Leading / Thinking / Performing®
Basics of Stock Option Valuation for
the Non-Valuation Professional
This article was authored by Nancy M. Czaplinski, CPA/ABV/CGMA, CFA, ASA, and Kevin M. Hagemeier,
ASA, CPIM, Milwaukee, Wisconsin.
This article discusses the nature of stock options; examines the factors affecting their
value; and explores the methods available for valuing these instruments, including
options for stock that has no established market price (i.e., those for private, nonpublic
companies). The discussion of the strengths and weaknesses of the various optionvaluation models is in general terms, without reference to the intricate statistical formulas
underlying each model. For purposes of this article, stock option and option will be used
interchangeably.
Before discussing the accounting treatment and valuation of options, a few terms
must be defined. First, an option gives the holder the right to purchase or sell a stated
number of shares of stock at a fixed price within a predetermined period. The holder of
the option, however, is not obligated to exercise the option. The fixed price at which the
option is exercisable is called the exercise price, or sometimes the strike price.
There are two types of options: call options and put options. Call options provide the
holder the right to buy a security. Put options, on the other hand, provide the holder the
right to sell a security. For simplicity, this article will focus on call options, only using put
options to present and highlight differences between the option rights.
The value of a stock option can be broken down into two components called intrinsic
value and time value. Intrinsic value is the difference between the value of the stock
and the exercise price, or price at which the stock can be purchased by the holder of
the option. For example, if the share price is $95 and the exercise price of the option
is $100, the intrinsic value is zero ($95 minus $100 produces a negative number).
However, if the figures were reversed and the exercise price of the option was $95 and
the share price was $100, the intrinsic value of the option would be $5 ($100 minus
$95). The intrinsic value of a stock option may be either positive or zero, but it can never
be negative, since the contract involves no liability on the part of the option holder; the
option holder can walk away without exercising the option. In our first example, where
the value of the stock was less than the option exercise price, the option is referred to
as being out of the money. In our second example, where the value of the stock was
greater than the option exercise price, the option is referred to as being in the money.
Where the value of the stock is equal to the option exercise price, the option is referred
to as being at the money.
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Options are often used in lieu of cash compensation for start-up companies or to retain key executives
and employees. Options can be granted to purchase stock of either a publicly traded or a privately held
(nonpublic) company. In the former, the option may or may not have a public trading market; in the latter,
neither the stock nor the options are publicly traded. As appraisers, we are often asked how a privately held,
nonpublic company can issue options because the stock associated with the option does not have a current
public market and, therefore, a published market price. Many privately held start-up companies issue options
to their employees to compensate them for the current inability to pay a market wage, hoping this added
incentive will reduce turnover of key employees and thereby spur growth and profitability. Executives of
privately held companies may receive options as an added incentive for reaching certain financial objectives.
Owners of privately held companies may also issue options to key employees as a way of broadening the
business ownership. From a valuation standpoint, before we can determine a value for the option, we must
first value the privately held company and determine its stock value. Once the value of the underlying stock
has been determined, the appropriate option valuation models may be used.
ACCOUNTING TREATMENT OF STOCK OPTIONS PURSUANT TO ASC 718
Financial Accounting Standards Board (“FASB”) Accounting Standards Codification ("ASC") Topic 718,
Compensation – Stock Compensation (“ASC 718”), establishes a method of accounting for stock-based
compensation that is based on the fair value of stock options and similar instruments. When applying this
general principle to stock compensation, equity instruments must be measured and recognized at their fair
value, and the compensation cost is recognized over the vesting period of each option grant.
The objective of the measurement process in ASC 718 is to recognize in the financial statements the
employee services received in exchange for equity instruments issued or liabilities incurred and the related
cost to the entity as those services are consumed. This Topic uses the terms compensation and payment in
their broadest senses to refer to the consideration paid for employee services (please refer to ASC 718-1010-1). The fair value of a stock option granted by an entity shall be estimated in accordance with ASC 718
by an option-pricing model that takes into consideration, as of the grant date, the following factors:
• Exercise price of the option
• Expected life or term of the option, taking into account both the contractual term of the option and the
effects of employees’ expected exercise and post-vesting employment termination behavior
• Current price of the underlying stock
• Expected volatility of the underlying stock
• Expected dividends on the stock
• The risk-free interest rate for the expected term of the option
ASC 718 suggests use of a closed-form model/Black-Scholes or a lattice model/binomial, which may
include Monte Carlo simulation (discussed later in this article) for valuation of options relating to employee
share-based compensation. An entity should change the valuation technique it uses to estimate fair value if it
concludes that a different technique is likely to result in a better estimate of fair value. A change in either the
valuation technique or the method of determining appropriate assumptions used in a valuation technique is
a change in accounting estimate for purposes of applying APB Opinion No. 20, Accounting Changes, and
should be applied prospectively to new awards.
OPTION VALUATION VARIABLES
The valuation of an option encompasses many variables. Following is a discussion
of some of the key option-valuation model inputs and how they impact the valuation
process for both put and call options. All of the exhibits that follow are based on our
previous example of an option with a $100 exercise price, an initial risk-free interest rate
of 5%, one month until expiration, and volatility of 22%. By using the Black-Scholes
method of valuing options, we ran varying scenarios that are summarized in the following
graphics.
Stock Price
In general, as the value of the underlying stock increases, the value of the call option also
increases. Referring to the blue solid line in Figure 1 below, if the exercise price is $100
and the underlying stock price is $95 and all other variables remain constant, the option
will have a greater value than if the underlying stock price was less than $95.
Figure 1. General Relationship Between Price of the Stock and Value of the Option
Exercise Price
As defined on a previous page, the exercise price is the price at which the holder of the
option may purchase the stock. Using the example from the previous paragraph, if the
exercise price of the option is $125 rather than $100, and the underlying stock price is
still $95, and all other variables remain constant, the value of the option would be less.
This situation occurs because the probability of the stock achieving a share price greater
than $125 to make the exercise of the option beneficial would be far less than the
probability of the stock achieving a price of $100 or above.
Risk-Free Rate
The risk-free rate of return is used to determine an opportunity value for the funds
required to fund the purchase of the option. Higher interest rates in the economy tend to
produce higher option values. As interest rates go up, the required rate of return on all
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investments, including common stock, rises. Concurrently, stock values tend to decline, and their expected
total rates of return to the investor, including dividends and capital appreciation, will equate to rates of return
available in the market on other investments of comparable risk. Said another way, to the extent that values
of the underlying common stocks reflect efficient capital markets, the higher the level of interest rates, the
higher the expected rate of appreciation in the underlying stock’s value. The higher the rate of appreciation in
the underlying stock, the greater the value of the option. Figure 2 depicts this relationship.
Figure 2. Relationship of Option Value to Interest Rates
Volatility Factor
The volatility factor quantifies the fluctuations in the value of the underlying stock over time. The wider the
fluctuations (or greater the volatility), the greater the option’s time value. Fluctuations add to the value of
the option because upside fluctuations theoretically enhance the option’s value infinitely, while downside
fluctuations cannot drive the option value below zero. Assuming our exercise price of $100 and the current
stock price of $100 with one month until expiration of the option, Figure 3 below depicts the changes in
value of the option given varying volatility levels of the underlying stock.
Figure 3. Relationship of Option Value to Volatility
Dividends
The payment of dividends on the underlying stock detracts from the value of an option,
because the holder of the option does not receive the dividends. In addition, the monies
a company pays out in dividends might otherwise be available for reinvestment in the
company, which would contribute to the growth in value of the underlying stock. For
put options, dividends have the opposite impact, increasing value with higher dividend
returns.
Time to Expiration
Time to expiration is the actual time from the present, or value date, until the expiration
date of the option. The longer the time to expiration, the greater the underlying stock’s
opportunity to appreciate in value, thereby enhancing the value of the option. Using
the example from above, the option with the $100 exercise price would have a greater
value, all other variables remaining constant, if the time to expiration was 180 days rather
than 30 days. This would allow for a longer period for the price of the underlying stock
to increase to a value greater than the $100 exercise price. This relationship is below in
Figure 4.
Figure 4. Relationship of Option Value to Time to Expiration
COMMONLY USED OPTION-VALUATION MODELS
Now that some of the key variables to the various option-valuation models have been
defined and discussed, the remainder of this article addresses the most commonly used
valuation methods and their unique strengths and weaknesses.
Black-Scholes Model
The Black-Scholes model is probably the most widely used and best-known theoretical
option-valuation model. A theoretical model is a forward-looking model that attempts
to determine what the option should sell for in the market given the option terms and
the underlying stock’s salient points. This option pricing model is a partial differential
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equation that computes an option price based on the difference between the adjusted stock price and
the adjusted present value exercise price. Adjustments in stock price and exercise price are based on a
lognormal distribution relationship between the change in option price relative to the change in stock price.
The relationship is called the hedge ratio, which is used to compute the number of options necessary to
equate the return on a portfolio for risk-free securities to the return of a portfolio of options and stocks.
The Black-Scholes model is appropriate for options with expiration dates that are relatively short term in
nature, largely due the underlying assumptions of the model. However, it is important to note that the BlackScholes model is often inappropriately used to value options that have a fairly long time until expiration. It is
important to be aware of the underlying assumptions of the Black-Scholes model to avoid misapplication.
The assumptions are as follows:
1. The short-term interest rate is known and is constant through time.
2. The stock price follows a random walk in continuous time with a rate of variance in proportion to the
square of the stock price.
3. The distribution of possible stock prices at the end of any finite interval is lognormal.
4. The variance of the rate of return on the stock is constant.
5. The stock pays no dividends and makes no other distributions.
6. The option can be exercised only at maturity.
7. No commissions or other transaction costs are incurred in buying or selling the stock or the option.
8. It is possible to borrow any fraction of the price of a security, or to buy or hold a security, at the shortterm interest rate.
9. A seller who does not own a security (a short seller) will simply accept the price of the security from the
buyer and agree to settle with the buyer on some future date by paying him or her an amount equal to
the price of the security on that date. While this short sale is outstanding, the short seller will have the
use of, or interest on, the proceeds of the sale.
10.The tax rate, if any, is identical for all transactions and all market participants.
It may be difficult to apply the Black-Scholes model on a 10-year option, for example, since it may be
improbable that the short-term interest rate will remain constant for a 10-year period. In addition, the
assumption that the option can be exercised only at maturity may not hold true, especially for a long-term
option on a highly volatile stock. For example, the stock price may exceed the exercise price in Year 5, and
the holder may exercise his or her right to purchase stock. However, the price of the option developed under
the Black-Scholes model assumes the option will not be exercised until expiration, which would be a full 10
years rather than five years. Given the above assumptions, the Black-Scholes model must be very carefully
applied on a situation-by-situation basis.
Lattice Model
A more integrated and complex model used to value options is called the lattice model. This model is a
theoretical model like the Black-Scholes model; however, it uses either a binomial or a trinomial distribution
process to derive value by separating the total maturity period of the option into discrete periods. When
progressing from one time period, or node, to another, the underlying common stock price is assumed to
have an equal probability of increasing and/or decreasing by upward and downward price movements.
Visually, a binomial lattice model would appear as follows:
As each node of the binomial lattice represents the value of the option, the model
considers the probabilities of various triggering events and early exercise and
incorporates the value of the option if such an event occurs. If none of the triggering
events occurs, the value of the option at each node will be calculated as the expected
value of the two successive nodes using the discount factor.
The lattice model is a more flexible valuation technique that can account for early
exercise behavior and various market and performance conditions.
Monte Carlo Simulation
Monte Carlo simulation methods are often used to value complex derivative instruments
including stock options by simulating various path-dependent conditions. This approach
simulates share price movements using assumptions of lognormally distributed prices,
averages the payoff values over the range of resultant outcomes, and then discounts the
expected payoff at the risk-free rate to get an estimate of the value of the option.
The advantage of Monte Carlo simulation is that it can be used when the payoff depends
on the path followed by the underlying common stock value as well as when it depends
only on the final value of common stock. Payoffs can occur at several times during the
life of the derivative rather than all at the end.
CONCLUSION
Valuations are needed for stock options issued by both public and nonpublic
companies. As the preceding indicates, the issuance and valuation of stock options is
very complex. Option values can fluctuate depending upon the volatility of the underlying
stock and the level of interest rates, among other factors. In addition, the type of
valuation model utilized will depend upon the complexity of the underlying option and
the level of model flexibility required. An added level of valuation is required when the
option is issued by a nonpublic company, as the value of the underlying stock must first
be determined before the option can be valued (especially for Section 409A compliance
purposes).
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Significant thought and input from legal, tax and qualified independent valuation advisors is required as to
establishment of a stock option plan, the tax ramifications of the option to the individual and to the company
and selection of the appropriate option-valuation model.
REFERENCES AND SUGGESTED READINGS
Financial Accounting Standards Board (“FASB”) Accounting Standards Codification ("ASC") Topic 718,
Compensation – Stock Compensation
Pratt, Shannon P. with Alina V. Niculita, Introduction to Valuing Stock Options, in Valuing a Business, 5th ed.,
ch. 25 (The McGraw Hill Companies, Inc., 2008)
American Institute of Certified Public Accountants Audit and Accounting Practice Aid, Valuation of PrivatelyHeld-Company Equity Securities Issued as Compensation
Internal Revenue Code Sections 83, 409A, 421, and 422
Beaton, Neil J., BVR’s Practice Guide to Valuations for IRC 409A Compliance (Business Valuation
Resources, 2009)
This newsletter is provided for general informational purposes only and is based upon the information available as of the time it was written. In
addition, it is intended for US-based companies and may not be appropriate for companies with a significant share of revenues originating outside
the United States.
About American Appraisal:
American Appraisal is a leading valuation and related advisory services firm that provides expertise in
areas such as fairness and solvency opinions and all classifications of tangible and intangible assets.
It comprises 900 employees operating from major financial cities throughout Asia-Pacific, Europe,
North America and South America. American Appraisal's portfolio of services focuses on four key
competencies: valuation, transaction consulting, real estate advisory, and fixed asset management.
Nancy M. Czaplinski, CPA/ABV/CGMA, CFA, ASA
Senior Managing Director and Senior Vice President
tel: 414 225 1035
email: [email protected]
Kevin M. Hagemeier, ASA, CPIM
Senior Manager
tel: 414 225 2019
email: [email protected]
To learn more about us, visit our website:
www.american-appraisal.com
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