The Black-Scholes Model

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The Black-ScholesModel
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• Applying Itos Lemma, we can find
• Therefore, the geometric rate of return is
  r-0.5sigma2.
• The arithmetic rate of return is r

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The 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

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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)

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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

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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

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Fischer Black (Continued)

• Looking for something practical
• Join ADL, meet Jack Treynor, learn finance and
  economics from him
• Developed Black-Scholes

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• 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)

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• 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)

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• 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?

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• 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)

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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.

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Effect of Variables on Option Pricing

c
p
C
P






















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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.

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The Derivation of the Black-Scholes
Differential Equation
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The Derivation of the Black-Scholes Differential
Equation continued
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The Derivation of the Black-Scholes Differential
Equation continued
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The 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 ƒ S K when t T
• The solution to the equation is
• ƒ S K er (T t )

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The payoff structure

• When the call option contract matures, the payoff
  is
• Solving the equation with the end condition, we
  obtain the Black-Scholes formula

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The Black-Scholes Formulas
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How 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.

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The basic property of Black-Schoels formula
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Rearrangement of d1, d2
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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.
• 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.

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Similar properties for put options
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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?

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Solution

• S 42, K 40, r 6, s25, T0.5
• 0.5341
• 0.3573

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Solution (continued)

• 4.7144
• 1.5322

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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

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Estimating 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

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Implied Volatility

• The implied volatility of an option is the
  volatility for which the Black-Scholes price
  equals the market price
• The is a one-to-one correspondence between prices
  and implied volatilities
• Traders and brokers often quote implied
  volatilities rather than dollar prices

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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

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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

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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?

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Solution

• 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

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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.
• Option is a highly leveraged investment

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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.

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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
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• 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

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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
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Investment 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?

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S 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
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final 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
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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

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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

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An alternative way to derive Put-Call Parity

• From the Black-Scholes formula

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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 ?

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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.

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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 doesnt distribute
  dividend, what are the call and put prices?

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Homework2

• 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?

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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?

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Homework4

• 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?

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Homework5

• 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