Berk_DeMarzo_cf5_ppt_20_accessible.pptx

April 9th, 2022

Corporate Finance

Fifth Edition

Chapter 20

Financial Options

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Chapter Outline

20.1 Option Basics

20.2 Option Payoffs at Expiration

20.3 Put-Call Parity

20.4 Factors Affecting Option Prices

20.5 Exercising Options Early

20.6 Options and Corporate Finance

Learning Objectives (1 of 3)

Define the following terms: call option, put option, exercise price, strike price, exercising the option, expiration date, American option, European option, in-the-money, and out-of-the-money.

Compute the value of a call or a put option at expiration.

List the rights and obligations of the buyer of the option and the seller of the option.

Learning Objectives (2 of 3)

Use put-call parity to solve for the call premium, the put premium, the stock price, the strike price, or the dividend.

Discuss the following factors that influence call and put option values: stock price, strike price, and volatility.

Describe arbitrage bounds for option prices.

Learning Objectives (3 of 3)

Explain why it is never optimal to exercise an American call option early on a non-dividend-paying stock, and why it is sometimes optimal to exercise an American put option early.

Explain the use of option modeling to value equity.

Describe how corporate debt can be viewed as a portfolio of riskless debt and a short position in a put option.

20.1 Option Basics (1 of 2)

Financial Option

A contract that gives its owner the right (but not the obligation) to purchase or sell an asset at a fixed price as some future date

Call Option

A financial option that gives its owner the right to buy an asset

20.1 Option Basics (2 of 2)

Put Option

A financial option that gives its owner the right to sell an asset

Option Writer

The seller of an option contract

Understanding Option Contracts (1 of 3)

Exercising an Option

When a holder of an option enforces the agreement and buys or sells a share of stock at the agreed-upon price

Strike Price (Exercise Price)

The price at which an option holder buys or sells a share of stock when the option is exercised

Expiration Date

The last date on which an option holder has the right to exercise the option

Understanding Option Contracts (2 of 3)

American Option

Options that allow their holders to exercise the option on any date up to, and including, the expiration date

European Option

Options that allow their holders to exercise the option only on the expiration date

Note: The names American and European have nothing to do with the location where the options are traded

Understanding Option Contracts (3 of 3)

The option buyer (holder)

Holds the right to exercise the option and has a long position in the contract

The option seller (writer)

Sells (or writes) the option and has a short position in the contract

Because the long side has the option to exercise, the short side has an obligation to fulfill the contract if it is exercised.

The buyer pays the writer a premium

Interpreting Stock Option Quotations (1 of 3)

Stock options are traded on organized exchanges.

By convention, all traded options expire on the Saturday following the third Friday of the month.

Open Interest

The total number of contracts of a particular option that have been written

Table 20.1 Option Quotes for eBay Stock

Source: Chicago Board Options Exchange at

A table displays the option quotes for E Bay incorporated on September 10 2018 at 12 33 E T, 33.75, negative 0.235. Bid 33.75, Ask 33.76. Size 18 by 14. Volume. 2108011. The table for calls is as follows. The table has 10 Rows and 7 columns. The columns have the following headings from left to right. Calls, Last sale, Net, Bid, Ask, Volume, Open. The Row entries are as follows. Row 2. 20 18 September 21 32.00, E BAY 1821132, 1.86, blank 1.86, 1.96, blank 534. Row 3. 20 18 September 21 33.00, E BAY 1821133, 1.04, blank 1.03, 1.08, 15, 827. Row 4. 20 18 September 21 34.00, E BAY 1821134, 0.45, negative 0.09, 0.42, 0.43, 110, 4959. Row 5. 20 18 September 21 35.00, E BAY 1821135, 0.15, negative 0.06, 0.14, 0.15, 258, 7199. Row 6. 20 18 September 21 36.00, E BAY 1821136, 0.05, negative 0.03, 0.05, 0.06, 268, 13279. Row 7. 20 19 January 1 30.00, E BAY 1821130, 4.75, blank 4.7, 4.8, blank 737. Row 8. 20 19 January 1 33.00, E BAY 1821133, 2.7, negative 0.12, 2.62, 2.68, 2, 467. Row 9. 20 19 January 1 35.00, E BAY 1821135, 1.73, 0.1, 1.61, 1.66, 6, 2079. Row 10. 20 19 January 1 37.00, E BAY 1821137, 0.96, blank 0.91, 0.96, 1, 2324. Row 11. 20 19 January 1 40.00, E BAY 1821140, 0.39, blank 0.34, 0.38, blank 3455. The table for puts is as follows. The table has 10 Rows and 7 columns. The columns have the following headings from left to right. Puts, Last Sale, Net, Bid, Ask, Volume, Open I n t. The Row entries are as follows. Row 1. 20 18 September 21 32.00, E BAY 1821132, 0.09, negative 0.02, 0.09, 0.1, 2, 2122. Row 2. 20 18 September 21 33.00, E BAY 1821133, 0.25, negative 0.06, 0.24, 0.26, 2, 2927. Row 3. 20 18 September 21 34.00, E BAY 1821134, 0.62, 0.03, 0.63, 0.65, 35, 5631. Row 4. 20 18 September 21 35.00, E BAY 1821135, 1.13, negative 0.22, 1.34, 1.39, 1, 3594. Row 5. 20 18 September 21 36.00, E BAY 1821136, 2.28, negative 0.07, 2.15, 2.3, 2, 1481. Row 6. 20 19 January 1 30.00, E BAY 1821130, 0.74, blank 0.65, 0.68, blank 7912. Row 7. 20 19 January 1 33.00, E BAY 1821133, 1.55, negative 0.05, 1.55, 1.58, 129, 7562. Row 8. 20 19 January 1 35.00, E BAY 1821135, 2.47, blank, 2.51, 2.57, blank 15795. Row 9. 20 19 January 1 37.00, E BAY 1821137, 3.8, 0.65, 3.85, 3.9, 79, 17202. Row 10. 20 19 January 1 40.00, E BAY 1821140, 5.6, Blank 6.3, 6.4, Blank 6093.

Interpreting Stock Option Quotations (2 of 3)

At-the-money

Describes an option whose exercise price is equal to the current stock price

In-the-money

Describes an option whose value, if immediately exercised, would be positive

Out-of-the-money

Describes an option whose value, if immediately exercised, would be negative

Interpreting Stock Option Quotations (3 of 3)

Deep in-the-money

Describes an option that is in-the-money and for which the strike price and the stock price are very far apart

Deep out-of-the-money

Describes an option that is out-of-the-money and for which the strike price and the stock price are very far apart

Textbook Example 20.1 (1 of 2)

Purchasing Options

Problem

It is the afternoon of September 10, 2018, and you have decided to purchase 10 January call contracts on eBay stock with an exercise price of $35. Because you are buying, you must pay the ask price. How much money will this purchase cost you? Is this option in-the-money or out-of-the-money?

Textbook Example 20.1 (2 of 2)

Solution

From Table 20.1, the ask price of this option is $1.66. You are purchasing 10 contracts and each contract is on 100

shares, so the transaction will cost

(ignoring any brokerage fees). Because this is a call option and the exercise price is above the current stock price ($33.75), the option is currently out-of-the-money.

Alternative Example 20.1 (1 of 3)

Problem

You have decided to purchase 2/15/2019 put contracts on the D J I A with an exercise price of $246.

Source:

As on January 21, 2019 at 15:52 ET, LAST 247. 06

There are three tables shown. One table shows the Calls values and another shows the Puts value. In between both the tables a table shows the Strike values as on February 02, 2015.

The Calls table has nine columns with four rows. The information given in the Calls table is as follows:

Last: 4.1

• Net: positive 1.405

• Bid: 4.35

• Ask: 4.55

• Vol: 3

• IV: 0.1432

• Delta: 0.5427

• Gamma: 0.0406

• Int: 15

Last: 4

• Net: positive 1.73

• Bid: 3.75

• Ask: 3.95

• Vol: 401

• IV: 0.1406

• Delta: 0.5016

• Gamma: 0.0416

• Int: 17

Last: 3.2

• Net: positive 1.31

• Bid: 3.25

• Ask: 3.45

• Vol: 14

• IV: 0.1403

• Delta: 0.46

• Gamma: 0.0415

• Int: 2

Last: 2.7

• Net: positive 1.14

• Bid: 2.77

• Ask: 2.92

• Vol: 1

• IV: 0.1411

• Delta: 0.4197

• Gamma: 0.0406

• Int: 5

The second table titled February 02, 2019. The table consists of one column and four rows.

The Strike values are as follows:

• DJX 246.000

• DJX 247.000

• DJX 248.000

• DJX 249.000

The Puts table has nine columns consisting of four rows. The last column has a negative symbol over it. The information given in the Puts table is as follows:

Last: 3.75

• Net: negative 1.6

• Bid: 3.45

• Ask: 3.65

• Vol: 10

• IV: 0.1514

• Delta: negative 0.457

• Gamma: 0.0384

• Int: 50

Last: 4.32

• Net: negative 1.58

• Bid: 3.85

• Ask: 4.1

• Vol: 2

• IV: 0.1499

• Delta: minus 0.4957

• Gamma: 0.039

• Int: 0

Last: 0

• Net: 0

• Bid: 4.3

• Ask: 4.6

• Vol: 0

• IV: 0.1487

• Delta: minus 0.5351

• Gamma: 0.0392

• Int: 0

Last: 0

• Net: 0

• Bid: 4.85

• Ask: 5.1

• Vol: 0

• IV: 0.147

• Delta: minus 0.5748

• Gamma: 0.0391

• Int: 3

Alternative Example 20.1 (2 of 3)

Problem

How much money will this purchase cost you?

Is this option in-the-money or out-of-the-money?

Alternative Example 20.1 (3 of 3)

Solution

The ask price is $3.65 per contract.

The total cost is

Because the strike price ($246) is less than the current price ($247.06), the put option is out-of-the-money.

Options on Other Financial Securities (1 of 2)

Although the most commonly traded options are on stocks, options on other financial assets, like the S&P 100 index, the S&P 500 index, the Dow Jones Industrial index, and the N Y S E index, are also traded.

Options on Other Financial Securities (2 of 2)

Hedge

To reduce risk by holding contracts or securities whose payoffs are negatively correlated with some risk exposure

Speculate

When investors use contracts or securities to place a bet on the direction in which they believe the market is likely to move

20.2 Option Payoffs at Expiration (1 of 2)

Long Position in an Option Contract

The value of a call option at expiration is

Where S is the stock price at expiration, K is the exercise price, C is the value of the call option, and max is the maximum of the two quantities in the parentheses.

Figure 20.1 Payoff of a Call Option with a Strike Price of $20 at Expiration

The x-axis shows the stock price from 0 to 60 in increments of 10. The y-axis shows the payoff in dollars from 0 to 40 in increments of 10. The graph shows that the curve starts at (0, 0) and at the strike price (20, 0) it starts to rise and stops at (60, 40).

20.2 Option Payoffs at Expiration (2 of 2)

Long Position in an Option Contract

The value of a put option at expiration is

Where S is the stock price at expiration, K is the exercise price, P is the value of the put option, and max is the maximum of the two quantities in the parentheses.

Textbook Example 20.2 (1 of 2)

Payoff of a Put Option at Maturity

Problem

You own a put option on Oracle Corporation stock with an exercise price of $20 that expires today. Plot the value of this option as a function of the stock price.

Textbook Example 20.2 (2 of 2)

Solution

Let S be the stock price and P be the value of the put option. The value of the option is

Plotting this function gives

The x-axis shows the stock price from 0 to 40 in increments of 10. The y-axis shows the payoff in dollars from 0 to 20 in increments of 10. The graph shows that the curve starts at (0, 20) and at the strike price (20, 0) it becomes parallel to the x-axis and stops at (40, 0).

Alternative Example 20.2 (1 of 2)

Problem

You own a put option on Dell stock with an exercise price of $12.50 that expires today. Plot the value of this option as a function of the stock price.

Alternative Example 20.2 (2 of 2)

Solution

Let S be the stock price and P be the value of the put

option. The value of the option is

The y-axis ranges from 0 through 14. The graph starts at (0, 12.5), falls diagonally through (6, 6.5) to a point at (12.5, 0) labeled, strike price, then moves horizontally through (20, 0). All values estimated.

Short Position in an Option Contract

An investor that sells an option has an obligation.

This investor takes the opposite side of the contract to the investor who bought the option.

Thus the seller’s cash flows are the negative of the buyer’s cash flows.

Figure 20.2 Short Position in a Call Option at Expiration

The x-axis shows the stock price from 0 to 60 in increments of 10. The y-axis (downward) shows payoff in dollars from 0 to minus 40 in reduction of 10. The graph shows that the curve stars at (0, 0) and is parallel to x-axis, till (20, 0) when it starts to drop and ends at (60, minus 40).

Textbook Example 20.3 (1 of 2)

Payoff of a Short Position in a Put Option

Problem

You are short in a put option on Oracle Corporation stock with an exercise price of $20 that expires today. What is your payoff at expiration as a function of the stock price?

Textbook Example 20.3 (2 of 2)

Solution

If S is the stock price, your cash flows will be

If the current stock price is $30, then the put will not be exercised and you will owe nothing. If the current stock price is $15, the put will be exercised and you will lose $5. The figure plots your cash flows:

The x-axis shows the stock price from 0 to 40 in increments of 10. The y-axis (downward) shows payoff in dollars from 0 to minus 20 in reduction of 5. The graph shows that the curve stars at (40, 0) and is parallel to x-axis, till (20, 0) when it starts to drop and ends at (0, minus 20).

Profits for Holding an Option to Expiration

Although payouts on a long position in an option contract are never negative, the profit from purchasing an option and holding it to expiration could be negative because the payout at expiration might be less than the initial cost of the option.

Figure 20.3 Profit from Holding a Call Option to Expiration

The y-axis shows profit at expiration in dollars from minus 5 to 10 in increments of 5 with x-axis originating at 0. The graph shows the following data:

• 2019 Jan 18 30.00 Call: The curve originates at (25, minus 5), runs parallel to x-axis till (30, minus 5) after which it starts to increase and ends at (45, 10).

• 2019 Jan 18 33.00 Call: The curve originates at (25, minus 2.5), runs parallel to x-axis till (33, minus 2.5) after which it starts to increase and ends at (45, 9).

• 2019 Jan 18 35.00 Call: The curve originates at (25, minus 1.5), runs parallel to x-axis till (35, minus 1.5) after which it starts to increase and ends at (45, 8).

• 2019 Jan 18 37.00 Call: The curve originates at (25, minus 1), runs parallel to x-axis till (35.5, minus 1) after which it starts to increase and ends at (45, 7).

• 2019 Jan 18 40.00 Call: The curve originates at (25, minus 0.5), runs parallel to x-axis till (40, minus 0.5) after which it starts to increase and ends at (45, 5).

Textbook Example 20.4 (1 of 2)

Profit on Holding a Position in a Put Option Until Expiration

Problem

Assume you decided to purchase each of the January put options quoted in Table 20.1 on September 10, 2018, and you financed each position by shorting a two-month bond with a yield of 2.5%. Plot the profit of each position as a function of the stock price on expiration.

Textbook Example 20.4 (2 of 2)

Solution

Suppose S is the stock price on expiration, K is the strike price, and P is the price of each put option on September 10th. Then your cash flows on the expiration date will be

The plot is shown below. Note the same trade-off between the maximum loss and the potential for profit as for the call options.

expiration in dollars from minus 10 to 10 in increments of 5 with x-axis originating at 0. The graph shows the following data:

• 2019 Jan 18 40.00 Put: The curve starts at (25, 8) and drops to (40, minus 6), after which it runs parallel to x-axis till (45, minus 6).

• 2019 Jan 18 37.00 Put: The curve starts at (25, 7.5) and drops to (36.5, minus 5), after which it runs parallel to x-axis till (36.5, minus 5).

• 2019 Jan 18 35.00 Put: The curve starts at (25, 7) and drops to (35, minus 3), after which it runs parallel to x-axis till (45, minus 3).

• 2019 Jan 18 33.00 Put: The curve starts at (25, 6.5) and drops to (32.5, minus 2), after which it runs parallel to x-axis till (45, minus 2).

• 2019 Jan 18 30.00 Put: The curve starts at (25, 4) and drops to (30, minus 0.5), after which it runs parallel to x-axis till (45, minus 0.5).

Returns for Holding an Option to Expiration (1 of 2)

The maximum loss on a purchased call option is 100% (when the option expires worthless).

Out-of-the money call options are more likely to expire worthless, but if the stock goes up sufficiently, it will also have a much higher return than an in-the-money call option.

Call options have more extreme returns than the stock itself.

Returns for Holding an Option to Expiration (2 of 2)

The maximum loss on a purchased put option is 100% (when the option expires worthless).

Put options will have higher returns in states with low stock prices.

Put options are generally not held as an investment, but rather as insurance to hedge other risk in a portfolio.

Figure 20.4 Option Returns from Purchasing an Option and Holding It to Expiration

• 2019 Jan 18 40.00 Call: The line originates at (25, minus 100) and runs parallel to the x-axis till (40, minus 100) after which it increases to (41, 500). • 2019 Jan 18 37.00 Call: The line originates at (37, minus 100) and increases to (42, 500). • 2019 Jan 18 35.00 Call: The line originates at (35, minus 100) and increases to (45, 500). • 2019 Jan 18 33.00 Call: The line originates at (34, minus 100) and increases to (45, 350). • 2019 Jan 18 30.00 Call: The line originates at (30, minus 100) and increases to (45, 200).

• 2019 Jan 18 30.00 Put: The line originates at (45, minus 100) and runs parallel to the x-axis till (40, minus 100) after which it increases to (25, 130). • 2019 Jan 18 33.00 Put: The line originates at (40, minus 100) and runs parallel to the x-axis till (37, minus 100) after which it increases to (25, 200). • 2019 Jan 18 35.00 Put: The line originates at (37, minus 100) and runs parallel to the x-axis till (34, minus 100) after which it increases to (25, 299). • 2019 Jan 18 37.00 Put: The line originates at (34, minus 100) and runs parallel to the x-axis till (33, minus 100) after which it increases to (25, 405). • 2019 Jan 18 40.00 Put: The line originates at (33, minus 100) and runs parallel to the x-axis till (30, minus 100) after which it increases to (27, 500).

Combinations of Options (1 of 4)

Straddle

A portfolio that is long a call option and a put option on the same stock with the same exercise date and strike price

This strategy may be used if investors expect the stock to be very volatile and move up or down a large amount but do not necessarily have a view on which direction the stock will move.

Figure 20.5 Payoff and Profit from a Straddle

The x-axis shows the Stock price in dollars with strike price K marked on the x-axis. The lines for Put and call payoff originate from different ends of the graph and coincide at K on the x-axis, forming a V. The lines for Put and call payoff originate from different ends of the graph and coincide below K on the x-axis, forming a V.

Combinations of Options (2 of 4)

Strangle

A portfolio that is long a call option and a put option on the same stock with the same exercise date but the strike price on the call exceeds the strike price on the put

Textbook Example 20.5 (1 of 2)

Strangle

Problem

You are long both a call option and a put option on Hewlett-Packard stock with the same expiration date. The exercise price of the call option is $40; the exercise price of the put option is $30. Plot the payoff of the combination at expiration.

Textbook Example 20.5 (2 of 2)

Solution

The red line represents the put’s payouts and the blue line represents the call’s payouts. In this case, you do not receive money if the stock price is between the two strike prices. This option combination is known as a strangle.

The x-axis shows the Stock price in dollars from 0 to 80 in increments of 20. The y-axis shows the payoff in dollars from 0 to 40 in increments of 10. The graph shows that the line for put originates at 30 on the y-axis and coincides with the x-axis on 30. The line for call originates at (80, 40) and coincides with the x-axis on 40.

Combinations of Options (3 of 4)

Butterfly Spread

A portfolio that is long two call options with differing strike prices and is short two call options with a strike price equal to the average strike price of the first two calls

Although a straddle strategy makes money when the stock and strike prices are far apart, a butterfly spread makes money when the stock and strike prices are close.

Figure 20.6 Butterfly Spread

The x-axis shows the stock price in dollars from 0 to 45. The y-axis shows the payoff in dollars from minus 30 to 30 in increments of 5 and the x-axis drawn at 0. The graph shows that the line for 20 call originates at 0 and runs along the x-axis till 20, after which it starts to increase and ends at (45, 20). The line for 40 call originates at 0 and runs along the x-axis till 40, after which it starts to increase and ends at (45, 5). The line for payoff originates at 0 and runs along the x-axis till 30, after which it decreases to (45, minus 30). The line for payoff of the entire combination originates at 0 and runs along the x-axis till 20, after which it increases to (30, 10) before dropping again to (40, 0).

Combinations of Options (4 of 4)

Protective Put

A long position in a put held on a stock you already own

Portfolio Insurance

A protective put written on a portfolio rather than a single stock

When the put does not itself trade, it is synthetically created by constructing a replicating portfolio.

Portfolio insurance can also be achieved by purchasing a bond and a call option

Figure 20.7 Portfolio Insurance

The plots show two different ways to insure against the possibility of the price of Tripadvisor stock falling below $45. The orange line in (a) indicates the value on the expiration date of a position that is long one share of Amazon stock and one European put option with a strike of $45 (the blue dashed line is the payoff of the stock itself). The orange line in (b) shows the value on the expiration date of a position that is long a zero-coupon riskfree bond with a face value of $45 and a European call option on Tripadvisor with a strike price of $45 (the green dashed line is the bond payoff).

A line graph a, plots pay off in dollars versus stock price in dollars. The Vertical axis shows payoff in dollars from 0 to 75 in increment of 15, and the Horizontal axis shows stock price in dollars from 0 to 75 in increment of 15. A doted curve starts from origin and ends at point (75, 75). Curve labeled as stock plus put starts from point (0, 45), becomes parallel with the horizontal axis till point (45, 45). Thereafter, it converges with the doted curve. All values are estimated.

A line graph b, plots pay off in dollars versus stock price in dollars. The Vertical axis shows payoff in dollars from 0 to 75 in increment of 15, and the Horizontal axis shows stock price in dollars from 0 to 75 in increment of 15. A doted curve labeled as riskless bond starts from the point (0, 45), becomes parallel with the horizontal axis till point (75, 45). Curve labeled as riskless bond plus call starts from point (0, 45), and converges with the horizontal axis till point (45, 45). Thereafter, the curve is upward sloping till point (75, 75). All values are estimated.

20.3 Put-Call Parity (1 of 4)

Consider the two different ways to construct portfolio insurance discussed previously

Purchase the stock and a put

Purchase a bond and a call.

Because both positions provide exactly the same payoff, the Law of One Price requires that they must have the same price.

20.3 Put-Call Parity (2 of 4)

Therefore,

Where K is the strike price of the option (the price you want to ensure that the stock will not drop below), C is the call price, P is the put price, and S is the stock price.

20.3 Put-Call Parity (3 of 4)

Rearranging the terms gives an expression for the price of a European call option for a non-dividend-paying stock:

This relationship between the value of the stock, the bond, and call and put options is known as put-call parity.

Textbook Example 20.6 (1 of 3)

Using Put-Call Parity

Problem

You are an options dealer who deals in non-publicly traded options. One of your clients wants to purchase a one-year European call option on H A L Computer Systems stock with a strike price of $20. Another dealer is willing to write a one-year European put option on H A L stock with a strike price of $20, and sell you the put option for a price of $3.50 per share. If H A L pays no dividends and is currently trading for $18 per share, and if the risk-free interest rate is 6%, what is the lowest price you can charge for the option and guarantee yourself a profit?

Textbook Example 20.6 (2 of 3)

Solution

Using put-call parity, we can replicate the payoff of the one-year call option with a strike price of $20 by holding the following portfolio: Buy the one-year put option with a strike price of $20 from the dealer, buy the stock, and sell a one-year risk-free zero-coupon bond with a face value of $20. With this combination, we have the following final payoff

depending on the final price of H A L stock in one year,

The columns have the following headings from left to right. Blank, S sub 1 is less than $20, S sub 1 is greater than $20. The Row entries are as follows. Row 1. Buy Put Option, 20 minus S sub 1, 0. Row 2. Buy Stock, S sub 1. S sub 1.. Row 3. Sell Bond, Negative 20, Negative 20. Row 4. Portfolio, 0, S sub 1 minus 20. Row 5. Sell Call Option, 0, Negative left parenthesis S sub 1 minus 20 right parenthesis. Row 6. Total Payoff, 0, 0.

Textbook Example 20.6 (3 of 3)

Note that the final payoff of the portfolio of the three securities matches the payoff of a call option. Therefore, we can sell the call option to our client and have future payoff of zero no matter what happens. Doing so is worthwhile as long as we can sell the call option for more than the cost of the portfolio, which is

Alternative Example 20.6 (1 of 2)

Problem

Assume

You want to buy a one-year call option and put option on Dellibar.

The strike price for each is $15.

The current price per share of Dellibar is $14.79.

Dellibar does not pay a dividend.

The risk-free rate is 2.5%.

The price of each call is $2.23.

Using put-call parity, what should be the price of each put?

Alternative Example 20.6 (2 of 2)

Solution

Put-Call Parity states

20.3 Put-Call Parity (4 of 4)

If the stock pays a dividend, put-call parity becomes

Textbook Example 20.7 (1 of 2)

Using Options to Value Near-Term Dividends

Problem

It is February 2016 and you have been asked, in your position as a financial analyst, to compare the expected dividends of several popular stock indices over the next several years. Checking the markets, you find the following closing prices for each index, as well as for options expiring December 2018.

Index	Feb 2016 Index Value	Dec 2018 Index Options Strike Price	Dec 2018 Index Options Call Price	Dec 2018 Index Options Put Price
DJIA	164.85	160	19.78	22.73
S&P 500	1929.80	1900	243.25	278.00
Nasdaq 100	4200.66	4200	636.35	666.85
Russell 2000	1022.08	1000	150.10	166.80

If the current risk-free interest rate is 0.90% for a December 2018 maturity, estimate the relative contribution of the near-term dividends to the value of each index.

Textbook Example 20.7 (2 of 2)

Solution

Rearranging the Put-Call Parity relation gives

Applying this to the D J I A, we find that the expected present value of its dividends over the next 34 month is

Therefore, expected dividends through December 2018 represent

of the current value of D J I A index. Doing a similar

calculation for the other indices, we find that near-term dividends account for 5.8% of the value of the S&P 500, 3.2% of the N A S D A Q 100, and 6.2% of the Russell 2000.

20.4 Factors Affecting Option Prices (1 of 2)

Strike Price and Stock Price

The value of a call option increases (decreases) as the strike price decreases (increases), all other things held constant.

The value of a put option increases (decreases) as the strike price increases (decreases), all other things held constant.

20.4 Factors Affecting Option Prices (2 of 2)

Strike Price and Stock Price

The value of a call option increases (decreases) as the stock price increases (decreases), all other things held constant.

The value of a put option increases (decreases) as the stock price decreases (increases), all other things held constant.

Arbitrage Bounds on Option Prices (1 of 3)

An American option cannot be worth less than its European counterpart.

A put option cannot be worth more than its strike price.

A call option cannot be worth more than the stock itself.

Arbitrage Bounds on Option Prices (2 of 3)

Intrinsic Value

The amount by which an option is in-the-money, or zero if the option is out-of-the-money.

An American option cannot be worth less than its intrinsic value.

Arbitrage Bounds on Option Prices (3 of 3)

Time Value

The difference between an option’s price and its intrinsic value.

An American option cannot have a negative time value.

Option Prices and the Exercise Date

For American options, the longer the time to the exercise date, the more valuable the option.

An American option with a later exercise date cannot be worth less than an otherwise identical American option with an earlier exercise date.

However, a European option with a later exercise date can be worth less than an otherwise identical European option with an earlier exercise date.

Option Prices and Volatility

The value of an option generally increases with the volatility of the stock.

Textbook Example 20.8 (1 of 2)

Option Value and Volatility

Problem

Two European call options with a strike price of $50 are written on two different stocks. Suppose that tomorrow, the low-volatility stock will have a price of $50 for certain. The high-volatility stock will be worth either $60 or $40, with each price having equal probability. If the exercise date of both options is tomorrow, which option will be worth more today?

Textbook Example 20.8 (2 of 2)

Solution

The expected value of both stocks tomorrow is $50 ̶ the low-volatility stock will be worth this amount for sure, and the high-

high-volatility stock has an expected value of

However, the options have very different values. The option on the low-volatility stock is worth nothing because there is no chance it will expire in-the-money (the low-volatility stock will be worth $50 and the strike price is $50). The option on the high-volatility stock is worth a positive amount because there is a 50% chance that it

will be worth

and a 50% chance that it will be

worthless. The value today of a 50% chance of a positive payoff (with no chance of a loss) is positive.

Alternative Example 20.8 (1 of 2)

Problem

You are considering investing in Finray, which currently trades at $13.10 per share.

A one-year call option with a strike price of $13 costs $0.87, while a put option with the same strike price and expiration costs $0.98.

If the risk-free rate is 0.10%, what is Finray’s expected dividend?

Alternative Example 20.8 (2 of 2)

Solution

Put-Call Parity states

Solving for PV(Div) yields

20.5 Exercising Options Early

Although an American option cannot be worth less than its European counterpart, they may have equal value.

Non-Dividend-Paying Stocks (1 of 5)

For a non-dividend paying stock, Put-Call Parity can be written as

Where dis(K) is the amount of the discount from face value of the zero-coupon bond K.

Non-Dividend-Paying Stocks (2 of 5)

Because dis(K) and P must be positive before the expiration date, a European call always has a positive time value.

Because an American option is worth at least as much as a European option, it must also have a positive time value before expiration.

Thus, the price of any call option on a non-dividend-paying stock always exceeds its intrinsic value prior to expiration.

Non-Dividend-Paying Stocks (3 of 5)

This implies that it is never optimal to exercise a call option on a non-dividend paying stock early.

You are always better off just selling the option.

Because it is never optimal to exercise an American call on a non-dividend-paying stock early, an American call on a non-dividend paying stock has the same price as its European counterpart.

Non-Dividend-Paying Stocks (4 of 5)

However, it may be optimal to exercise a put option on a non-dividend paying stock early

Non-Dividend-Paying Stocks (5 of 5)

When a put option is sufficiently deep in-the-money, dis(K) will be large relative to the value of the call, and the time value of a European put option will be negative.

In that case, the European put will sell for less than its intrinsic value.

However, its American counterpart cannot sell for less than its intrinsic value, which implies that an American put option can be worth more than an otherwise identical European option.

Textbook Example 20.9 (1 of 2)

Early Exercise of a Put Option on a Non-Dividend-Paying Stock

Problem

Table 20.2 lists the quotes from the C B O E on July 20, 2018, for options on Alphabet stock (Google’s holding company) expiring in September 2018. Alphabet will not pay a dividend during this period. Identify any option for which exercising the option early is better than selling it.

Textbook Example 20.9 (2 of 2)

Solution

Because Alphabet pays no dividends during the life of these options (July 2018 to September 2018), it should not be optimal to exercise the call options early. In fact, we can check that the bid price for each call option exceeds that option’s intrinsic value, so it would be better to sell the call than to exercise it. For

example, the payoff from exercising a call with a strike of 1000 early is

while the option can be sold for $204.90.

On the other hand, an Alphabet shareholder holding a put option with a strike price of $1360 or higher would be better off exercising—rather than selling—the option. For example, by exercising the 1400 put the shareholder would receive $1400 for her stock, whereas by selling the stock and the option she would only

receive

The same is not true of the puts with strikes

below $1360, however. For example, the holder of the 1320 put option who exercises it early would receive $1320 for her stock, but would net

by selling the stock and the put instead. Thus, early exercise

is only optimal for the deep in-the-money put options.

Table 20.2 Alphabet Option Quotes

Source: Chicago Board Options Exchange at

GOOGL (ALPHABET INC (A)) 1197.88 minus 1.22

Jul 20 2018 @ 17:59 ET Bid 1197 Ask 1198.78 Size 1 into 1 Vol 1896554

Calls Bid Ask Open Int Puts Bid Ask Open Int

2018 Sep 21 1000.00 (GOOGL182111000) 204.9 206.4 576 2018 Sep 21 1000.00 (GOOGL1821U1000) 3.2 3.6 834

2018 Sep 21 1040.00 (GOOGL182111040) 166.9 169.7 132 2018 Sep 21 1040.00 (GOOGL1821 U1040) 5.3 6 141

2018 Sep 21 1080.00 (GOOGL182111080) 127.5 136.9 226 2018 Sep 21 1080.00 (GOOGL1821U1080) 9.3 9.9 342

2018 Sep 21 1120.00 (GOOGL182111120) 95.1 104.5 196 2018 Sep 21 1120.00 (GOOGL1821U1120) 16.1 16.9 277

2018 Sep 21 1160.00 (GOOGL182111160) 70 71 345 2018 Sep 21 1160.00 (GOOGL1821U1160) 27.2 28.1 176

2018 Sep 21 1200.00 (GOOGL182111200) 46.3 47.3 1295 2018 Sep 21 1200.00 (GOOGL1821U1200) 43.6 44.5 345

2018 Sep 21 1280.00 (GOOGL182111280) 16.1 16.6 202 2018 Sep 21 1280.00 (GOOGL1821U1280) 89.7 99 10

2018 Sep 21 1320.00 (GOOGL182111320) 8.4 8.9 150 2018 Sep 21 1320.00 (GOOGL1821U1320) 127 128 10

2018 Sep 21 1360.00 (GOOGL182111360) 4.1 4.6 109 2018 Sep 21 1360.00 (GOOGL1821U1360) 159 169 1

2018 Sep 21 1400.00 (GOOGL182111400) 2.1 2.35 265 2018 Sep 21 1400.00 (GOOGL1821U1400) 198 207.5 0

Dividend-Paying Stocks (1 of 3)

The put-call parity relationship for a dividend-paying stock can be written as

If PV(Div) is large enough, the time value of a European call option can be negative, implying that its price could be less than its intrinsic value.

Because an American option can never be worth less than its intrinsic value, the price of the American option can exceed the price of a European option.

Dividend-Paying Stocks (2 of 3)

With a dividend paying stock, it may be optimal to exercise the American call option early.

When a company pays a dividend, investors expect the price of the stock to drop.

When the stock price falls, the owner of a call option loses.

Unlike the owner of the stock, the option holder does not get the dividend as compensation.

However, by exercising early and holding the stock, the owner of the call option can capture the dividend.

Textbook Example 20.10 (1 of 2)

Early Exercise of Options on a Dividend-Paying Stock

Problem

General Electric (G E) stock went ex-dividend on December 22, 2005 (only equity holders on the previous day are entitled to the dividend). The dividend amount was $0.25. Table 20.3 lists the quotes for G E options on December 21, 2005. From the quotes, identify the options that should be exercised early rather than sold.

Textbook Example 20.10 (2 of 2)

Solution

The holder of a call option on G E stock with a strike price of $32.50 or less is better off exercising ̶ rather than selling ̶ the option. For example, exercising

the 06 January 10 call and immediately selling the stock would net

The option itself can be sold for $25.40, so the holder is better off by

$0.12 by exercising the call rather than selling it. To understand why early exercise can be optimal in this case, note that interest rates were about 0.33% per month, so the value of delaying payment of the $10 strike price until January was worth only $0.033, and the put option was worth less than $0.05. Thus, from Eq. 20.7, the benefit of delay was much less than the $0.25 value of the dividend.

On the other hand, all of the put options listed have a positive time value and thus should not be exercised early. In this case, waiting for the stock to go ex-dividend is more valuable than the cost of delaying the receipt of the strike price.

Table 20.3 Option Quotes for G E on December 21, 2005 (G E paid $0.25 dividend with ex-dividend date of December 22, 2005)

Source: Chicago Board Options Exchange at

GE 35.52 minus 0.02

Dec 21, 2005 @ 11:50 ET Vol 8103000

Calls Bid Ask Open Interest Puts Bid Ask Open Interest

06 Jan 10.00 (GE AB-E) 25.4 25.6 738 06 Jan 10.00 (GE MB-E) 0 0.05 12525

06 Jan 20.00 (GEAD-E) 15.4 15.6 1090 06 Jan 20.00 (GE MD-E) 0 0.05 8501

06 Jan 25.00 (GE AE-E) 10.4 10.6 29592 06 Jan 25.00 (GE ME-E) 0 0.05 36948

06 Jan 30.00 (GEAF-E) 5.4 5.6 37746 06 Jan 30.00 (GE MF-E) 0 0.05 139548

06 Jan 32.50 (GEAZ-E) 2.95 3.1 13630 06 Jan 32.50 (GE MZ-E) 0 0.05 69047

36 Jan 35.00 (GEAG-E) 0.7 0.75 146682 06 Jan 35.00 (GE MG-E) 0.3 0.35 140014

06 Jan 40.00 (GEAH-E) 0 0.05 84366 06 Jan 40.00 (GE MH-E) 4.7 4.8 4316

06 Jan 45.00 (GEAI-E) 0 0.05 7554 06 Jan 45.00 (GE Ml-E) 9.7 9.8 767

06 Jan 50.00 (GEAJ-E) 0 0.05 17836 06 Jan 50.00 (GE MJ-E) 14.7 14.8 383

06 Jan 60.00 (GEAL-E) 0 0.05 7166 06 Jan 60.00 (GE ML-E) 24.7 24.8 413

Dividend-Paying Stocks (3 of 3)

The put-call parity relationship for puts can be written as

As stated earlier, European options may trade for less than their intrinsic value.

On the next slide, note that all the puts with a strike price of $1400 or higher trade for less than their exercise value.

Table 20.4 Two-Year Call and Put Options on the S&P 500 Index