The Black-Scholes Model

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The Black-ScholesModel
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Randomness matters in nonlinearity

• An call option with strike price of 10.
• Suppose the expected value of a stock at call
  options 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.

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• Applying Itos Lemma, we can find
• Therefore, the average rate of return is
  r-0.5sigma2. (But there could be problem because
  of the last term.)

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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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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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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

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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
• 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 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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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.
• 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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Effect of Variables on Option Pricing

c
p
C
P






















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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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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

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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

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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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Application to corporate liabulities

• Black, Fischer Myron Scholes (1973). "The
  Pricing of Options and Corporate Liabilities

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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
  companys 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

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• Value of debt Kexp(-rT) put
• Asset value is equal to the value of financing
  from equity and debt
• Asset call Kexp(-rT) put
• Rearrange the formula in a more familiar manner
• call Kexp(-rT) put Asset

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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?

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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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Homework

• 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? If
  the stock is expected to distribute 1.5 dividend
  after 5 months, what are the call and put prices?

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Homework

• 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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Homework

• 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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Homework

• 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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Homework

• 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?

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Homework

• 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?