CHAPTER 23 OPTIONS

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CHAPTER 23 OPTIONS

• Topics
• 23.1 Background
• 23.5 Stock Option Quotations
• 23.2 - 23.4 Value of Call and Put Options at
  Expiration
• 23.6 Combinations of Options
• 23.7 Valuing Options
• 23.8 An Option Pricing Formula

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23.1 Background

• A derivative security is simply a contract which
  has a value that is dependent upon (or derived
  from) the value of some other asset(s)
• Derivatives that we consider in this course
• Futures and forward contracts (Chapter 26)
• Options
• there are many different kinds of options stock
  options, index options, futures options, foreign
  exchange options, interest rate caps, callable
  bonds, convertible bonds, retractable/extendable
  bonds, etc.

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Why option is important in corporate finance

• Important financial markets besides stock and
  bond markets
• Many employees are compensated in stock options
• You need to what theyre worth
• You need to know how their value might depend
  upon the actions chosen by those employees
• Many corporate finance projects have implicit
  real options in them. A static valuation (i.e.,
  that ignores this option value) can give
  misleading results
• In fact, almost any security can be thought of in
  terms of options (including stocks and corporate
  bonds)

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Terminology

• Option Gives the owner the right to buy or sell
  an asset on or before a specified date for a
  predetermined price
• Call the right to buy (C)
• Put the right to sell (P)

The underlying asset (Current price Spot price,
S)
Expiry date, T
Exercise price, K or X
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Terminology contd

• Owner of an option
• Buy
• Long position
• Seller of an option
• Short
• Write
• The act of buying or selling the underlying asset
  via the option contract is exercising the option
• European options can only be exercised at the
  expiry date, while American options can be
  exercised at any time up to and including the
  expiry date
• (unless otherwise stated, the options we
  consider will be European options)
• Value of the option is called premium

Has a right
Has an obligation (A contingent obligation)
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Terminology contd

• Option contracts are either exchange-traded or
  available on a customized basis in the
  over-the-counter market
• Standardized exchange-traded option contracts (on
  the CBOE)
• Usually are for 100 options (1 contract)
• Have strike prices 2.50 apart in the range of 5
  to 25, 5 apart in the range of 25 to 200, and
  10 apart for strike prices above 200
• Expiration months are two near term months plus
  two additional months from the Jan, Feb, or Mar
  quarterly cycle
• The expiration date is usually the Saturday
  following the third Friday of the expiration
  month (which is the last trading day)

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23.5 Stock Option Quotations
RBCs Nov. 08 options

• Strike strike price
• Exp Expiry month
• Bid the bid price (The price at which somebody
  bids to buy)
• Ask the ask price (The price at which somebody
  asks to sell)
• Open int. Open interest, or number of
  outstanding contracts (one contract usually
  equals 100 options)

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Option quotations contd

• Options are not usually protected against cash
  dividends but are protected against stock-splits
  and stock dividends
• Suppose in Oct. you buy 1 call option of RY with
  expiry of Jan. and a strike of 30.
• In Nov. RY pays 50 cash dividend.
• In Nov. RY does a 21 split. Your 1 call option
  will become 2 options with strike price halved

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23.2-4 Value of Options
Value of a Call Option at Expiration Date (T)
Example An option on a common stock ST
market price of the common stock on the
expiration date, T. Suppose the exercise price, X
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Payoff on the expiration date (The third Friday in June) Payoff on the expiration date (The third Friday in June) Payoff on the expiration date (The third Friday in June)
ST gt 8, let ST 9 ST 8 ST lt 8 let ST 7
Call option value

• If ST 9 at the expiry date, exercise option,
  purchase share at 8 and then sell share for 9.
  Payoff
• C max(ST X, 0)

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Value of a Put Option at Expiration Date (T)

• A put. Exercise price X 11
• If ST 10 at the expiry date, exercise option,
  purchase share at 10 and then sell share for
  11. Payoff
• P max(X - ST, 0)

Payoff on the expiration date (The third Friday in Sept.) Payoff on the expiration date (The third Friday in Sept.) Payoff on the expiration date (The third Friday in Sept.)
ST gt 11, let ST 12 ST 11 ST lt 11 let ST 10
Put option value
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Moneyness

• an in the money option is one that would lead to
  a positive cash flow if exercised immediately
• an at the money option is one that would lead to
  a zero cash flow if exercised immediately
• an out of the money option is one that would lead
  to a negative cash flow if exercised immediately
• Let X denote the strike price of the option and
  St denote the current price of the underlying
  asset

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Value components

• Intrinsic Value the intrinsic value of an option
  is the maximum of zero and the options immediate
  exercise value
• Time Value
• The difference between the option premium and the
  intrinsic value of the option.

Time value
Intrinsic Value
OPTION VALUE (PREMIUM)


E.g. (1) The Nov. RY 30 Call. Price 8.20.
Spot 37.10.
(2) The Nov. RY 30 Put. Price 1.05.
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Option payoffs contd
Profit/loss from Buying a stock, e.g., at 15
Payoff
0
Share Price
15
-15
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• Payoff/profit of Long Call
• Let X 15. Whats your payoff/profit of buying
  a call at T?
• Payoff is just the final value of the option CT,
  profit takes the option premium into account.
• When you long an option, you buy the option at a
  cost (the option premium)

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15 35
Share Price
15
Payoff/profit to buy a Put option, given a 15
exercise price.
5
10 15
Share Price
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The buyer and seller always have mirror image
payoffs
Long call
Share Price
15

• Profit-
• When you go short in an option, you sell the
  option (and receive premium)

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Exercise

• Plot the payoff/profit diagrams for (1) short
  stock (suppose the sell price is 15) and (2)
  selling put (suppose X 15).

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23.6 Combinations of Options and stocks

• Portfolios of stocks and options.
• Firms and individuals portfolios (asset
  holdings) may consist of different options and
  stocks on the same underlying asset. These
  positions may offset each other or compound the
  payoff possibilities.
• You want to combine them and consider the risk of
  changes in the price of the underlying asset to
  the aggregate portfolio.

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• Short call Long put with same strike and expiry
  (called synthetic short sale)
• Equivalent to

• Long Call Short put with same strike and expiry
  (let X 55)
• Equivalent to

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Writing a Covered Call Long stock short call

• X 55, and cost of stock (S0) 55. Option
  premium 5.
• Payoff and Profit

ST 45 50 55 60
Buy 1 share -5
Short 1 call 0
Total payoff -5
Profit 0 5 5

• Long stock position covers or protects a trader
  from the payoff on the short call that becomes
  necessary if there is a sharp increase in stock
  price.

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Covered call contd

• Payoff and profit (When strike purchase price)
• Break-even point
• Maximum profit
• What if strike ? purchase price?

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Protective Put Long stock long put

• Use puts to limit downside
• Often used with index options to provide
  portfolio insurance
• Let X 40, S040

ST 35 40 45 50
Buy 1 share 10
Buy 1 put 0
Total payoff 10
Profit 5

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Protective put contd
Profit You start to make when Otherwise, you
lose at most P.
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Straddle Buy 1 call and 1 put (same X, T)

• e.g. S0 40, C 10, P 10, X 50

ST 35 40 40 45 50 55 60 65
Buy 1 call 0 0 0 10 15
Buy 1 put 10 5 5 0 0
Total payoff 10 5 5 10 15
Profit -10 -15 -15 -10 -5
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Profit Diagram of Long Straddle
You do not make if
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22.7 (Arbitrage) Bounds on Call Option

• Objective derive restrictions on option values
  which (i) any reasonable option pricing model
  must satisfy and (ii) do not depend on any
  assumptions about the statistical distribution of
  the price of the underlying asset
• To simplify matters, we will assume that no
  dividends are paid by the underlying asset during
  the life of the option
• Lets look at a call only
• Fact 1 A call option with a lower exercise price
  is worth more.
• A call option with an exercise price of zero is
  effectively the same financial security as the
  stock itself.
• An upper bound for the call option price is the
  price of the stock.

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Contd

• Fact 2 The option value is at least the payoff
  if exercised immediately
• The action of exercising gives you the underlying
  asset
• However, if you want the asset
• If you exercise now you pay exercise price now
• Or better, you can exercise on the expiry day and
  pay the exercise price later.
• Equivalently, if you exercise immediately, you
  only need to pay the present value of (Exercise
  price)
• Lower bound of call St PV(X)
• Fact 3 As the stock price gets large (relative
  to exercise price)
• the probability that the option expires worthless
  vanishes
• the value of the call approaches
• Value of stock PV (Exercise Price) St - X

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Bounds for Call option
ST
Call
Time value
Intrinsic value
0
ST
PV(X)
The value of a call option Ct must fall within
Max(St PV(X),0) lt Ct lt St
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Implications of Call Bounds

• If St 0 then Ct 0.
• If X 0 then Ct St.
• If T-t 8, then Ct St.
• For an American option, it is not optimal to
  exercise a call option before its expiry date (if
  the underlying stock will not pay any dividends
  over the life of the option).
• Note that this implies that the value of an
  American call option will be equal to that of a
  corresponding European option (same underlying
  asset, X, T).

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Its never optimal to exercise the American call
early

• Why?
• If you exercise a call at any time t, you receive
  the intrinsic value (St X)
• But the option is always worth at least the
  intrinsic value.
• Example St 60, T - t 3 months, X 55, r
  10
• if you exercise you receive 5 but you lose
  time value of the call.
• The option is worth at least
  
• At least a dominant strategy is to put 55 in
  bank and wait until the expiry to exercise.
• You get interest income on the 55.
• If you want the underlying, hold the option and
  you can buy it any time you want.

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Call bounds example

• What is the lower bound for the price of a
  six-month call option on a non-dividend-paying
  stock when the stock price is 80, the strike
  price is 75, and the risk-free interest rate is
  10 percent per annum?

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Arbitrage Bounds on Put Option Values

• For put options, we will show that
• X Pt max(0, X St)

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Arbitrage Bounds on Put Option Values (contd)
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Implications of Put Option Bounds

• If St 0 then Pt X.
• If X 0 then Pt 0.
• For a European put,
• and if T-t 8, then Pt 0.
• It may be optimal to exercise an American put
  option before its expiry date, even if the
  underlying asset does not pay dividends during
  the life of the option (consider what happens if
  S ? 0)

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An aside Continuously compounding

• In option pricing, people generally use
  continuously compounding instead of discrete
  compounding that is used in Chapter 8.
• Continuous compounding assumes that compounding
  period for effective return is as short as
  possible (think of milliseconds). A dollar with
  nominal return of r over a period of T will
  become the following continuous-time proceeds

where m is the compounding frequency.
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European options Put-Call Parity (no dividends)
Suppose you buy one share of stock, buy one put
option on the stock, and sell one call option on
the stock (same X, T). Cashflow at expiry
ST lt X
1. Buy stock
2. Buy put
3. Write call
Cashflow at expiry

• No matter what happens, you end up with a payoff
  of X
• Todays value of your synthetic investments must
  equal PV(cf _at_ expiry)

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Put-call parity contd

• Given any three of risk free bond, stock, put,
  and call, the fourth can always be created
  synthetically. (In fact, this was done before
  1977 on the CBOE to construct puts)
• Holding stock and put today is equal to holding a
  call the PV(X) amount of riskfree bond.
• Note that the LHS is the value of protective put.
  
• You can also use the profit graph of protective
  put to prove put-call parity

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Put call parity with dividend paying stocks

• We can also derive parity relationships when
  there are dividends paid by the underlying asset,
  when the options are American, etc.
• In the case of European options on a stock paying
  known dividends before the expiry date, the
  result is
• where It is the present value (at t) of the
  dividends paid before T

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Put-call parity example 1

• The price of a European call which expires in six
  months and has a strike price of 30 is 2. The
  underlying stock price is 29, and there is no
  dividend payment. The term structure is flat,
  with all risk-free interest rates being 10
  percent. What is the price of a European put
  option that expires in six months and has a
  strike price of 30?

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Put-call parity example 2

• Several years ago, the Australian firm Bond
  Corporation sold some land that it owned near
  Rome for 110 million and as a result boosted its
  reported earnings for that year by 74 million.
  The next year it was revealed that the buyer was
  given a put option to sell the land back to Bond
  for 110 million and also that Bond had paid 20
  million for a call option to repurchase the land
  for the same price of 110 million.
• 1. What happens if the land is worth more than
  110 million when the options expire? What if it
  is worth less than 110 million?
• 2. Assuming that the options expire in one year,
  what is the interest rate?
• 3. Was it misleading to record a profit from
  selling the land?

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How do we value options?

• Why the discounted cash flow approach will not
  work?
• Expected cash flows?
• Opportunity cost of capital?
• We are going to use a replicating portfolio
  method
• Owning the option is equivalent to owning some
  number of shares and borrowing money
• Therefore the price of the option should be the
  same as the value of this replicating portfolio.

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The Binomial Model (Two-State Option Pricing)

• In order to use the replicating portfolio method,
  we need to make some assumptions about how the
  underlying stock behaves
• The simplest possible setting is one in which
  there are only two possible outcomes of stock
  price at the expiry date of 1 period (binomial)
  
• Also assume
• the underlying stock wont pay a dividend over
  the life of the option
• interest rates are constant over the life of the
  option

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Example I

• Lets assume a stocks current price is 25. It
  may go up (u) or down (d) by 15 next year, with
  equal probabilities.
• The discount rate for the stock is 10.
• The stock has a beta of 1.
• T-bills are yielding 5.
• A call option on the stock is traded on the NYSE.
  The strike price of this option is 25, and the
  maturity is one year.
• What is the option worth?

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The method Replicate the payoffs of the option
using only the stock and loan

• Step 1 Find the payoffs of the option in both
  good and bad states.
• Step 2 Choose the right number of shares to
  make the difference in outcomes equal to the
  option.
• Step 3 Replicate the payoffs by adding the
  right size loan to your shares

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Example contd Matching cash flows
Investment strategy State 1 (future price State 2 (future price
1. buy a call  
2.
(i) Buy 0.5(delta) shares of stock
(ii) Borrow 10.119 _at_ 5 a year
Total from replicating strategystrategy 2
Value of the option
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A generalization of option replicating
portfolios deciding delta and loan

• Form a portfolio with ? shares of stock and B
  borrowed or invested in the risk-free bond. The
  current value of the portfolio
• V0 S0 ? B
• The basic idea is to pick ? and B to replicate
  the payoffs of the call option at its expiry
• The number of shares bought, ?, is called the
  hedge ratio or option delta
• If B lt 0, then it is borrowing if B gt 0, then it
  is lending (investing in T-bills).

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Solve for ? and B Graph representation
t 0
t 1

• To match the cashflows

Note rf is interest rate for the length of one
period.
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Contd

• Solve for ?, B
• (1) (2),
• and
• In words
• ? (Spread of possible option prices)/ (Spread
  of possible stock prices)
• In the previous example

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Possibly more intuitive

• You might think of option deltas as solving
• (Option delta) (Spread in Share price)
  Spread in option price
• That is, the delta scales the amount of
  variation in the stock price so that the spread
  of outcomes in your replicating portfolio is the
  same as in the option.
• (The implicit loan in options) Note that we just
  showed the call option is like a levered version
  of stock.
• Its like buying the stock bundled with
  personal debt.
• Levered equity higher risk or higher beta.

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Optional Example continued Call Options are
riskier than the stock.

• Find the options beta.
• C S0 ? B
• Portfolio beta. (Answer 5.25)

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Valuing the put option

• Method 1 Same as before.
• Find the options delta
• Option Delta (Spread of possible option
  prices)/(Spread of possible stock prices)
• Then find the bank loan that equates the payoff
  of option and the payoff of (stock position
  loan)
• Method 2
• Since we already have the call option value, just
  use put-call parity to infer the put option value.

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• Exercise Suppose Example I is a put option
  instead of a call. Use the binomial model to find
  its delta, replicate the payoffs, and find the
  value of the put.

Put graph here.
P S0 ? B
Interpret ? and B for put
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Implication of irrelevance of the underlyings
expected return in the real world

• The option value does not directly depend on
• Probabilities and expected returns of the
  underlying
• Individual preferences about risk aversion
• If there are just options traded, expected return
  from the option payoffs should be the same, which
  is the riskfree rate.
• Indirectly, these factors do affect the option
  value, but only by determining the current stock
  price S0
• A very useful trick that can be used to simplify
  calculations is to exploit the fact that
  investors risk aversion doesnt matter and just
  assume that they are risk-neutral
• In this case, all assets should earn the risk
  free rate of return
• 1) Expected returns of stock riskfree rate
• 2) Expected option payoffs option value (1
  riskfree rate)

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Binomial option pricing risk neutral pricing
Let S1,u u S0 and S1,d d S0

• The key is to find risk-neutral probabilities for
  stock price movements. Let q be the
  risk-neutral probability of an up move.
• To find q, by meaning (1) of previous slide, we
  have

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Risk neutral pricing

• A minor bit of algebra yields
• By meaning (2) of previous slide, value of option
  can now be derived as
• A big advantage of risk-neutral pricing is we can
  circumvent the dynamic hedging nuisance in
  multi-period pricing.

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Example of the Risk-Neutral Valuation of a Call

• Suppose a stock is worth 25 today and in one
  period will either be worth 15 more or 15 less.
  The risk-free rate is 5 for one-period. What is
  the value of an at-the-money call option with a
  maturity of one-period?
• The binomial tree

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Example contd

• Step 1 Compute risk neutral probabilities
• Step 2 Find the value of the call in the up
  state and down state
• Step 3 Find the value of the call at time 0.

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Exercise

• Suppose the option is a put in Example I. Use
  risk-neutrality to value the put. (You should be
  able to do this without the help of graphs.)

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Expanding the binomial model to allow more
possible price changes
Binomial to Black Scholes
1 step 2 steps 4 steps
(2 outcomes) (3 outcomes) (5
outcomes) etc. etc.
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The Black-Scholes Model

• As the number of intervals N ??, underlying
  stock price follows a random walk with a positive
  growth rate.
• In this case, solving the option value using the
  portfolio replicating method yields the famous
  Black-Scholes equations.
• Solving the Black-Scholes equations for the basic
  version of European call options with no
  dividends yields the Black-Scholes formula

Where
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Inputs to Black-Scholes

• 5 inputs
• X Strike
• S Spot (current stock price)
• s annual standard deviation of underlying
  stocks return
• r continuously compounded annual risk free
  interest rate
• T-t Time to maturity expressed as fraction of
  year
• N(.) is the cumulative standard normal
  distribution function, i.e.,
• N(d1) is the probability that a variable
  distributed normally with mean zero and standard
  deviation of one will be less than or equal to d1
  
• Note that we dont need expected return of the
  underlying asset to figure out the stock option
  value.

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Black-Scholes and Binomial Methods

1. N(d1) is delta, the number of shares held in the
   replicating portfolio
2. The amount borrowed is

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Some properties of Standard Normal Distribution
N(d1 3) 0.9987 There is 99.87 probability
that a drawing from a standard normal
distribution will be below 3. N(0) 0.5,
N(8)1 N(-d1) 1 - N(d1) N(d1 0) 0.5
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Table 23.3 Cumulative Probabilities of Standard
Normal Distn
d 0.00 0.01 0.02 0.03 0.09
0.00 0.0000 0.0040 0.0080 0.0120 0.0359
0.10
0.20 0.0793 0.0910


3.00 0.4987     0.4988   0.4990
N(0.23) N(-0.23) In excel, the function is
normsdist( ).
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Linear Interpolation

• N (-0.0886) ?
• From Table 23.3
• N ( - 0.08) 0.5 0.0319 0.4681
• N ( - 0.09) 0.5 0.0359 0.4641
• Linear interpolation
• Exercise
• N(0.0886)

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Black-Scholes Value of Put
The corresponding Black-Scholes formula for puts
is

• Exercise Prove that the B-S call and put
  formulae satisfy the put-call parity.

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Example
Call option. Inputs T-t 0.25 (time to expiry 3
months), St 53, X 55, r .05, ? 0.30.
Whats the value of the call? (2.5895)
N(d2)
N(d1)
? C
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Example contd

• What is the value of the corresponding put
  option? (3.9063)

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Factors determining option values
Volatility is probably the most important
consideration in options, since option value is
very sensitive to volatility.
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The Black-Scholes Model with Dividends (European
options)

• Options are protected against stock splits but
  not cash dividends.
• Two possible adjustment to B-S value for
  dividends
• 1. Discrete dividend payments
• For options on a single stock, one approach is to
  assume that dividends are known up to the option
  expiry date. The underlying stock is then viewed
  as consisting of two components
• A risk free series of dividends known up until T
• The risky stock value
• Effectively the spot price is lower from the
  perspective of option holder.
• ? Replace in the B-S formula S with the new
  after-dividend S, i.e., with S PV(Dividend).

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Example

• S 50, X 45, s 20, rf 6, t 3 mths,
  div. 0.50
• (payable in 2 mths)
• Adjust the stock price
• S 50 0.50 e(-0.06 2/12) 49.505
• Recalculate the BS call value using S instead of
  S throughout. (C 5.494)

Div 2
X 45
S 50
0
3
2
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• 2. Continuous Dividend Yield
• In some cases it is reasonable to assume that
  dividends are paid continuously as a known
  percentage of the value of the underlying asset.
• Examples include stock indexes, foreign
  currencies and futures contracts
• If an asset pays a continuous dividend yield q,
  we would do the similar adjustment.
• Adjust back dividend to St
• ? Replace St with

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Example

• S 50, X 45, s 20, rf 6, t 3 mths, q
  4
• Adjust the stock price
• S 50 e(-0.04 3/12) 49.502
• Recalculate the BS call value using S instead
  of S throughout. ( C 5.491)

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Review Questions

• 1. Two stocks have identical firm-specific risks
  but different betas. All else being equal, a put
  option on the high-beta stock is worth more than
  one on the low-beta stock.

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• 2. Value of portfolio of options and option on
  portfolios
• Suppose you have two stocks (stocks 1 and 2) in a
  portfolio. The return correlation of the two
  stocks is 0.5. Currently, stocks 1 and 2 are both
  trading at 20, and both stocks have same
  volatility. The weight of stock 1 in the
  portfolio is 0.5, and the weight of stock 2 is
  also 0.5.
• There are three call options, written on stock 1,
  stock 2 and the portfolio respectively, all with
  same X and T.
• Which option is the most expensive?
• Which option is the least expensive?

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• Assigned problems 23.3, 5-7, 10, 13, 14, 17,
  19, 21, 24, 25, 26, 29, 32, 33

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