressive and invests all her money in call options
with strike price at 13000. Suppose both options will
mature in six months. The interest rate is 4% per
annum, compounded continuously. The implied
volatility of options is 15% per annum. For simplicity we
assume the dividend yield of the index is zero. If
S&P/TSX index ends up at 12000, 13500 and 15000
respectively after six months. What is the final wealth of
each investor? What conclusion can you draw from the
results?
Homework3
• The price of a non-dividend paying stock is
$19 and the price of a 3 month European
call option on the stock with a strike price
of $20 is $1. The risk free rate is 5% per
annum. What is the price of a 3 month
European put option with a strike price of
$20?
Homework4
• A 6 month European call option on a
dividend paying stock is currently selling
for $5. The stock price is $64, the strike
price is $60. The risk free interest rate is
8% per annum for all maturities. What
opportunities are there for an arbitrageur?
Homework5
• Use Excel to demonstrate how the change
of S, K, T, r and σ affect the price of call
and put options. If you don’t know how to
use Excel to calculate Black-Scholes
option prices, go to COMM423 syllabus
page on my teaching website and click on
Option calculation Excel sheet
Homework6
• A company has 3 million dollar asset, of
which 1 million is financed by equity and 2
million is finance with zero coupon bond
that matures in 10 years. Assume the risk
free rate is 3% and the volatility of the
company asset is 25% per annum. What
should the bond investor require for the
final repayment of the bond? What is the
interest rate on the debt? How about the
volatility of the company asset is 35%?
Homework 7
• The asset values of companies A, B are both at
1000 million dollars. Each companies is purely
financed with equity, with100 million shares
outstanding. The stock prices of companies A, B
are both at 10 dollars per share. Company A
provides its CEO 1 million shares. What is the
value of these shares?
Homework 7
• Company B provides its CEO call options
on 5 million shares with strike price at 10
dollars and a maturity of 5 years. Assume
the risk free rate is 2% per annum and the
volatility is 25% per annum. Please
calculate the total value of the call options.
Is the value of the option provided to the
CEO of company B higher or lower than
the value of shares provided to the CEO of
company A?
The influence of Beat the Market
• Practical experience is not merely the
ultimate test of ideas; it is also the ultimate
source. At their beginning, most ideas are
dimly perceived. Ideas are most clearly
viewed when presented as abstractions,
hence the common assumption that
academics --- who are proficient at
presenting and discussing abstractions --are the source of most ideas. (p. 6,
Treynor, 1973) (quoted in p. 49)
Why Black and Scholes
• Jack Treynor, developed CAPM theory
• CAPM theory: Risk and return is the same
thing
• Black learned CAPM from Treynor. He
understood return can be dropped from
the formula
Fischer Black (1938 – 1995 )
•
•
•
•
•
Start undergraduate in physics
Transfer to computer science
Finish PhD in mathematics
Looking for something practical
Join ADL, meet Jack Treynor, learn finance and
economics
• Developed Black-Scholes
• Move to academia, in Chicago then to MIT
• Return to industry at Goldman Sachs for the last
11 years of his life, starting from 1984
• Fischer never took a course in either economics or
finance, so he never learned the way you were
supposed to do things. But that lack of training proved to
be an advantage, Treynor suggested, since the
traditional methods in those fields were better at
producing academic careers than new knowledge.
Fischer’s intellectual formation was instead in physics
and mathematics, and his success in finance came from
applying the methods of astrophysics. Lacking the ability
to run controlled experiments on the stars, the
astrophysist relies on careful observation and then
imagination to find the simplicity underlying apparent
complexity. In Fischer’s hands, the same habits of
research turned out to be effective for producing new
knowledge in finance. (p. 6)
• Both CAPM and Black-Scholes are thus much
simpler than the world they seek to illuminate,
but according to Fischer that’s a good thing, not
a bad thing. In a world where nothing is
constant, complex models are inherently fragile,
and are prone to break down when you lean on
them too hard. Simple models are potentially
more robust, and easier to adapt as the world
changes. Fischer embraced simple models as
his anchor in the flux because he thought they
were more likely to survive Darwinian selection
as the system changes. (p. 14)
• John Cox, said it best, ‘Fischer is the only
real genius I’ve ever met in finance. Other
people, like Robert Merton or Stephen
Ross, are just very smart and quick, but
they think like me. Fischer came from
someplace else entirely.” (p. 17)
• Why Black is the only genius?
• No one else can achieve the same level of
understanding?
• Fischer’s research was about developing clever
models ---insightful, elegant models that
changed the way we look at the world. They
have more in common with the models of
physics --- Newton’s laws of motion, or
Maxwell’s equations --- than with the
econometric “models” --- lists of loosely
plausible explanatory variables --- that now
dominate the finance journals. (Treynor, 1996,
Remembering Fischer Black)
The objective of this course
• We will learn Black-Scholes theory.
• Then we will develop an economic theory
of life and social systems from basic
physical and economic principles.
• We will show that the knowledge that
helps Black succeed will help everyone
succeed.
• There is really no mystery.
Effect of Variables on Option
Pricing
Variable
S0
K
T
s
r
D
c
+
–
?
+
+
–
p
–
+?
+
–
+
C
+
–
+
+
+
–
P
–
+
+
+
–
+
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
• The portfolio is instantaneously riskless and must
instantaneously earn the risk-free rate
• This leads to the Black-Scholes differential equation
• Thorp and Kassouf (1967): Beat the market, long
stock and short warrant. This provided the stimulus
for this line of thinking.
The Derivation of the BlackScholes Differential Equation
S  mS t  sS z
 ƒ
ƒ
2 ƒ 2 2 
ƒ
ƒ   mS 
 ½ 2 s S t 
sS z
t
S
S
 S

W e set up a portfolio consisting of
 1 : derivative
ƒ
+
: shares
S
The Derivation of the Black-Scholes
Differential Equation continued
The value of the portfolio  is given by
ƒ
  ƒ 
S
S
The change in its value in time t is given by
ƒ
  ƒ 
S
S
The Derivation of the Black-Scholes
Differential Equation continued
The return on the portfolio must be the risk - free
rate. Hence
  r t
We substitute for ƒ and S in these equations
to get the Black - Scholes differenti al equation :
2
ƒ
ƒ

ƒ
2 2
 rS
½ s S
 rƒ
2
t
S
S
The Differential Equation
• Any security whose price is dependent on the
stock price satisfies the differential equation
• The particular security being valued is determined
by the boundary conditions of the differential
equation
• In a forward contract the boundary condition is
ƒ = S – K when t =T
• The solution to the equation is
ƒ = S – K e–r (T
–t)
The payoff structure
• When the contract matures, the payoff is
C ( S ,0)  max( S  K ,0)
• Solving the equation with the end condition,
we obtain the Black-Scholes formula
The Black-Scholes Formulas
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
How they found the solution
• The equation had been obtained quite
awhile ago. But they could not find a
solution for some time.
• Later they use formulas from others which
contains expected rate of return. They set
the return to be the risk free rate. That was
the formula.
• It can be solved directly from the equation
and the initial condition.
The basic property of BlackSchoels formula
C  S  Ke
 rT
Rearrangement of d1, d2
S
ln(  rT )
1
Ke
d1 
 s T
2
s T
S
ln(  rT )
1
Ke
d2 
 s T
2
s T
Properties of B-S formula
• When S/Ke-rT increases, the chances of
exercising the call option increase, from
the formula, d1 and d2 increase and N(d1)
and N(d2) becomes closer to 1. That
means the uncertainty of not exercising
decreases.
• When σ increase, d1 – d2 increases,
which suggests N(d1) and N(d2) diverge.
This increase the value of the call option.
Similar properties for put options
 rT
Ke
ln(
)
1
S
 d2 
 s T
2
s T
 rT
Ke
ln(
)
1
S
 d1 
 s T
2
s T
Calculating option prices
• The stock price is $42. The strike price for
a European call and put option on the
stock is $40. Both options expire in 6
months. The risk free interest is 6% per
annum and the volatility is 25% per
annum. What are the call and put prices?
Solution
• S = 42, K = 40, r = 6%, σ=25%, T=0.5
ln( S0 / K )  (r  s 2 / 2)T
d1 
s T
• = 0.5341
ln( S0 / K )  (r  s 2 / 2)T
d2 
s T
• = 0.3573
Solution (continued)
c  S 0 N (d1 )  K e
 rT
N (d 2 )
• =4.7144
pKe
 rT
• =1.5322
N (d 2 )  S 0 N (d1 )
The Volatility
• The volatility of an asset is the standard
deviation of the continuously
compounded rate of return in 1 year
• As an approximation it is the standard
deviation of the percentage change in the
asset price in 1 year
Estimating Volatility from
Historical Data
1. Take observations S0, S1, . . . , Sn at
intervals of t years
2. Calculate the continuously compounded
return in each interval as:
 Si 

ui  ln
 Si 1 
3. Calculate the standard deviation, s , of
the ui´s
4. The historical volatility estimate is: sˆ 
s
t
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
Causes 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
Dividends
• European options on dividend-paying
stocks are valued by substituting the stock
price less the present value of dividends
into Black-Scholes
• 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
Calculating option price with
dividends
• Consider a European call option on a
stock when there are ex-dividend dates in
two months and five months. The dividend
on each ex-dividend date is expected to
be $0.50. The current share price is $30,
the exercise price is $30. The stock price
volatility is 25% per annum and the risk
free interest rate is 7%. The time to
maturity is 6 month. What is the value of
the call option?
Solution
• The present value of the dividend is
• 0.5*exp (-2/12*7%)+0.5*exp(-5/12*7%)=0.9798
• S=30-0.9798=29.0202, K =30, r=7%,
σ=25%, T=0.5
• d1=0.0985
• d2=-0.0782
• c= 2.0682
Investment strategies and
outcomes
• With options, we can develop many
different investment strategies that could
generate high rate of return in different
scenarios if we turn out to be right.
• However, we could lose a lot when market
movement differ from our expectation.
Example
• Four investors. Each with 10,000 dollar
initial wealth.
• One traditional investor buys stock.
• One is bullish and buys call option.
• One is bearish and buy put option.
• One believes market will be stable and
sells call and put options to the second
and third investors.
Parameters
S
K
R
T
sigma
d1
d2
c
p
20
20
0.03
0.5
0.3
0.1768
-0.035
1.8299
1.5321
• Number of call options the second investor
buys
10000/ 1.8299 = 5464.84
• Number of put options the second investor
buys
10000/ 1.5321 = 6526.91
Final wealth for four investors with
different levels of final stock price.
Final stock price
20
15
30
10000
7500
15000
Second investor
0
0
54648.4
Third investor
0
32635
0
30000
-2635
-24648.4
First investor
Fourth investor
American Calls
• An American call on a non-dividend-paying
stock should never be exercised early
– Theoretically, what is the relation between an
American call and European call?
– Which one customers prefer? Why?
• An American call on a dividend-paying stock
should only ever be exercised immediately
prior to an ex-dividend date
Put-Call Parity; No Dividends
• Consider the following 2 portfolios:
– Portfolio A: European call on a stock + PV of the
strike price in cash
– Portfolio C: European put on the stock + the stock
• Both are worth MAX(ST , K ) at the maturity of the
options
• They must therefore be worth the same today
– This means that
c + Ke -rT = p + S0
An alternative way to derive PutCall Parity
• From the Black-Scholes formula
C  P  SN (d )  Ke rT N (d )  {Ke rT N (d )  SN (d )}
1
2
2
1
 S  Ke rT
Arbitrage Opportunities
• Suppose that
c =3
S0 = 31
T = 0.25
r = 10%
K =30
D=0
• What are the arbitrage
possibilities when
p = 2.25 ?
p=1?
Application to corporate liabulities
• Black, Fischer; Myron Scholes (1973).
"The Pricing of Options and Corporate
Liabilities
Put-Call parity and capital structure
• Assume a company is financed by equity and a zero
coupon bond mature in year T and with a face value of
K. At the end of year T, the company needs to pay off
debt. If the company value is greater than K at that time,
the company will payoff debt. If the company value is
less than K, the company will default and let the bond
holder to take over the company. Hence the equity
holders are the call option holders on the company’s
asset with strike price of K. The bond holders let equity
holders to have a put option on their asset with the strike
price of K. Hence the value of bond is
• Value of debt = K*exp(-rT) – put
• Asset value is equal to the value of
financing from equity and debt
• Asset = call + K*exp(-rT) – put
• Rearrange the formula in a more familiar
manner
• call + K*exp(-rT) = put + Asset
Example
• A company has 3 million dollar asset, of
which 1 million is financed by equity and 2
million is finance with zero coupon bond
that matures in 5 years. Assume the risk
free rate is 7% and the volatility of the
company asset is 25% per annum. What
should the bond investor require for the
final repayment of the bond? What is the
interest rate on the debt?
equity financing
1million
debt financing
2million
total asset
3million
debt maturity
5years
risk free rate
volatility
7%
25%
S
K
R
T
sigma
c
p
value of debt
debt rate
3
3.253908
0.07
5
0.25
1
0.29299
2
0.097342
Discussion
• From the option framework, the equity
price, as well as debt price, is determined
by the volatility of individual assets. From
CAPM framework, the equity price is
determined by the part of volatility that covary with the market. The inconsistency of
two approaches has not been resolved.
Homework1
• The stock price is $50. The strike price for
a European call and put option on the
stock is $50. Both options expire in 9
months. The risk free interest is 6% per
annum and the volatility is 25% per
annum. If the stock doesn’t distribute
dividend, what are the call and put prices?
Homework2
Three investors are bullish about Canadian stock market.
Each has ten thousand dollars to invest. Current level
of S&P/TSX Composite Index is 12000. The first
investor is a traditional one. She invests all her money
in an index fund. The second investor buys call options
with the strike price at 12000. The third investor is very
aggressive and invests all her money in call options
with strike price at 13000. Suppose both options will
mature in six months. The interest rate is 4% per
annum, compounded continuously. The implied
volatility of options is 15% per annum. For simplicity we
assume the dividend yield of the index is zero. If
S&P/TSX index ends up at 12000, 13500 and 15000
respectively after six months. What is the final wealth of
each investor? What conclusion can you draw from the
results?
Homework3
• The price of a non-dividend paying stock is
$19 and the price of a 3 month European
call option on the stock with a strike price
of $20 is $1. The risk free rate is 5% per
annum. What is the price of a 3 month
European put option with a strike price of
$20?
Homework4
• A 6 month European call option on a
dividend paying stock is currently selling
for $5. The stock price is $64, the strike
price is $60. The risk free interest rate is
8% per annum for all maturities. What
opportunities are there for an arbitrageur?
Homework5
• Use Excel to demonstrate how the change
of S, K, T, r and σ affect the price of call
and put options. If you don’t know how to
use Excel to calculate Black-Scholes
option prices, go to COMM423 syllabus
page on my teaching website and click on
Option calculation Excel sheet
Homework6
• A company has 3 million dollar asset, of
which 1 million is financed by equity and 2
million is finance with zero coupon bond
that matures in 10 years. Assume the risk
free rate is 3% and the volatility of the
company asset is 25% per annum. What
should the bond investor require for the
final repayment of the bond? What is the
interest rate on the debt? How about the
volatility of the company asset is 35%?
Homework 7
• The asset values of companies A, B are both at
1000 million dollars. Each companies is purely
financed with equity, with100 million shares
outstanding. The stock prices of companies A, B
are both at 10 dollars per share. Company A
provides its CEO 1 million shares. What is the
value of these shares?
Homework 7
• Company B provides its CEO call options
on 5 million shares with strike price at 10
dollars and a maturity of 5 years. Assume
the risk free rate is 2% per annum and the
volatility is 25% per annum. Please
calculate the total value of the call options.
Is the value of the option provided to the
CEO of company B higher or lower than
the value of shares provided to the CEO of
company A?