chapter 12 an options primer - wiley-blackwell€¦ · ppt file · web view · 2006-10-12chapter...

Chapter 12 1

CHAPTER 12AN OPTIONS PRIMER

In this chapter, we provide an introduction to options. This chapter is organized into the following sections:

1. Options and Options Markets

2. Options Pricing

3. The Option Pricing Model

4. Speculating with Options

5. Hedging with Options

Chapter 12 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 1970’s.

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.

Chapter 12 3


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.

Chapter 12 4


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.

Chapter 12 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.

Chapter 12 6


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.

Chapter 12 7


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.

Chapter 12 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:

0 2 3

-$2.50

1

Chapter 12 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.


Chapter 12 10

Call 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 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

Chapter 12 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:

0 2 3

-$2.50

1

Chapter 12 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.


Chapter 12 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.

Chapter 12 14

Option ExchangesTable12.1

Principal Options Exchanges and Option Traded Exchange Option Traded

Panel 1: Options Exchanges in the United States Chicago Board Options Exchange (CBOE) Options on Individual Stocks LongBTerm Options on Individual Stocks Options on Stock Indexes Options on Interest Rates American Stock Exchange (AMEX) Options on Individual Stocks LongBTerm Options on Individual Stocks Options on Stock Indexes Options on Exchange Traded Funds Philadelphia Stock Exchange (PHLX) Options on Individual Stocks LongBTerm Options on Individual Stocks Options on Stock Indexes Options on Foreign Currency Pacific Exchange (PSE) Options on Individual Stocks LongBTerm Options on Individual Stocks International Securities Exchange (ISE Options on Individual Stocks Boston Options Exchange (BOX) Options on Individual Stocks Chicago Mercantile Exchange (CME) Options on futures traded at the CME Chicago Board of Trade (CBOT) Options on futures traded at the CBOT New York Mercantile Exchange (NYMEX) Options on futures traded at NYMEX New York Board of Trade (NYBOT) Options on futures traded at the NYBOT Kansas City Board of Trade (KCBT) Options on futures traded at the KCBT Minneapolis Grain Exchange (MGE) Options on futures traded at the MGE Panel 2: Key Options Exchanges Outside the United States Eurex (Germany and Switzerland) Options on individual stocks Options on futures Euronext (Brussels, Amsterdam, Paris,& London) Options individual stocks Options on Stock Indexes Option on Interest rates Euronext.liffe (Brussels, Amsterdam, Paris, & London) Options on futures

Chapter 12 15

Option Quotations

Insert Figure 12.1 here

Chapter 12 16

Option Pricing

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

Factor Name Call Option

Put Option

S Stock Price + - E Exercise Price - + T Time + + σ Volatility of Underline Stock (Standard Deviation) + + r Risk-Free Interest Rates + -

This section considers the effects of the first three factors. Thus, a call price can be expressed using:

C(S, E, t)

Chapter 12 17

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 ES > E

Where t = 0.

Chapter 12 18


First Possibility: S E

Example:A call option with an exercise price of $80 on astock 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

Chapter 12 19


Second Possibility: S > 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

Chapter 12 20


Example 2:

A call option with an exercise price of $40 on astock trading at $50. The option is now selling$15. The option is about to expire.



Write a call option $ 15Buy the stock - 50Initial Cash Flow -$35

If owner of the call option exercises the option. The arbitrageur’s transactions are:


Initial cash flow -$ 35Deliver stock 0Collect exercise price +$40Net Cash Flow $ 5

Chapter 12 21


If the owner of the call option allows the option to expire. The arbitrageur’s transactions are:

Initial cash flow -$ 35Sell Stocks +$50Net Cash Flow $ 15

In order for these arbitrage opportunities not to exist, principle 1 must hold.

Chapter 12 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

Chapter 12 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

Chapter 12 24

Option Values and Profits Expiration

Insert Figure 12.3b here

Chapter 12 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.

Chapter 12 26

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

Insert Figure 12.4 here

Chapter 12 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 < E2, C(S, E1, t) C(S, E2, t)

If this relationship does not hold, an arbitrage profit can be earned.

Chapter 12 28


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.



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.

Chapter 12 29


Insert figure 12.5a here

Insert figure 12.5b here

Chapter 12 30


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

Profit or Loss on the Option Position

Stock Price at

Expiration

For E = $90

For E = $100

For Both

80 90 95

100 105 110 120

- $5 - $5

0 + $5 +$10 +$15 +$25

+$10 +$10 +$10 +$10 + $5

0 - $10

+ $5 + $5 +$10 +$15 +$15 +$15 +$15

The result is graphed in figure 12.5c.

Chapter 12 31


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.

Chapter 12 32


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 > 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.

Chapter 12 33


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:


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

Chapter 12 34


Insert figure 12.6 here

Chapter 12 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.

Chapter 12 36


Table 12.2 Portfolio Values in One Year

Stock Price Change

+10%

B10%

Portfolio A Stock

$11,000

$9,000

Portfolio B Maturing Bond Call Option

$10,000 $1,000

$10,000 0

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:

$1,071 = (1.12)

$10,000 - $10,000 C

(E) Value PresentSC

Chapter 12 37


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 > 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:

$1,667 = (1.20)

$10,000 - $10,000 C

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:

(E) Value PresentSC

Chapter 12 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 σ1 > σ2, C(S, E, t, r, σ1) C(S, E, t, r, σ2)

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.

Chapter 12 39

Prices of Call Options and The Riskiness of Stocks

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

Table 12.3

Portfolio Values in One Year

Stock Price Change

+10%

B10% Portfolio B

Maturing Bond Call Option

In Portfolio B, the call option must be worth at least $1,071.

$10,000 $1,000

$10,000 0

Stock Price Change

+20%

B20% Portfolio A

Stock Portfolio B

Maturing Bond Call Option

$12,000

$10,000 $2,000

$8,000

$10,000 0

Chapter 12 40

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, σ) = S - Present Value(E) + I

Option pricing models can be used to determined the insurance policy value.

Chapter 12 41

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.

Chapter 12 42



Chapter 12 43


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:

t

trESd

2

15.)/ln(

tdd 12

2121 d and d of y valuesprobabilit normal cumulative)(),( dNdN

Where+ S = stock price- E = exercise price+ t = time to expiration+ r = risk-free interest rate+ σ = variability of the stock

Chapter 12 44


Example: assume the following values:

S = $100E = $100t = 1 yearr = 12%σ = 10%

Step 1: calculate the values for d1 and d2.

t

trESd

2

15.)/ln(

)1)(1(.

1)01(.5.12.)100/100ln(1

d

25.11.1250.0

1

d

tdd 12

15.1)1)(1(.25.12 d

Chapter 12 45


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.

Chapter 12 46

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:

CS

rEP t

1

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:

$1.13 = $11.84 + $100 - (1.12)$100 = P

Chapter 12 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 C$10.95

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.

Chapter 12 48


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.

Table 12.5

Payoffs for Calls, Puts, and a Straddle at Expiration Stock Price at Expiration

Call Profit/Loss

Put Profit/Loss

Straddle Profit/Loss

$50

B$10

+$43

+$33 $80

B$10

+$13

+$ 3

$83

B$10

+$10

0 $85

B$10

+$ 8

B$ 2

$90

B$10

+$ 3

B$ 7 $95

B$10

B$ 2

B$12

$100

B$10

B$ 7

B$17 $105

B$ 5

B$ 7

B$12

$110

0

B$ 7

B$ 7 $115

+$ 5

B$ 7

B$ 2

$117

+$ 7

B$ 7

0 $120

+$10

B$ 7

+$ 3

$150

+$40

B$ 7

+$33 Note: Profit/Loss figures reflect the amounts paid for the instruments:

Call = $10 Put = $7 Straddle = Call + Put = $17

Chapter 12 49


The payoff is graphically displayed in Figure 12.9.


Chapter 12 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.

Table 12.6

A Hedged Portfolio Original Portfolio: S = $100 C = $11.84 8,944 shares of stock $894,400 A short position for options on 10,000 shares (100 contracts) B$118,400 Total Value $776,000 Stock Price Rises by 1%: S = $101 C = $12.73 8,944 shares of stock $903,344 A short position for options on 10,000 shares (100 contracts) B$127,300 Total Value $776,044 Stock Price Falls by 1%: S = $99 C = $10.95 8,944 shares of stock $885,456 A short position for options on 10,000 shares (100 contracts) B$109,500 Total Value $775,956

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

chapter 12 an options primer - wiley-blackwell€¦ · ppt file · web view · 2006-10-12chapter...

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