CHAPTER 12 AN OPTIONS PRIMER

1
CHAPTER 12AN OPTIONS PRIMER

• In this chapter, we provide an introduction to
  options. This chapter is organized into the
  following sections
• Options and Options Markets
• Options Pricing
• The Option Pricing Model
• Speculating with Options
• Hedging with Options

2
Options and Options Markets

• Options
• Options are specialized financial instruments
  that give the purchaser the right but not the
  obligation to do something.
• That is, the purchaser can do something if he/she
  wants to, but he/she does not have to do it.
• Options are a relatively new financial
  instruments dating back to the 1970s.
• IBM Example
• IBM common stocks trades at 120, an investor
  has an option to buy a IBM stock for 100
  through August in the current year.

3
Options and Options Markets

• There are two classes of options referred to as
  put and call options. You may purchase or sell
  either a call option or a put option.
• Put and call options each give buyers and sellers
  different rights and responsibilities as follows
  
• Call Options
• The buyer of a call option has the right but not
  the obligation to purchase a pre-specified amount
  of a pre-specified asset at a pre-specified price
  during a pre-specified time period.
• The seller of a call option has the obligation to
  sell a pre-specified amount of a pre-specified
  asset at a pre-specified price if asked to do so
  during a pre-specified time period.

4
Options and Options Markets

• Put Options
• The buyer of a put option has the right but not
  the obligation to sell a pre-specified amount of
  a pre-specified asset at a pre-specified price
  during a pre-specified time period.
• The seller of a put option has the obligation to
  purchase a pre-specified amount of a
  pre-specified asset at a pre-specified price if
  asked to do so during a pre-specified time
  period.

5
Options and Options MarketsTerminology

• The Premium
• The buyer of an option pays the seller of the
  option a premium on the day that the
  agreement is entered into.
• The Strike Price or the Exercise Price
• The pre-specified price is referred to as the
  strike or the exercise price.
• Expiration
• The amount of time specified in the options
  contract.
• Exercise
• The option buyer elects to utilize his/her right.
  
• In the case of a call option, the buyer utilizes
  his/her right to buy the stock.
• In the case of a put option, the buyer utilizes
  her/his right to sell the stock.

6
Options and Options MarketsTerminology

• Option Writer
• The seller of an option.
• Writing an Option
• The act of selling an option.
• European Options
• European options can be exercised only on the
  maturity date.
• American Options
• American options can be exercised any time prior
  to maturity.
• Covered Call
• Writing call options against stock that the
  writer owns.
• Naked Option
• Writing a call option on a stock that the writer
  does not own.

7
Options and Options MarketsTerminology

• Intrinsic Value
• The value of an option if it is exercised
  immediately.
• Option Clearing Corporation (OCC)
• Oversees the conduct of the market and helps to
  make the market orderly. Option buyers and
  sellers only obligations are to the OCC. If an
  option is exercised, the OCC matches buyers and
  sellers, and manages the completion of the
  exercise process.

8
Call Option Example

• You buy a call option on 100 shares of IBM stock
  with a strike price of 50 per share and a
  premium of 2.50 per share. The option has 3
  months to maturity.
• Suppose that at the time you enter into the
  contract, the price of IBM stock is 49.50
• The buyer pays the seller a 2.50 premium on the
  day they enter into the agreement.
• Timeline

9
Call Option Example

• After three months the stock price will have
  either gone up, gone down, or stayed the same.
• A. Suppose that after three months the price of
  IBM stock has gone down to 47 per share.
• The option will expire worthless that is, the
  buyer will not exercise his/her right to purchase
  the shares for 50 per share.
• The purchaser of the option loses the 2.50 per
  share premium.
• B. Suppose that after three months, the price of
  IBM stock has stayed at 49.50 per share.
• The option will expire worthless that is, the
  buyer will not exercise her/his right to purchase
  the shares for 50 per share.
• The purchaser of the option loses the 2.50 per
  share premium.

10
Call Option Example

• Suppose that after three months, the price of
  IBM stock has gone up to 70 per share. In
  this case, the buyer of the call option
  will exercise his option.
• The purchaser of the option will exercise his/her
  right to purchase 100 shares for 50 per share.
• The purchaser will then go to the market to sell
  his/her share of stock for 70 per share. Thus
  the purchaser makes a profit

11
Put Option Example

• You buy a put option on 100 shares of IBM stock
  with a strike price of 50 per share and a
  premium of 2.50 per share. The option has 3
  months to maturity.
• Suppose that at the time you enter into the
  contract, the price of IBM stock is 50.50
• The buyer pays the seller a 2.50 premium on the
  day they enter into the agreement.
• Timeline

12
Put Option Example

• After three months the stock price will have
  either gone up, gone down, or stayed the same.
• A. Suppose that after three months the price of
  IBM stock has gone down to 45 per share.
• The buyer of option will exercise her/his option
  to sell the stock for 50.
• Thus, the buyer will purchase the stock for 45
  in the market and sell it using the put option
  for 50.
• The purchaser of the put option makes a profit
• B. Suppose that after three months, the price
  of IBM stock has stayed at 50.50 per
  share.
• The option will expire worthless that is, the
  buyer will not exercise her/his right to sell the
  shares for 50 per share.
• The purchaser of the option loses the 2.50 per
  share premium.

13
Put Option Example

• C. Suppose that after three months, the price of
  IBM stock has gone up to 70 per share. In
  this case, the buyer of the put option will
  not exercise his/her option.
• The option will expire worthless that is, the
  buyer will not exercise his/her right to sell
  the shares for 50 per share.
• The purchaser of the option loses 2.50 per share
  premium.

14
Option Exchanges
15
Option Quotations

• Insert Figure 12.1 here

16
Option Pricing

• Five factors affect the price of options on
  stocks without cash dividends

This section considers the effects of the first
three factors. Thus, a call price can be
expressed using C(S, E, t)
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Pricing Call Options at Expiration

• First principle of option pricing
• At expiration, a call option must have a value
  that is equal to zero or to the difference
  between the stock price and the exercise price,
  whichever is greater. This quantity is referred
  to as the intrinsic value of the option
• C(S, E, 0) max(0, S - E)
• If this condition does not hold, an arbitrage
  opportunity exists.
• Two possibilities may arise, regarding the
  relationship between the exercise price (E) and
  the stock price (S).
• S ? E
• S gt E
• Where t 0.

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Pricing Call Options at Expiration

• First Possibility S ? E
• Example
• A call option with an exercise price of 80 on
  a stock trading at 70. The option is about to
  expire.
• If an option is at expiration and the stock price
  is less than or equal to the exercise price, the
  call option has no value.
• Max(0, S-E)
• Max(0, 70-80) 0

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Pricing Call Options at Expiration

• Second Possibility S gt E
• If the stock price is greater than the exercise
  price, the call option must have a price equal to
  the difference between the stock price and the
  exercise price.
• Max(0, S-E) S E
• If this relationship did not hold, there would be
  an arbitrage opportunity.
• Example 1
• Consider a call option that is selling with an
  exercise price of 40 on a stock
  trading at 50. The option is
  selling for 5.
• An arbitrageur would make the following trades
• Transaction Cash Flow
• Buy a call option -5Exercise the option
  to buy the stock -40Sell the stock
  50Net Cash Flow 5

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Pricing Call Options at Expiration

• Example 2
• A call option with an exercise price of 40 on
  a stock trading at 50. The option is now
  selling 15. The option is about to expire.
• An arbitrageur would make the following trades
• Transaction Cash Flow
• Write a call option 15Buy the stock -
  50Initial Cash Flow -35
• If owner of the call option exercises the option.
  The arbitrageurs transactions are
• Transaction Cash Flow
• Initial cash flow - 35Deliver stock
  0Collect exercise price 40Net Cash
  Flow 5

21
Pricing Call Options at Expiration

• If the owner of the call option allows the option
  to expire. The arbitrageurs transactions are
• Initial cash flow - 35Sell Stocks 50Net
  Cash Flow 15
• In order for these arbitrage opportunities not to
  exist, principle 1 must hold.

22
Graphical Analysis of Option Values and Profits
Expiration

• Assume a call and a put option both with 100
  striking price. We can graph the payoff on these
  options as demonstrated in Figure 12.2.

• Insert Figure 12.2 Here

23
Option Values and Profits Expiration

• Assume a call and a put option both with 100
  striking price. Trades had taken place for the
  options with premiums of 5 on each of the put
  and call options.
• Figure 12.3 illustrates the alternatives outcomes.

• Insert Figure 12.3a here

24
Option Values and Profits Expiration

• Insert Figure 12.3b here

25
Pricing Prior to Maturity

• Second principle of option pricing
• A call option with a zero exercise price and an
  infinite time to maturity must sell for the same
  price as the stock. This is because the buyer of
  the call option can convert his/her option into
  the stock and sell it.
• C(S, 0, ?) S
• Combining principle 1 and 2, we can establish
  bounds for the price of an option. That is,
  establish upper and lower limits for the price of
  the option.
• The bounds for the price of a call option are a
  function of the stock price, the exercise price,
  and the time to expiration. Figure 12.4 presents
  these boundaries.

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A Call Option with Zero Exercise Price and an
Infinite Time until Expiration

• Insert Figure 12.4 here

27
Relationship Between Option Prices

• Third principle of option pricing
• If two call options are alike, except the
  exercise price of the first is less than that of
  the second, then the option with the lower
  exercise price must have a price that is equal to
  or greater than the price of the option with the
  higher exercise price.
• The relationship can be defined as follows
• If E1 lt E2, C(S, E1, t) ? C(S, E2, t)
• If this relationship does not hold, an arbitrage
  profit can be earned.

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Relationship Between Option Prices

• Example
• You have two identical options. The first option
  has an exercise price of 100 and sells for 10.
  The second option has an exercise price of 90
  and sells for 5.
• An arbitrageur would make the following trades
• Transaction Cash Flow
• Sell the option with the 100 exercise price
  10Buy the option with the 90 exercise price
  - 5
• Net Cash Flow 5
• Figures 12.5a creates an arbitrage opportunity
  and figure 12.5b graphs the combined positions.

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Relationship Between Option Prices

• Insert figure 12.5a here

• Insert figure 12.5b here

30
Relationship Between Option Prices

• Consider the profit and loss position on each
  option and the overall position for alternative
  stock prices that might prevail at expiration.

The result is graphed in figure 12.5c.
31
Relationship Between Option Prices

• Insert figure 12.5c here

Notice in figure 12.5c that regardless of the
ultimate stock price, a positive profit is
earned. In order to avoid this arbitrage
principle 3 must hold.
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Relationship Between Option Prices

• Fourth principle of option pricing (expiration
  date principle)
• If there are two options that are otherwise
  alike, the option with the longer time to
  expiration must sell for an amount equal to or
  greater than the option that expires earlier.
• If t1 gt t2, C(S, t1, E) ? C(S, t2, E)
• If the option with the longer period to
  expiration sold for less than the option with the
  shorter time to expiration, there would also be
  an arbitrage opportunity.

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Relationship Between Option Prices

• Example
• Two options on the same stock both having a
  striking price of 100. The first option has a
  time to expiration of 6 months and trades for
  8. The second option has 3 months to expiration
  and trades for 10.
• An arbitrageur would make the following
  transactions
• Transaction Cash Flow
• Buy the 6-month option for 8 - 8Sell the
  3-month option for 10 10Net Cash Flow
  2
• The option with the longer time to expiration
  must be worth more than the option with the
  shorter time to expiration.
• Figure 12.6 illustrate this

34
Relationship Between Option Prices

• Insert figure 12.6 here

35
Call Option Prices and Interest Rates

• Example
• Assume that a stock now sells for 100. Over the
  next year, its value can change by 10 in either
  direction (100 shares equal to 9,000 or
  11,000) The risk-free rate of interest is 12.
  A call option exists with a striking price of
  100/share and expiration one year from now.
• Assume two portfolios
• Portfolio A 100 shares of stock, current value
  10,000.
• Portfolio B A 10,000 pure discount bond maturing
  in one year, with a current value of 8,929
  (PV with 12 interest rate). One option
  contract, with an exercise price of 100/share
  (10,000/ entire contract)
• Table 12.2 illustrates the impact of price
  changes on each portfolio.

36
Call Option Prices and Interest Rates

• Portfolio B is the best portfolio to hold. If the
  stock price goes down, Portfolio B is worth
  1,000 more than Portfolio A. If the stock price
  goes up, Portfolios A and B have the same value.
• This implies that the value of the option should
  be at least

37
Call Option Prices and Interest Rates

• Fifth principle of option pricing
• Other things being equal, the higher the
  risk-free rate of interest, the greater must be
  the price of a call option.
• Thus, the interest rate principle can be
  expressed as
• If r1 gt r2, C(S, E, t, r1) ? C(S, E, t, r2)
• Recall that the price of the call must be either
  zero or S - E at expiration or

The call price must be greater than or equal to
the stock price minus the present value of the
exercise price. So the higher the interest rate,
the higher the value of call option. Using the
data from previous example, now assume that
interest rates goes up to 20. The new value of
the option should be
38
Prices of Call Options and The Riskiness of Stocks

• Sixth principle of option pricing (the risk
  principle)
• The riskier the stock on which an option is
  written, the greater will be the value of a call
  option. Thus the sixth principle can be stated
  as
• If s1 gt s2, C(S, E, t, r, s1) ? C(S, E, t, r,
  s2)
• Other things being equal, a call option on a
  riskier good will be worth at least as much as a
  call option on a less risky good.

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Prices of Call Options and The Riskiness of Stocks

• Table 12.3 shows the impact that stock price
  changes have on option prices.

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Option Pricing Model

• Recall that the price of an option must be at
  least as great as the stock price minus the
  present value of the exercise price. However,
  options have an inherent insurance policy.
• The insurance character of the option can be seen
  by comparing the payoffs from Portfolio A and B
  from Table 12.3. Holding the options insures that
  the worst outcome from the investment will be
  10,000.
• To reflect this, the value of the option must be
  equal to the stock price minus the present value
  of the exercise price, plus the value of the
  insurance policy (I) inherent in the option or
• C(S, E, t, r, s) S - Present Value(E) I
• Option pricing models can be used to determined
  the insurance policy value.

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Option Pricing Model (OPM)

• The Black and Scholes Option Pricing Model (OPM)
  assumes that stock prices follow a stochastic
  process or Wiener process. Where a stochastic
  process is a mathematical description of the
  change in the value of some variable through
  time.
• Wiener process shows that the changes over any
  given time interval are distributed normally.
• Figure 12.7 shows a graph of the path that stock
  prices might follow if they followed a Wiener
  process.

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Option Pricing Model (OPM)

• Insert figure 12.7 here

43
Option Pricing Model (OPM)

• The Black-Scholes OPM is given by
• C SN(d1) - E e-rt N(d2)
• The Black-Scholes OPM can be used to calculate
  the theoretical price of an option. If we know
  the value of the following variables

Where S stock price- E exercise price
t time to expiration r risk-free interest
rate s variability of the stock
44
Option Pricing Model (OPM)

• Example assume the following values
• S 100E 100t 1 yearr 12s 10
• Step 1 calculate the values for d1 and d2.

45
Option Pricing Model (OPM)

• Step 2 calculate N(d1) and N(d2)
• The cumulative normal probability can be obtained
  from tables that are widely available or by using
  the excel function normsdist(d)
• Using a standardized normal probability
  distribution table
• N(d1) N(1.25) .8944N(d2) N(1.15)
  .8749
• Step 3 calculate the call option price using OPM
• C S N(d1) - E e-rt N(d2)
• C 100 (.8944) - 100 e-(.12)(1) (.8749)
• C 89.44 - 100 (.8869) (.8749)
• C 89.44 - 77.60 11.84
• The value of the option is 11.84
• Recall form Table 12.2 that option value was
  10.71.
• The difference is due to the value of the
  insurance policy that is captured by the
  Black-Scholes OPM.

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The Value of Put Options and Put-Call Parity

• While the Black-Schole OPM applies to call
  options, we can infer the corresponding value of
  a put option by utilizing a concept called
  Put-Call Parity.
• The Put-Call Parity tells us that the value of a
  put option can be computed as follows

For example, suppose a stock is trading for 100
per share. Using the Black-Scholes OPM, we have
computed the value of a call option with a 100
striking price to be 11.84. The interest rate is
12. The value of the put option is computed as
47
Speculating with Options

• Using our prior calculations, what would be the
  effect of a 1 change in stock prices? What are
  the speculating opportunities?
• Original Values 1 Increase 1 Decrease
• S 100 S 101 S 99
• C 11.84 C 12.73 C10.95
• Options can be used to take very low risk
  speculative positions by using options in
  combinations. The combinations are virtually
  endless, including combinations called strips,
  straps, spreads and straddles.

48
Speculating with Options

• A straddle is a combination of positions
  involving a put and a call option on the same
  stock. To buy a straddle, the investor buys both
  call and put options. Consider a call and put
  option, both with an exercise price of 100. The
  call trades for 40 and the put for 7. Table
  12.5 shows the payoff on the straddle at various
  stock prices.

49
Speculating with Options

• The payoff is graphically displayed in Figure
  12.9.

• Insert figure 12.9 here

50
Hedging with Options

• Options can be used to control risk. Consider an
  original portfolio comprised of 8,944 shares of
  stock selling at 100 per share and assume that a
  trader sells 100 option contracts, or options on
  10,000 shares, at 11.84. The entire portfolio
  would have a value of 776,000.
• Table 12.6 shows a hedged portfolio.

Notice that by hedging, the value of the
portfolio did not change as a result of the
change in stock prices.