Financial valuation Applications and Models phần 10 pot

I: Valuation Issues Related to Stock Options 919 more in tune with the real world. These models are not only applicable to the valu- ation of stock options but are used to value capital investments, intangible assets, and entire divisions or companies. Before these valuation techniques are introduced, it is important to understand the characteristics of stock options and the terminology used in describing them. The following paragraphs review those key issues and create the foundation for the work that follows. Definitions 1. American option. An option that can be exercised at any time during its life. 2. Binomial option pricing model. An option pricing model based on the assump- tion that stock prices can move only to two values over a short period of time. 3. Black-Scholes model. A model used to calculate the value of a European call option. Developed in 1973 by Fisher Black and Myron Scholes, it uses the stock price, strike price, expiration date, risk-free return, and the standard deviation (volatility) of the stock’s return to estimate the value of the option. 4. Call option. A provision that gives the holder the right, but not the obligation, to buy a stock, bond, commodity or other instrument at a specified price within a specific time period. 5. Carrying value. Also know as “book value,” it is a company’s total assets minus liabilities, such as debt. 6. Employee stock option (ESO). Stock options granted to specified employees of a company. These options carry the right but not the obligation to buy a cer- tain amount of shares in the company at a predetermined price. 7. European option. An option that can be exercised only at the end of its life. 8. Incentive stock option (ISO). A type of employee stock option with various tax benefits granted under Section 422 of the Internal Revenue Code of 1986. These options may be granted only to individuals who are employees of the granting company or a parent or subsidiary of the granting company. A num- ber of restrictions under Section 422 may disqualify an ISO, in which case it becomes a nonqualified stock option. 9. Long-term equity anticipation securities (LEAPS). An options contract that expires more than nine months in advance and can last as long as two years. 10. Nonqualified stock option (NSO). A type of employee stock option which is less advantageous for the employer from a tax standpoint than an ISO, but which is less restrictive and generally easier to set up and administer. Any stock Employee stock options (ESOs) are particularly attractive given the ability to defer the recognition of these grants as compensation and the fact that cash is not normally involved. Vesting rights often are embed- ded in these ESOs to promote employee retention by rewarding longevity. ValTip 920 OTHER VALUATION SERVICES AREAS option granted to an employee that is not an incentive stock option is, by default, an NSO. 11. Put option. A provision giving the holder the right, but not the obligation, to sell a stock, bond, commodity or other instrument at a specified price within a specific time period. 12. Strike price. The stated price per share for which underlying stock may be pur- chased (for a call) or sold (for a put) by the option holder upon exercise of the option contract. Option Basics Stock options generally grant the holder the right, but not the obligation, to acquire stock in a corporation. The lack of an obligatory purchase requirement dis- tinguishes stock options from forward or future contracts where final purchase is mandatory. When granted, stock options usually carry an exercise price and a stated option term. Stock option plans usually are set up to promote the long-term success of the company granting the options by attracting and retaining employees, outside directors, and consultants. Options encourage these individuals to focus on the company’s long-range goals by granting them an ownership interest in the company. Most stock options can be characterized as call options where the holder pos- sesses the right to buy the underlying stock at a specified price and date. In contrast, a put option allows the holder to sell the underlying stock at a specified price and date. Many employee stock ownership plans (ESOPs) contain put provisions (not options) for stock owned by employees of the company. The strike price, also known as the exercise price, is a fixed price at which the holder may purchase the underlying stock. The exercise price is set upon the grant- ing of the specific option and, under most circumstances, cannot be changed with- out triggering somewhat onerous reporting requirements. Employee stock options are classified as either incentive stock options or nonqualified stock options. ValTip Options that can be exercised during and up to the expiration date are known as American options. Options that can be exercised only upon the expiration date are known as European options. ValTip I: Valuation Issues Related to Stock Options 921 Most stock options lapse after a certain time period. Incentive stock options cannot have an expiration date more than 10 years after the granting date. However, publicly traded options generally have expiration dates that are measured in months rather than years. Contract The purchaser of an option typically is referred to as the option holder. Sellers of options typically are referred to as option writers, since they “write” the option con- tract. In exchange for the contract, the option holder pays a premium to the option writer. There are seven items specified in the option contract: 1. Underlying instrument. The instrument that may be bought or sold. 2. Contract size. The number of shares of underlying stock that the contract involves. 3. Exercise price (or strike price). The price at which the underlying stock will transact if the option is exercised. 4. Settlement date. The date on which money is received for the contract. 5. Expiration date (expiry). The date that the option expires. 6. Style. The ability to exercise prior to expiry (i.e., American or European). 7. Premium. Price paid for the contract. While options on stock indices, foreign exchange, agricultural commodities, precious metals, futures, and interest rates exist, the discussion in this chapter is lim- ited to stock options granted to employees of the issuing corporation. Option parameters specify how the option can be exercised. The most common styles are American and European options. American options can be exercised at any time on or before the expiration date. European options can be exercised only on the expiration date. There is a third type of option that is exercisable only on predetermined dates, such as every month, or every quarter. They are referred to as Bermuda options. Most publicly traded options have expiration dates of less than a year, but the Chicago Board Options Exchange (CBOE) now lists longer-term options on several blue-chip stocks. Known as long-term equity anticipation securities, or LEAPS for short, these options have longer-term expiration dates. In contrast, ISOs normally have a 10-year life. Descriptive Terminology Options have a particular vocabulary, especially in describing their status. The fol- lowing are some of most common terms and definitions related to options. If not exercised before the expiration date, the option simply expires with no additional value. ValTip 922 OTHER VALUATION SERVICES AREAS At the money. The term used when the exercise price and the underlying price are equal. In the money. The term used when an option’s strike price is less than the cur- rent price of the underlying stock. Out of the money. The term used when an option’s strike price is greater than the underlying stock price. Warrants A warrant is a particular type of call option issued by the company itself. When the warrant is settled, the company issues additional shares, increasing the number of shares outstanding; in contrast, a call option is settled with the delivery of previ- ously issued shares. In addition, the cash flows of the company increase with the exercise of a warrant since the exercise price is paid to the company. Therefore, the dilution created by the issuance of stock is partially offset by the cash received. Options Trading Standardized options contracts were first traded on a national exchange in 1973, when the Chicago Board Options Exchange began listing call options. Option con- tracts now are traded on a number of exchanges, including the CBOE, Philadelphia Stock Exchange, American Stock Exchange, New York Stock Exchange, and Pacific Exchange. Some options trade in the over-the-counter (OTC) markets as well. Over- the-counter stocks do not have standardized terms but are customized for each transaction. Due to the customization, the market is limited, thus increasing the cost of establishing an OTC option contract. Most public option trading occurs on organized exchanges. Components of Value Two basic components make up the price paid for an option: intrinsic value and time value. The most obvious component of an option’s value is its intrinsic value. This intrinsic value is the amount of money available from the immediate exercise of the option, or the amount the option is in the money. For a call option, this amount reflects the value of the stock less the exercise price. Volatility is the expected standard deviation of the underlying stock. As volatil- ity increases, so does the probability that the stock will increase (calls) or decrease (puts) by a large enough magnitude to allow the option to be “in the money” before it expires. To illustrate this concept, consider the holder of a call option. The holder of a call option is not exposed to the downside risk of the stock. The holder’s loss if the As a result of dilution, the value of a warrant may vary somewhat from the value of a call option with identical terms. ValTip stock price declines is limited to the price paid for the call option, no matter what the likelihood that the stock price will decline. However, the more a stock’s price can increase over a given period, the higher the option holder’s potential profit. This makes the option holder prefer high volatility since it increases the chance that the stock’s price will increase above the exercise price. The final determinant of an options price is the risk-free rate. The risk-free rate represents the interest rate that could be earned by investing the exercise price over the time period from option purchase to exercise. Assuming the option holder had perfect knowledge that the stock price would increase, the holder would, in effect, be getting a risk-free loan for the length of the option. For a put option, it is just the opposite, since the option holder gives up the potential to invest. The length of time before an option expires is a fairly straightforward concept. The longer the period until expiration, the greater the chance the option will end up above or below the exercise price of the underlying stock. Exhibit 23.11 summarizes the effects that a change in one variable has on the value of an underlying call or put option, all else being equal. Exhibit 23.11 Effect of an Increase in Variable on Option Value Variable Call Option Put Option ________ __________ __________ Market value of stock + – Exercise price – + Volatility + + Risk-free rate + – Expiration date* + + *For European options on dividend paying stocks, value may not increase with time due to the dividend effect. The Divided Effect Dividends represent a cash return to the investors. A company has the choice of either paying dividends or reinvesting that money in the business. The reinvestment of that cash could allow the business to earn more in the future, thus increasing its I: Valuation Issues Related to Stock Options 923 Even without being in the money, an option may have value. This value is created by the possibility that the option could be exercised prof- itably in the future. Three factors determine time value: 1. Volatility of the stock underlying the option 2. Risk-free rate of interest over the option period 3. Length of time before the option expires ValTip stock price. Paying out the dividend effectively reduces the stock price by the divi- dend amount on the ex-dividend date (the date that the shareholders of record are determined for dividend payment). By reducing the stock price on the ex-dividend date, the value of a call option decreases and the value of a put option increases. Valuation Tools With the introduction of stock options and the components that drive their value, tools have been developed to calculate their value. The following models were designed to value publicly traded options. Each has its own virtues and limitations. Understanding those limitations and adjusting for them is the key to valuing a wide variety of options. A discussion of the complex mathematical assumptions used to derive these for- mulas is beyond the scope of this chapter. Software programs of option models are available from numerous sources or can be modeled using the provided equation. The focus here is on the benefits of each model and the selection of appropriate inputs for the option valuation models. Black-Scholes Model The most widely recognized option-pricing model is known as the Black-Scholes model. Developed by Fisher Black and Myron Scholes in 1973, the Black-Scholes model was the first model used to calculate a theoretical call price (ignoring divi- dends paid during the life of the option). The model (shown in Exhibit 23.12) uses the five key determinants of an option’s price: 1. Underlying stock price 2. Exercise price 3. Volatility 4. Time to expiration 5. Short-term (risk free) interest rate While advanced mathematical techniques were used to develop the Black- Scholes model, it is not necessary to understand the formula’s derivation in order to use it. Normally, each component of the formula is readily available. The stock price is based on the closing price of the stock as of the day of valuation or, when the stock is restricted or in a private company, on the estimated price of the stock. The exercise price is given in the contract. The time remaining until expiration can be expressed as a percentage of a year for options with expirations of less than a year or in years for those options with expiries greater than a year. The risk-free rate is approximated by using rates paid for U.S. Treasury bills, matching the length of the option maturity to the U.S. Treasury bill period. Volatility is the expected volatility of the underlying stock. Volatility is measured using the annualized standard deviation of the underlying stock price movements. Generally, the expected volatility can be calculated from the histori- cal volatility in the stock or the implied volatility from publicly traded stock options. 924 OTHER VALUATION SERVICES AREAS When using historical volatility, it is generally best to review the latest 12- month period although longer or shorter periods are sometimes used. When long- term options exist or nonrecurring events have occurred, adjustments may be made to reflect expectations of future performance. When the stock is lightly traded or not publicly traded, it may be necessary to use an average of the historical volatilities of similar stocks in the marketplace as a proxy for anticipated volatility. It is important to average the volatilities and not cal- culate volatility based on the standard deviation of a portfolio of these guideline stocks, since diversification among the stocks will lower volatility and not be reflec- tive of the anticipated volatility of an individual stock. Industry volatilities also can be used. Implied volatility is calculated for publicly traded options by adjusting the Black-Scholes formula to solve for volatility. Assuming identical options, this implied volatility should represent the market’s indication of expected volatility. As with historical volatility, when the stock is lightly traded, it may be neces- sary to use an average of the implied volatilities of similar stocks in the marketplace. I: Valuation Issues Related to Stock Options 925 Exhibit 23.12 Black-Scholes Option Pricing Model The original formula for calculating the theoretical call option price is as follows: C = S ϫ N(d 1 )ϪXe –rt ϫ N(d 2 ) Where: s ␴ 2 ln __ ϩ r ϩ __ T x2 d 1 ϭ __________________________ ␴͌ʳʳ T d 2 ϭ d 1 Ϫ␴͌ʳʳ T C = Call price S = Stock price X = Exercise price T = Time remaining until expiration r = Current risk-free interest rate ␴ = Expected annual volatility of stock price ln = Natural logarithm N(x) = Standard normal cumulative distribution function e = Exponential function ΄ ΅ ΄ ΅ A key limitation of the Black-Scholes model is that it was developed to price European call options in the absence of dividends. ValTip The natural logarithm, the standard normal cumulative distribution function and the exponential function are all mathematical constants. The Black-Scholes model also gives a reasonable price for an American call. Earlier, we introduced the two components to option value, intrinsic value and time value. The early exercise of an American option would forfeit the time value com- ponent. Adjusting Black-Scholes for European Puts. The Black-Scholes formula can be adjusted to calculate the value of a European put option by applying the put-call parity theorem. The concept of put-call parity is that the payoff for a put could be replicated using a combination of call options, shorting stock, and borrowing. A general formulation of put-call parity is: P = C – S + Xe –rt Applying this to the Black-Scholes option pricing formula and simplifying the equation, the formula for valuing a put option is: P = S ϫ N(-d 1 ) + Xe –rt ϫ N(-d 2 ) While this formula adjusts the basic Black-Scholes formula for a European put, it does not address the value of an American put. Refer back to Exhibit 23.11. One of the components of time value is the risk-free rate. The risk-free rate has a nega- tive value effect on a put option. So there is the possibility that an American put option will have negative time value, thus making early exercise valuable. Adjusting Black-Scholes for Dividends. As previously discussed, the Black- Scholes model assumes that dividends are not paid. Since some stocks do pay divi- dends, the model needs adjustment to properly value the options on these stocks. To understand this adjustment, one needs to review the effect of dividends on stock price. Basic valuation theory states that a stock is worth the present value of its future cash flows. Cash flows retained in the business are reinvested, creating higher poten- tial future cash flows. When dividends are paid, the stockholder receives the cash and can determine whether to reinvest it in the company or in other ventures. The stockholder is equally well off in either case, but cash has come out of the company, reducing its value directly in line with the amount of dividend paid. There are two methods for adjusting the Black-Scholes model for dividends. For short-lived options, the option’s underlying share price could be lowered by the present value of the dividends. When valuing a longer-lived option, the Black- Scholes model can be adjusted for the expected long-term dividend yield of the stock. The formulas in Exhibit 23.13 show the Black-Scholes model adjusted for dividends. While the formulas adjust the Black-Scholes model to estimate the value of European options in the presence of dividends, American options are not specifically addressed. The ability to exercise early and avoid the lost value of the stock due to dividends (calls) or take advantage of the decline (puts) has additional value over and above the European option. Adjusting Black-Scholes to Price Warrants. The difference between a warrant and a call option is the dilution created by the warrant, which gives the warrant a lower value than an option with the same terms. The effect of dilution can be cal- culated to derive the value of the warrant. The formula is: 926 OTHER VALUATION SERVICES AREAS I: Valuation Issues Related to Stock Options 927 1 Warrant Value ϭ ________________________ ϫ Value of Option Equivalent Number of Warrants ΄ 1ϩ __________________ ΅ Number of Shares The option equivalent is the value of an option with the same terms as the warrant. The Binomial Model The Black-Scholes model allows for the rapid calculation of option value. In most instances, with the proper adjustments, it yields a fairly accurate estimate of value to standardized stock options. However, its accuracy is more limited in certain situ- ations, the most notable of which is an American put option. A more robust option valuation model was created in 1979 by John Cox, Stephen Ross, and Mark Rubinstein, when they developed a binomial model for pricing stock options. The binomial model breaks down the time to expiration into time intervals, or steps. At each step, the stock price will either move up or down. How much the stock will move up or down is related to the stock’s volatility and the option’s time to expira- tion. Charting these possible movements at each step produces a binomial tree rep- resenting all of the possible paths the stock price could take during the life of the option. This makes the binomial model more rigorous to apply than the Black- Scholes model. The option prices are calculated at each step of the tree, working from expira- tion to the present. The option prices at each step are calculated using the option Exhibit 23.13 Dividend Adjusted Black-Scholes Option Pricing Model The original formula for calculating the theoretical call option price is as follows: C = S a e –bT ϫ N(d 1 ) – Xe –bT ϫ N(d 2 ) P = –S a e –bT ϫ N(-d 1 ) + Xe –bT ϫ N(–d 2 ) Where: s ␴ 2 ln __ ϩ b ϩ __ T x2 d 1 ϭ __________________________ ␴͌ʳʳ T d 2 ϭ d 1 Ϫ␴͌ʳʳ T S = Stock price X = Exercise price T = Time remaining until expiration b = Cost of carry (the risk-free rate minus the dividend yield) ␴ = Expected annual volatility of stock price ln = Natural logarithm N(x) = Standard normal cumulative distribution function e = Exponential function ΄ ΅ ΄ ΅ prices from the previous step of the tree using the probabilities of the stock prices moving up or down, the risk-free rate and the time interval of each step. Any adjust- ments are put into the model as needed to reflect ex-dividend dates or the optimal exercise for American options. Exhibit 23.14 is a pictorial representation of the binomial tree with the stock price from time 0 through time 3. Exhibit 23.14 Binomial Tree 928 OTHER VALUATION SERVICES AREAS S 0 S uuu S ddd S uud S udd S u S uu S d S dd S ud ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ t0——— —— —t1———— —— t2———————t3 Exhibit 23.15 Binomial Model Formulas Cox-Ross-Rubinstein approach Equal Probability approach The variables are: t = Total time in years n = Number of periods ⌬t = Length of time period in years = t/n ␴ = Estimated annual volatility r f = Current risk-free interest rate u = Up ratio d = Down ratio p ϭ 0.5 2e r f ␴tϩ2s͌ʳʳ␴t u ϭ ________ e 2s͌ʳʳ␴t ϩ1 2e r f ␴t d ϭ ________ e 2s͌ʳʳ␴t ϩ1 e r f Dt Ϫ d p ϭ ________ u Ϫ d u ϭ e s͌ʳ␴ 1 d ϭ __ u [...]... discussing valuation issues related to ESOs, as with any valuation, the analyst must appropriately define the assignment: • Follow valuation procedures to define the assignment appropriately so that the right level/standard of value, valuation date, and procedures are used • Gather appropriate support • Analyze the data in a meaningful and appropriate manner before arriving at a conclusion and writing... increases the value of a call option and decreases the value of a put option 939 J: Real Option Valuations Black-Scholes Model The Black-Scholes model is currently the most recognized and widely used theoretical model for the valuation of options Black and Scholes assumed that it is possible to set up a hedged, or riskless, position consisting of owning a share of stock and selling the option on the stock... because they understand what value is, how value is determined, and they know the value drivers of a particular company However, the focus here is on what can be done to create additional value and how analysts can assist corporate managers to answer this question 942 OTHER VALUATION SERVICES AREAS Many public and private corporations have specific programs in place to measure and maximize their shareholder... of their decisions can be meas- 954 OTHER VALUATION SERVICES AREAS ured using the concept of economic profits, and their compensation can be tied to the value creation results of their decisions ValTip Valuation analysts are uniquely qualified to assist management with the design and implementation of value creation strategies Analysts can expand traditional valuation methodologies to incorporate the... of option pricing models applied to managerial decisions ValTip Real options is a managerial decision making tool that utilizes financial option pricing models such as the Black-Scholes and applies them to “real” rather than just specific financial decisions Many managerial decisions follow the structure of options In the example, Digicell has the right to acquire and further develop and manufacture the... than 10 percent of the voting power of all stock outstanding at the time of the grant unless the exercise price is at least 110 percent of the fair market value of the stock and the option is not exercisable more than five years from the time of the grant • The ISO agreement must specify in writing that the ISO cannot be transferred by the option holder other than by will or by the laws of decedent and. .. written and must list the restrictions placed on its exercise It must set forth an offer to sell the stock at the option price and the period of time that the option will remain open I: Valuation Issues Related to Stock Options 931 • The option must be granted within 10 years of the date of adoption or shareholder approval, whichever date is earlier • The option must be exercisable only within the 10- year... million capitalized at a market weighting of 20 percent debt and 80 percent equity Also assume that Company B has invested capital of $1 million and is weighted at market rates of 50 percent debt and 50 percent equity Each company has a pretax cost of debt of 8 percent and a cost of equity of 20 percent to its shareholders (see Exhibits 23.20 and 23.21) (For illustrative purposes, we have assumed equal... Exhibits 23.24 and 23.25, Companies A and B have the same operating performance Let’s assume both companies have revenue of $1 million Both companies’ revenues are expected to grow at 10 percent per year with a longterm growth rate of 6 percent The operating performance before depreciation and interest expense is expected to be exactly the same for both companies 30 James R Hitchner and Mark L Zyla,... public and the potential option payoffs for going public versus remaining private must be key among the factors considered In arriving at a conclusion of the volatility to apply, as when arriving at a conclusion of value, it is necessary to weigh all the evidence and the strengths and weaknesses of each part of the analysis to reach a reasonable conclusion Since volatility has the greatest potential . level/standard of value, valuation date, and procedures are used • Gather appropriate support • Analyze the data in a meaningful and appropriate manner before arriving at a conclusion and writing. calculate their value. The following models were designed to value publicly traded options. Each has its own virtues and limitations. Understanding those limitations and adjusting for them is the key. own more than 10 percent of the voting power of all stock outstanding at the time of the grant unless the exercise price is at least 110 percent of the fair market value of the stock and the option