# The Black-Scholes Model Chapter 13

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```					The Black-
Scholes
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  dz
s
• Applying Ito’s Lemma, we can find
1
( r   2 )t
 z (t )
S  S0e        2
e

• Therefore, the average rate of return is r-
0.5sigma^2. (But there could be problem
because of the last term.)
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 )
• 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
11 years of his life
• 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   c   p   C    P
S0       +   –   +    –
K        –   +   –    +
T

?   ?   +    +
+   +   +    +
r        +   –   +    –
D        –   +   –    +
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
The Derivation of the Black-
Scholes Differential Equation

S  S t  S z
 ƒ     ƒ     2 ƒ 2 2       ƒ
ƒ   S 
 S          ½ 2  S t 
        S z
         t    S             S
We set up a portfolio consistingof
 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 portfoliomust be the risk - free
rate. Hence
  r t
We substitute for ƒ and S in these equations
al
to get the Black - Scholes differenti equation:
ƒ      ƒ            2ƒ
 rS     ½ 2 S 2 2  r ƒ
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

 rT
c  S 0 N (d1 )  K e          N (d 2 )
p  K e  rT N (d 2 )  S 0 N (d1 )
ln( S 0 / K )  (r   2 / 2)T
where d1 
 T
ln( S 0 / K )  (r   2 / 2)T
d2                                   d1   T
 T
The basic property of Black-
Schoels formula

 rT
C  S  Ke
Rearrangement of d1, d2

S
ln(  rT )
1
d1     Ke        T
 T       2
S
ln(  rT )
1
d2      Ke       T
 T      2
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
 d2         S         T
 T          2
 rT
Ke
ln(          )
1
 d1        S          T
 T          2
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( S 0 / K )  (r   2 / 2)T
d1 
 T

• = 0.5341
ln( S 0 / K )  (r   2 / 2)T
d2 
 T
• = 0.3573
Solution (continued)

c  S 0 N (d1 )  K e  rT N (d 2 )
• =4.7144
 rT
pKe           N (d 2 )  S 0 N (d1 )
• =1.5322
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
s
4. The historical volatility estimate is:  
ˆ
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 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.
Parameters
S                         20
K                         20
R                       0.03
T                        0.5
sigma                    0.3
d1                   0.1768
d2                   -0.035
c                    1.8299
p                    1.5321
• Number of call options the second investor
10000/ 1.8299 = 5464.84

• Number of put options the second investor
10000/ 1.5321 = 6526.91
Final wealth for four investors with
different levels of final stock price.

Final stock price      20      15        30

First investor      10000    7500     15000

Second investor         0       0   54648.4

Third investor          0   32635         0

Fourth investor     30000   -2635   -24648.4
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?
– What are the market prices? 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
(Equation 8.3, page 174)
• 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 Put-
Call 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 tructure
• 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 there 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?
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 co-
vary 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?
If the stock is expected to distribute \$1.5
dividend after 5 months, what are the call
and put prices?
Homework2
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 and a dividend of \$0.80 is
expected in 1 month. 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 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? How about the
volatility of the company asset is 35%?

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