Title: The Black-Scholes Model
1The Black-ScholesModel
2- Applying Itos Lemma, we can find
- Therefore, the geometric rate of return is
r-0.5sigma2. - The arithmetic rate of return is r
3The history of option pricing models
- 1900, Bachelier, the purpose, risk management
- 1950s, the discovery of Bacheliers work
- 1960s, Samuelsons formula, which contains
expected return - Thorp and Kassouf (1967) Beat the market, long
stock and short warrant - 1973, Black and Scholes
4The 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)
5Why 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
6 Fischer Black (1938 1995 )
- Start undergraduate in physics
- Black found Julian Schwinger, his teacher in
quantum mechanics, extremely arrogant. Schwinger
got Nobel prize in physics in 1965, for work in
quantum electrodynamics - Transfer to computer science for masters program
- Followed Marvin Minsky on artificial
intelligence. AI is very hot today. But at
Blacks time, he found it lack of substance. - Got PhD in mathematics
- 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
7Fischer Black (Continued)
- Looking for something practical
- Join ADL, meet Jack Treynor, learn finance and
economics from him - Developed Black-Scholes
8- 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. Fischers 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 Fischers
hands, the same habits of research turned out to
be effective for producing new knowledge in
finance. (p. 6)
9- Both CAPM and Black-Scholes are thus much simpler
than the world they seek to illuminate, but
according to Fischer thats 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)
10- John Cox, said it best, Fischer is the only real
genius Ive 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?
11- Fischers 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 --- Newtons
laws of motion, or Maxwells equations --- than
with the econometric models --- lists of
loosely plausible explanatory variables --- that
now dominate the finance journals. (Treynor,
1996, Remembering Fischer Black)
12The 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.
13Effect of Variables on Option Pricing
c
p
C
P
14The 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.
15 The Derivation of the Black-Scholes
Differential Equation
16The Derivation of the Black-Scholes Differential
Equation continued
17The Derivation of the Black-Scholes Differential
Equation continued
18The Differential Equation
- Any security whose price is dependent on the
stock price satisfies the differential equation.
It is a very general equation. - The particular security being valued is
determined by the boundary conditions of the
differential equation - In a forward contract the boundary condition
is Æ’ S K when t T - The solution to the equation is
- Æ’ S K er (T t )
19The payoff structure
- When the call option contract matures, the payoff
is - Solving the equation with the end condition, we
obtain the Black-Scholes formula
20The Black-Scholes Formulas
21How they found the solution
- The equation had been obtained quite awhile ago
by Black. 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.
22The basic property of Black-Schoels formula
23Rearrangement of d1, d2
24Properties 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. - The difference between d1-d2 is one standard
deviation. - When s increase, d1 d2 increases, which
suggests N(d1) and N(d2) diverge. This increase
the value of the call option.
25Similar properties for put options
26Calculating 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?
27Solution
- S 42, K 40, r 6, s25, T0.5
- 0.5341
- 0.3573
28Solution (continued)
29The 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
30Estimating Volatility from Historical Data
- Take observations S0, S1, . . . , Sn at
intervals of t years - Calculate the continuously compounded return in
each interval as - Calculate the standard deviation, s , of the uis
- The historical volatility estimate is
31Implied 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
32Causes 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
33Dividends
- 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
34Calculating 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?
35Solution
- The present value of the dividend is
- 0.5exp (-2/127)0.5exp(-5/127)0.9798
- S30-0.979829.0202, K 30, r7, s25, T0.5
- d10.0985
- d2-0.0782
- c 2.0682
36Investment 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. - Option is a highly leveraged investment
37Example
- 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.
38Parameters
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
39- 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
40Final 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
41Investment with options
- Four investors are bullish about a stock. Each
has ten thousand dollars to invest. Current stock
price is 100 dollars per share. The first
investor is a traditional one. She invests all
her money to buy shares. The second investor buys
call options with the strike price at 100. The
third investor buy call options with strike price
at 110. Both options will mature in six months.
The fourth investor buy call options with strike
price at 110. The option will mature in three
months. The interest rate is 3 per annum,
compounded continuously. The implied volatility
of options is 20 per annum. For simplicity we
assume the dividend yield of the stock is zero.
If the stock price ends up at 100, 110 and 120
respectively after three months or after six
months. What is the final wealth of each
investor? What conclusion can you draw from the
results?
42S 100
K 100 110 110
T 0.5 0.5 0.25
sigma 20
R 3
d1 0.18 -0.50 -0.83
d2 0.04 -0.64 -0.93
C 6.37 2.61 1.09
total investment 10000
number of options bought 1569.61 3828.63 9163.01
43final stock price at three month 100 110 120
investor 4 0 0 91630.14
final stock price at six month 100 110 120
investor 1 10000 11000 12000
investor 2 0 15696.05 31392.11
investor 3 0 0 38286.27
44American 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
45Put-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
46An alternative way to derive Put-Call Parity
- From the Black-Scholes formula
47 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 ?
48Discussion
- 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.
49Homework1
- 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 doesnt distribute
dividend, what are the call and put prices?
50Homework2
- Three investors are bullish about Canadian stock
market. Each has ten thousand dollars to invest.
Current level of SP/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 SP/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?
51Homework3
- 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?
52Homework4
- A 6 month European call option on a non 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?
53Homework5
- Use Excel to demonstrate how the change of S, K,
T, r and s affect the price of call and put
options. If you dont 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