January 2, 2020

Chapter 16

Option Valuation

Slides

16-1. Chapter 16

16-2. Option Valuation

16-3. Learning Objectives

16-4. Option Valuation

16-5. Just What is an Option Worth?

16-6. A Simple Model to Value Options Before Expiration, I.

16-7. A Simple Model to Value Options Before Expiration, II.

16-8. The One-Period Binomial Option Pricing Model—The Assumptions

16-9. The One-Period Binomial Option Pricing Model—The Setup

16-10. The Value of this Portfolio (long Shares and short one call) is:

16-11. To Calculate Today’s Call Price, C:

16-12. Therefore, Our First Step is to Calculate

16-13. Sidebar: What is ?

16-14. The One-Period Binomial Option Pricing Model—The Formula

16-15. Now We Can Calculate the Call Price, C

16-16. The Two-Period Binomial Option Pricing Model

16-17. The Method

16-18. The Binomial Option Pricing Model with Many Periods

16-19. What Happens When the Number of Periods Gets Really, Really Big?

16-20. The Black-Scholes Option Pricing Model

16-21. The Black-Scholes Option Pricing Model

16-22. The Black-Scholes Option Pricing Formula

16-23. Formula Details

16-24. Example: Computing Prices for Call and Put Options

16-25. We Begin by Calculating d1 and d2

16-26. Using the =NORMSDIST(x) Function in Excel

16-27. The Call Price and the Put Price:

16-28. We can Verify Our Results Using Put-Call Parity Equation

16-29. Valuing the Options Using Excel

16-30. Using a Web-based Option Calculator

16-31. Varying the Option Price Input Values

16-32. Varying the Underlying Stock Price

16-33. Varying the Time Remaining Until Option Expiration

16-34. Varying the Volatility of the Stock Price

16-35. Varying the Interest Rate

16-36. Calculating the Impact of Stock Price Changes on Option Prices

16-37. Calculating Delta

16-38. Example: Calculating Delta with Excel

16-39. The "Delta" Prediction:

16-40. Hedging with Stock Options

16-41. Hedging Using Call Options—The Prediction

16-42. Hedging Using Call Options—The Results

16-43. Hedging Using Put Options—The Prediction

16-44. Hedging Using Put Options—The Results

16-45. Hedging a Portfolio with Index Options

16-46. Example: Calculating the Number of Option Contracts Needed to Hedge an

Equity Portfolio

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Option Valuation 16-2

16-47. Implied Standard Deviations

16-48. CBOE Implied Volatilities for Stock Indexes

16-49. Employee Stock Options, ESOs

16-50. Features of ESOs

16-51. Why are ESOs Granted?

16-52. ESO Repricing

16-53. ESO Repricing Controversy

16-54. ESOs Today

16-55. Valuing Employee Stock Options, I.

16-56. Valuing Employee Stock Options, II.

16-57. Example: Valuing Coca-Cola ESOs Using Excel

16-58. Summary: Coca-Cola Employee Stock Options

16-59. Useful Websites

16-60. Chapter Review, I.

16-61. Chapter Review, II.

Chapter Organization

16.1 A Simple Model to Value Options before Expiration

16.2 The One-Period Binomial Option Pricing Model

A. The One Period Binomial Option Pricing Model – The

Assumptions

B. The One Period Binomial Option Pricing Model – The Setup

C. The One Period Binomial Option Pricing Model – The Formula

D. What is Delta?

16.3 The Two-Period Binomial Option Pricing Model

A. Step 1: Build a Price Tree for Stock Prices Through Time

B. Step 2: Use the Intrinsic Value Formula to Calculate the

Possible Option Prices at Expiration

C. Step 3: Calculate the Fractional Share Needed to Form Each

Risk-Free Portfolio at the Next-to-Last Date

D. Step 4: Calculate All Possible Option Prices at the Next-to-Last

Date

E. Step 5: Repeat this Process by Working Back to Today

16.4 The Binomial Option Pricing Model with Many Periods

16.5 The Black-Scholes Option Pricing Model

16.6 Varying the Option Price Input Values

A. Varying the Underlying Stock Price

B. Varying the Option's Strike Price

C. Varying the Time Remaining Until Option Expiration

D. Varying the Volatility of the Stock Price

E. Varying the Interest Rate

16.7 Measuring the Impact of Stock Price Changes on Option Prices

A. Interpreting Option Deltas

16.8 Hedging Stock with Stock Options

A. Hedging Using Call Options – The Prediction

A. Hedging Using Call Options – The Results

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Option Valuation 16-3

B. Hedging Using Put Options – The Prediction

C. Hedging Using Put Options – The Results

16.9 Hedging a Stock Portfolio with Stock Index Options

16.10 Implied Standard Deviations

A. CBOE Implied Volatilities for Stock Indexes

16.11 Employee Stock Options

A. ESO Features

B. ESO Repricing

C. ESO at The Gap, Inc.

D. Valuing Employee Stock Options

16.12 Summary and Conclusions

Selected Web Sites

www.cboe.com (for a free option price calculator)

www.option-price.com (for an options pricing calculator)

www.ivolatility.com (for applications of implied volatility)

www.wsj.com

www.coca-cola.com (information on Coke related to in text example)

www.ino.com (Web Center for Futures and Options)

Annotated Chapter Outline

16.1 A Simple Model to Value Options before Expiration

If the put option is guaranteed to finish out of the money, then the put-call

parity equation can be used to value the call option.

16.2 The One-Period Binomial Option Pricing Model

A. The One Period Binomial Option Pricing Model Assumptions

u = the up factor, indicating the stock price move on the up side

d = down factor, indicating the stock price move on the down side

B. The One Period Binomial Option Pricing Model – The Setup

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Option Valuation 16-4

C. The One Period Binomial Option Pricing Model – The Formula

Call =

ΔS(1 + r − u )+ Cu

1+r

D. What is Delta?

Delta is a proportion of shares to calls that is needed to form a risk-free

portfolio.

16.3 The Two-Period Binomial Option Pricing Model

A. Step 1: Build a Price Tree for Stock Prices Through Time

B. Step 2: Use the Intrinsic Value Formula to Calculate the Possible

Option Prices at Expiration

C. Step 3: Calculate the Fractional Share Needed to Form Each Risk-

Free Portfolio at the Next-to-Last Date

D. Step 4: Calculate All Possible Option Prices at the Next-to-Last Date

E. Step 5: Repeat this Process by Working Back to Today

16.4 The Binomial Option Pricing Model with Many Periods

This is simply a repetitive approach of the single period model.

16.5 The Black-Scholes Option Pricing Model

The Black-Scholes Option Pricing Model, an important development in finance,

states the value of a European option on a non-dividend-paying stock as a

function of these five input factors:

The current price of the underlying stock.

The strike price specified in the option contract.

The risk-free interest rate over the life of the option contract.

The time remaining until the option contract expires, sometimes called

expiry.

The price volatility of the underlying stock.

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Option Valuation 16-5

The Black-Scholes-Merton formula for the price of a European call option on a

single share of common stock is:

The Black-Scholes-Merton formula for the price of a European put option on a

single share of common stock is:

d1 and d2 are calculated as:

Note that after calculating d1 and d2, the standard normal probability of the value

of dn is used as the input into the formulas for the call and put options.

Using this form of put-call parity, the value of a put option can be calculated from

the value of the call option, as follows:

P = C - Se-yT + Ke-rT

The Black-Scholes Model (no Merton) excludes dividends, so the e-yT term is

absent.

Lecture Tip: Once students understand the basics of options discussed in the

previous chapter, it is important that they have a sense of option pricing and how

changing each of the input variables affects the option price. Changing the input

variables will be discussed in the next section, but understanding option pricing is

important as well. The text indicates this model is very complicated and does not

need to be computed by hand calculator. There are many alternatives to

calculating this model by hand: web calculators (such as the CBOE and others

shown above in Selected Web Sites), programmable calculators, computer

software, and spreadsheet models.

Lecture Tip: Another way to give students a sense of the B-S-M model and show

them that it is not as complicated as they may think is to simplify the model. This

can be accomplished by starting with the basic call pricing model:

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C=S e−y T N(d1) − K e−r T N(d2)

P= − K e−r T N(−d2) − S e−y T N(−d1)

d1=

ln

(

S

K

)

+

(

r−y+σ2

2

)

T

σ

√

T

d2=d1−σ

√

T

Option Valuation 16-6

Explain how N(d1) and N(d2) reflect probabilities related to the variability of the

stock price and the probability of expiring in-the-money. Next, describe that we

will assume we know the future with certainty so that the probabilities are equal

to 100%. We also assume no dividends will be paid on this stock, so we have:

The result is very similar to the call pricing from the previous chapter; the value of

a call option is equal to the stock price minus the strike price. The only change is

that we have the present value of the strike price. This presents a much more

simplified version to allow students to understand the intuition.

Finally, the imbedded spreadsheet contained in the PPT slides is a very handy

way to show students that the B-S-M formula really is not too difficult for them to

handle.

16.6 Varying the Option Price Input Values

Table 16.3 summarizes the effects of each of the five input variables (excludes

dividend) on the price of call and put options. Note that the table can be

interpreted from the viewpoint of, "If the input variable increases, how does this

affect the call or put option price?" So, for example, if the stock price increases,

the value of a call option will increase and the value of a put option will decrease.

The two most important inputs are the stock price and the strike price. The next

six sections discuss the logic for these effects.

A. Varying the Underlying Stock Price

Figure 16.6 shows how the value of call and put options vary as the stock price

changes. As the stock price increases, S - K also increases, so the value of a call

option will increase. As the stock price increases, K - S decreases, so the value

of a put option will decrease.

B. Varying the Option's Strike Price

As the strike price increases, S - K decreases, so the value of a call option will

decrease. As the strike price increases, K - S increases, so the value of a put

option will increase.

C. Varying the Time Remaining Until Option Expiration

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C=S e−y T N(d1) − K e−r T N(d2)

C=S−K e−r T

Option Valuation 16-7

Figure 16.7 shows how the values of call and put options vary as the time to

expiration changes. An easy way to view changing the time to expiration is to ask

the question, "Would you rather own an option with one day until expiration or

three months until expiration?" Obviously owning an option with more time to

move in-the-money or move further in-the-money is more valuable. This is true

for both call options and put options.

D. Varying the Volatility of the Stock Price

Figure 16.8 shows how the values of call and put options vary as the volatility

changes. Start by asking the question, "Would you rather own an option that

experiences more price movement (volatility) or less price movement (volatility)?"

Owning an option with more volatility gives it a higher probability of being in-the-

money or moving further in-the-money, and that makes it more valuable.

Alternatively, an option with very little volatility may never move in-the-money.

This is true for both call options and put options.

E. Varying the Interest Rate

Figure 16.9 shows how the values of call and put options vary as the risk-free

rate changes. We start with the simplified B-S-M model:

Now we can vary the risk-free rate and observe the changes in option values. As

the risk-free rate increases, more discounting occurs, which decreases the

present value of K. Therefore S - K will be a larger value, so an increasing

interest rate will increase the value of a call option. Put options will react in the

opposite way. As the risk-free rate increases, more discounting occurs, but since

the discounting reduces the present value of K, the value K - S will decrease, so

the value of the put option will decrease as the risk-free rate increases.

Lecture Tip: Starting with put-call parity and varying the dividend yield:

C - P = S - Ke-rT - D

From the above equation, increasing the dividend yield will decrease the value of

a call option and increase the value of a put option. Also consider that when a

firm pays a dividend, its assets are reduced by the amount of the dividend, which

causes a decrease in the stock price. The lower stock price reflects a lower call

option value.

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C=S−K e−r T

Option Valuation 16-8

16.7 Measuring the Impact of Stock Price Changes on Option Prices

Delta: Measure of the dollar impact of a change in the underlying stock

price on the value of a stock option. Delta is positive for a call option and

negative for a put option.

Investment professionals that use options use the "Greeks" to assess the impact

of changes in input value on option prices. Delta is calculated as follows:

If the dividend yield is zero (non-Merton model), then the discounting term “falls”

away.

The imbedded spreadsheet in the PowerPoint helps students understand how to

calculate deltas for calls and puts.

A. Interpreting Option Deltas

Delta measures the impact of a change in the stock price on the value of the

option. A $1 change in stock price causes the option to change by delta dollars.

Delta is positive for calls and negative for puts.

Gamma, Theta, and Rho are also measures of option price sensitivity commonly

used by investment professionals – so-called option “greeks.”

16.8 Hedging Stock with Stock Options

Suppose that you want to protect yourself against declines in XYZ stock price.

That is, you want to hedge, i.e., you want to have a portfolio that does not

change in value if the stock price changes. Thus, you want changes in your

portfolio value from stock price changes to be equal to changes in the value of

your portfolio due to options.

Change in stock price × Shares = - Change in option price × Number of options

You know that the delta of an option is a prediction of how the option price will

change when the stock price changes. So, we can rewrite the equation above as:

Change in stock price × Shares = - Option delta × Change in stock price ×

Number of options

So, the number of options needed = -Shares held × (1/Option delta)

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Put option Delta = −e−y T N

(

−d1

)

<0

Call option Delta =e−y T N

(

d1

)

>0

Option Valuation 16-9

A. Hedging Using Call Options – The Prediction

To hedge declines in XYZ share prices using call options, you need to write, or

short sell, call options to protect against a price decline. This is because the price

of a call option and the underlying price move in the same direction. Because

delta is between zero and one, you will have to write more than one option

contract for each 100 shares hedged.

B. Hedging Using Call Options – The Results

The gain in the call options nearly offsets your loss of $1,000 in XYZ shares. Why

isn’t it exact? This is because delta changes as the underlying changes. This

“gamma” effect can also be hedged—but, the hedger needs to use call options

with two separate strikes.

C. Hedging Using Put Options – The Prediction

Underlying prices and put option prices are inversely related. This means the

hedger takes the same position in the put and the underlying. But, because a put

delta is between negative one and zero, the hedger will have to buy more than

one option contract for each 100 shares hedged.

D. Hedging Using Put Options – The Results

The gain in the put options more than offsets the loss in the shares. Why isn’t it

exact? The put delta also falls when the stock price falls. This means that too

many put options were purchased. This gamma effect can be neutralized using a

second set of put options with a strike that differs from the initial puts. Note

further that the puts used in this example were more than 10% out-of-the-money.

This means that one would expect the gamma effect to be greater for these than

for at-the-money puts.

16.9 Hedging a Stock Portfolio with Stock Index Options

Hedging using stock index options is very common among portfolio managers. To

calculate the number of stock index contracts necessary to hedge an equity

portfolio:

Number of option contracts = -

Portfolio beta × Portfolio value

Option delta × Option contract value

Note that when the dividend yield is zero, the option delta is equal to N(d1) and is

often referred to as the hedge ratio. To maintain an effective hedge over time, it is

necessary to rebalance the hedge on a regular basis, perhaps weekly.

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Option Valuation 16-10

Rebalancing requires recalculating the number of option contracts necessary to

maintain an effective hedge.

16.10 Implied Standard Deviations

Implied standard deviation (ISD): An estimate of stock price volatility obtained

from an option price.

Implied volatility (IVOL): Another term for implied standard deviation.

A. CBOE Implied Volatilities for Stock Indexes

VIX, VXO, VXN: Volatility indexes for the S&P 500, S&P 100, NASDAQ 100

stock indexes, respectively, based on stock index options.

16.11 Employee Stock Options

Employee stock option (ESO): An option granted to an employee by a

company giving the employee the right to buy shares of stock in the company at

a fixed price for a fixed time.

Companies issuing stock options to employees must report estimates of the

value of these ESOs. The Black-Scholes-Merton formula is widely used for this

purpose. The practice of granting options to employees has become widespread.

It is almost universal for upper management, but some companies, like The Gap

and Starbucks, have granted options to almost every employee.

A. ESO Features

A typical ESO has a 10-year life, which is much longer than ordinary options.

Unlike traded options, ESOs cannot be sold. They also have a “vesting”

period.

B. ESO Repricing

ESOs are almost always “at the money” when they are issued, meaning that

the stock price is equal to the strike price. If the stock falls significantly after

an ESO is granted, then the option is said to be “underwater.” Occasionally, a

company will lower the strike price on underwater options. Such options are

said to be “restruck” or “repriced.”

C. ESO at The Gap, Inc.

D. Valuing Employee Stock Options

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Option Valuation 16-11

The Financial Accounting Standards Board issued FASB 123 to tell

companies how to calculate the fair value of employee stock options. FASB

basically states that the fair value of ESOs should be determined using an

option pricing model that takes into account the:

Stock price at the grant date.

Exercise price.

Expected life of the option.

Volatility of the underlying stock.

Risk-free interest rate over the expected life of the option.

Expected dividends.

Companies issuing stock options to employees must report estimates of the

value of these ESOs. The Black-Scholes-Merton formula is widely used for this

purpose.

For example, in December 2002, the Coca-Cola Company granted ESOs with a

stated life of 15 years. However, to allow for the fact that ESOs are often

exercised before maturity, Coca-Cola also used a life of 6 years to value these

ESOs.

The embedded spreadsheet in the PPT slides shows students how to value

ESOs using the B-S-M formula.

16.12 Summary and Conclusions

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