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The Black-Scholes Model

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1.5321 p 1.8299 c -0.035 d2 0.1768 d1 0.3 sigma 0.5 T 0.03 R 20 K 20 S -24648.4 -2635 30000 Fourth investor 0 32635 0 Third investor 54648.4 0 0 Second investor ... – PowerPoint PPT presentation

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Title: The Black-Scholes Model


1
The 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

3
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

4
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)

5
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

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

7
Fischer 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)

12
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.

13
Effect of Variables on Option Pricing

c
p
C
P






















14
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.

15
The Derivation of the Black-Scholes
Differential Equation
16
The Derivation of the Black-Scholes Differential
Equation continued
17
The Derivation of the Black-Scholes Differential
Equation continued
18
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 Æ’ S K when t T
  • The solution to the equation is
  • Æ’ S K er (T t )

19
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

20
The Black-Scholes Formulas
21
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.

22
The basic property of Black-Schoels formula
23
Rearrangement of d1, d2
24
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.

25
Similar properties for put options
26
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?

27
Solution
  • S 42, K 40, r 6, s25, T0.5
  • 0.5341
  • 0.3573

28
Solution (continued)
  • 4.7144
  • 1.5322

29
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

30
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

31
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

32
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

33
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

34
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?

35
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

36
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

37
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.

38
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
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

40
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
41
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?

42
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
43
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
44
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

45
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

46
An 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 ?

48
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.

49
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?

50
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?

51
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?

52
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?

53
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
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