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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Education.
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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