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Chapter 36 Asset pricing models.

WHAT ARE THEY?

Chapter 35, "Securities Valuation," presented techniques for valuing equity securities (stocks). A key factor in valuing securities is a required rate of return. This chapter presents asset pricing models that help determine the appropriate required rate of return for equity securities. Additionally, this chapter presents valuation models for valuing stock options. Stock options are a type of derivative; a security whose value is derived from equity securities.

PORTFOLIO THEORY AND ASSET PRICING MODELS

Modern portfolio theory suggests that the rate of return required on a security must compensate investors for the risk involved. The theory began with the pioneering work of Harry Markowitz in the 1950s and basically states that given the choice between multiple assets (or portfolios) with equal expected returns, investors prefer the asset (or portfolio) with the lowest risk. Similarly, given the choice between multiple assets (or portfolios) of equal perceived risk, investors prefer the asset (or portfolio) with the highest expected return.

Markowitz Portfolio Theory

Markowitz believed investors expect returns on an asset approximating that asset's average (mean) returns. He defined risk as the variability (standard deviation) of returns. Using these concepts, investors can graph all available portfolios to examine the relationship between risk and return as depicted in Figure 36.1.

The area on and under the curve in Figure 36.1 represents all available portfolios. The curve from point A to point D is called the efficient frontier, and it includes those portfolios with the highest return for a particular level of risk.

Portfolios falling below the efficient frontier are considered inefficient. For example, portfolio E in Figure 36.1 would not be an optimal choice for investors. Portfolio B offers investors the same return as portfolio E but at a lower level of risk. Alternatively, Portfolio C offers a higher return than Portfolio E for the same level of risk. Therefore, Portfolios B and C are more efficient than Portfolio E.

[FIGURE 36.1 OMITTED]

Which efficient portfolio is preferred? That depends on the individual investor and their overall tolerance for risk. An investor who wants to minimize the standard deviation of expected returns might select portfolio A, while an investor who is willing to take risks to achieve the highest possible return might choose portfolio D.

Capital Market Line

The Markowitz model was expanded into the capital market theory by Sharpe, Lintner, and others. Capital market theory adds the concept of a risk-free asset (rf), typically represented by a government fixed income security. When the risk-free rate is added to the graph of the efficient frontier, a line can be drawn (see Figure 36.2) from rf tangent to the efficient frontier (intersecting at Portfolio M). This is called the capital market line (CML). If the investor can borrow and lend at the risk-free rate (by purchasing government bonds or by selling them short), a new efficient frontier is created along the CML. Portfolio M represents the market-weighted portfolio of all investable assets (the market portfolio).

[FIGURE 36.2 OMITTED]

An investor wholly invested in the market portfolio could expect the same risk and return as the market portfolio. Investors who put less than 100% in the market portfolio and place the remainder in the risk-free asset (a government security) would have a portfolio with less expected return and risk than the market portfolio.

Investors willing to accept more risk can borrow at the risk-free rate and invest more than 100% (their own money plus the borrowed money) of their assets in the market portfolio. Such a portfolio would have a higher expected return, but higher expected risk than the market portfolio. The expected return from a portfolio can be measured as:

Equation 36-1

E ([r.sub.p]) = [r.sub.f], [[sigma].sub.p] [[r.sub.m] - [r.sub.f]/[[sigma].sub.m]]

where: E([r.sub.p]) represents a portfolio's expected return [r.sub.m] is the expected return of the market portfolio

The expected return for a portfolio is therefore the risk-free rate plus a premium for the risk of the portfolio, based upon the market risk premium ([r.sub.m] - [r.sub.f]) and the risk of the portfolio relative to the risk of the market (as captured by the standard deviations of each). A portfolio with the same risk as the market portfolio would have an expected return equal to that of the market portfolio.

Capital Asset Pricing Model

The capital market line can be applied to individual equity securities by using Beta as a measure of risk instead of standard deviation. Beta measures the risk of an individual security relative to the market. It is determined by running a regression of a security's return against returns on a market index (such as the Standard and Poor's 500). A Beta of 1.0 means the security exhibited the same level of risk as the index over the period measured. A Beta greater (less) than 1.0 indicates higher (lower) risk than the index. Figure 36.3 depicts the security market line, which represents the relationship between expected return and risk measured by Beta.

[FIGURE 36.3 OMITTED]

The capital asset pricing model (CAPM) states that the expected return on an individual asset (E([r.sub.i)]) should be the risk-free rate plus a premium for risk, determined as:

Equation 36-2

E ([r.sub.i] ) = [r.sub.f] + [[beta].sub.i] ([r.sub.m] - [r.sub.f])

The risk premium for an individual security is based on the security's Beta and the overall equity market risk premium ([r.sub.m]- [r.sub.f]). Expected market and risk free returns are a matter of judgment. They are often based on historical market returns combined with current market conditions.

Example 1: Cornerstone Electronics, Inc. (CEI) has a Beta of 1.3. Assume that the expected return on the market is 12% and the expected risk-free rate is 4%. Under the CAPM, investors should require a return of:

4% + 1.3 (12% - 4%) = 14.4%.

As one would expect, CEI's higher risk results in a required return higher than that of the market. This required rate of return could then be used to assess the value of the equity using models such as the dividend discount model presented in Chapter 35. Assuming that CEI has an expected growth rate of 6% per year and a current dividend of $2.00, its value under a constant dividend growth model would be:

[V.sub.0] = [D.sub.0] (1+g)/ r - g = 2.00 (1.06)/(0.144 - 0.06) = $25.24

The capital asset pricing model assumes that investors own diversified portfolios. As such, it only accounts for the impact of systematic risk; that is the risk that is common to all market securities and cannot be mitigated by holding diverse securities. Beta captures the relative sensitivity of a particular security to this overall market risk. The CAPM does not consider non-systematic risk, also known as unique risk or diversifiable risk. In a well-diversified portfolio, non-systematic risks of individual securities should offset each other. For example, one stock in the portfolio could be affected by a loss of market share to a major competitor whose stock gains as a result of the change in market share. Since the diversified portfolio owns both, the two offset each other. If a portfolio is not well diversified, the required rate of return computed under the CAPM may not capture all of the risks in that portfolio.

The capital asset pricing model can help estimate the appropriate rate of return for valuing a security. But it should be viewed primarily as a starting point. Why? The CAPM relies on a number of assumptions, and research into the CAPM's ability to explain observed returns has been mixed. An investor should consider whether the required return computed under the CAPM makes sense and satisfies the investor's requirement for return commensurate with the risk of the investment.

The CAPM, in particular the SML, can also be useful in selecting securities for inclusion in a portfolio as is demonstrated in Chapter 38.

Arbitrage Pricing Theory

The capital asset pricing model captures risk in a single factor, the variability of returns relative to a market index (market risk). Arbitrage pricing theory (APT), developed by Stephen A. Ross, introduces multiple risk factors into the assessment of expected returns. Each of the factors can capture different aspects of risk. Unlike the CAPM, the individual factors that influence risk and expected return are not pre-specified. A generic form of an APT model can be specified as:

Equation 31-3

E ([r.sub.i]) = a + [b.sub.i1], [F.sub.1] + [b.sub.i2] [F.sub.2] + [b.sub.i3] [F.sub.3] + ... + [b.sub.in] [F.sub.n] + [epsilon]

Where a is some constant term (such as the risk-free rate), and [b.sub.in] represents the sensitivity of a security to a particular risk factor, [F.sub.n.sup.1]

Investment advisors and analysts can develop their own risk factors for use in an APT model or use those developed in research and practice by others. Burmeister, Roll, and Ross created a model with the following five individual factors, F: (2)

* [F.sub.1] Confidence Risk--Unexpected changes in investor's confidence about risk taking, measured as the differential in corporate versus government bond yields.

* [F.sub.2] Time Horizon Risk--Unexpected changes in an investors' preference for short versus long-term investments.

* [F.sub.3] Inflation Risk--Unexpected changes in inflation.

* [F.sub.4] Business Cycle Risk--Unexpected changes in real business activity.

* [F.sub.5] Market-Timing Risk--The market return not explained by the other four factors--commonly thought of as market risk, as in the CAPM.

In this model, the constant is represented by the short-term (30 day) Treasury bill rate. Each of the individual risk factors must be measured periodically, as must an individual security's (i) sensitivity to each factor.

Example 2: Dot Murphy, investment advisor, is assessing the expected return for Alpha Enterprises (AE), a large diversified conglomerate. Dot has computed the following coefficients and risk factor components for AE:
Risk Factor                 AE's         Risk (F)
                        Coefficient

Confidence Risk             0.60          2.50%
Time Horizon Risk           0.50         -0.70%
Inflation Risk             -0.40         -4.00%
Business Cycle Risk         1.50          1.50%
Market-Timing Risk          1.10          3.50%


The current Treasury bill rate is 3.0%. Dot computes the expected return on AE as:

3.0% + 0.60 x 2.50% + 0.50 x (-0.70%) - 0.40 x (-4.0%) + 1.5 x 1.5% + 1.10 x 3.50% = 11.85%

Option Pricing Models

Options are contracts involving the future trade of an underlying asset. In an options contract, only one party is obligated to complete the transaction but must do so only if the other party exercises the option. An option gives the holder (buyer of the option) the right, but not the obligation, to buy the underlying asset (a call option) or sell it (a put option) at a specified price for a given period of time.

The writer (seller of the option) collects a fee for selling the option and must fulfill their side of the contract if the option is exercised. Chapter 35 showed valuation methods for shares of stock. This discussion will demonstrate valuation models for stock options when the current value of the underlying stock is known.

Table 36-1 summarizes the rights and obligations to the parties in an options contract.

Options can be negotiated between two parties, though standardized options are traded on organized exchanges. Option values are determined by several important characteristics: the strike price, the expiration date, and the underlying security.

* Strike Price--The strike price, also known as the exercise price, is the price at which the holder of the option can buy (or sell) the underlying asset. Usually, there are standardized options contracts with several strike prices available above and below the current market price. If the market price moves above or below this range, new options series are added.

* Expiration Date--Standardized equity options cease trading on the third Friday of the indicated expiration month and expire the next day (Saturday.) Further, options are generally available for the two near term months plus two additional months, depending upon the security's designated quarterly cycle.

Most options traded in the U.S. are American style options, which can be exercised any time on or before the expiration date. By contrast, European style options can only be exercised on the expiration date.

* Underlying Security--This is the underlying asset on which the value of the option depends. For standardized equity options, one contract generally represents 100 shares of common stock or American Depository Receipts.

OPTIONS PRICING FUNDAMENTALS

Call Options Pricing

The option seller (writer) charges a fee (premium) to the option purchaser. How is the option premium deter mined? It is determined by the market, based upon the prospects for the underlying security and the terms o the option. Option value can be broken into two parts intrinsic value and time value. The option's intrinsic value is the amount by which the current market price exceeds (in the case of a call option) or falls short of (in the case of a put option) the strike price. Table 36-2 presents some hypothetical American style options prices in early July for XYZ Computers, assuming the current market price is $113.

At a current price of $113, a call option on XYZ with a $100 strike price has an intrinsic value of $13 per share (current market price of $113 less strike price of $100).3 This is the intrinsic value regardless of the time to expiration. An option that has intrinsic value is known as being "in the money," and the intrinsic value will vary based on the strike price. Table 36-3 presents intrinsic values by strike price for the July XYZ call options.

The difference between the current option price (premium) and the intrinsic value represents the time value. For the July 100 option, the intrinsic value is $13.00 and the time value is $1.30 ($14.30 - $13.00) (see

Table 36-2).

The time value of the option reflects the fact that the underlying price of the security can change before the expiration of the contract. Note that the longer the time to expiration, the higher is the time value. All of the options with a strike price of $100 have an intrinsic value of $13.00. The time values of the options with a strike price of $100 are presented in Table 36-4.

The longer the time until expiration, the greater is the likelihood that the option will confer additional value to the holder prior to expiration. Over time, the option's time value will be reduced to zero (known as decaying), and the option value should equal intrinsic value at expiration.

For the call options with a strike price of $125, the intrinsic value is zero. This makes sense because the option holder has the right to buy the underlying security for $125 when it is only worth $113. Therefore the premium for a call option with a strike price above the current stock price is comprised entirely of time value.

Such call options are referred to as "out of the money." Call options with a strike price below the current market price of the underlying security are known as "in the money." Call options with a strike price equal to the current market price are known as "at the money."

While determining the intrinsic value is relatively straightforward, it is difficult to determine what the time value, and hence, total value of the option should be. While an investor can observe the current market prices of the options contracts, the value could be higher or lower. Fortunately, options-pricing models, such as the binomial model and the Black-Scholes model, exist to help us understand what the intrinsic value of an option should be.

The Binomial Model

For simplicity, let's assume that an investor wants to value a European style option; one which can only be exercised at expiration. In order to assess the value of a European stock option today, expectations about future prices for the underlying security must be developed. In a binomial model, it is assumed that in the next period (which could be a day, week, month, or year) there are only two outcomes; the underlying stock price goes up by X% or goes down by Y%. The X and Y percentages are the investor's expectations for a particular security. Assume a call option on an underlying security where:

* Current market price of the underlying security is $30 (denoted as [S.sub.0]).

* Strike price is $30 (denoted as K).

* Time to maturity is 1 year (Denoted as T).

* In one year, the stock price is expected to either increase by 10.25% or decrease by 4.938%.

* The risk free rate of interest is 5.063%.

In one year, it is expected that the stock price (S1) will be either $33.08 or $28.52, depicted in Figure 36.4.

[FIGURE 36.4 OMITTED]

The value of the option at expiration will be the intrinsic value; the excess value of the underlying security over the strike price (there is no remaining time value, and the option cannot be worth less than zero). Therefore, if the price of the underlying security increases to $33.08, the option will be worth $3.08 at expiration. Conversely, if the price of the underlying security falls to $28.52, the option will expire worthless. The expected future value of the call option, [C.sub.1], is depicted in Figure 36.5.

In the binomial options pricing model, the current value of an option is the present value of the weighted average of the future call values:

[FIGURE 36.5 OMITTED]

Equation 36.4

[C.sub.0] = [pi] [C.sub.1.sup.+] + (1 - [pi]) ([C.sub.1.sup.-])/(1 + r)

The risk free rate, r, is the rate for the single period over which the option values are being discounted, and n is the weighting factor. The weighting factor is based upon the potential increase and decrease in the price of the underlying security:

Equation 36.5

[pi] = (1 + r) - d/u - d

The potential downside is represented by d, which is one minus the downside percentage (Y%) in decimal form. For the example, the potential downside is 4.938% or 0.04938, so d is 0.95062. The potential upside is represented by u, which is one plus the upside percentage (X%) in decimal form. For the example, the potential upside is 10.25% or 0.1025, so u is 1.1025. Computing [pi] results in:

[pi] = (1 + r) - d/u - d = (1.05063 - 0.95062)/(1.1025 - 0.95062) = 0.65848

The intrinsic value of the call option today should therefore be:

[C.sub.0] = [pi] ([C.sub.1.sup.+] ) + (1 - [pi]) ([C.sub.1.sup.-])/(1 + r) =

0.65848 (3.08) + 0.34152 (0.00)/ 1.05063 = $1.93

Figure 36.6 presents the completed call option diagram, known as a binomial tree.

[FIGURE 36.6 OMITTED]

This one-period binomial model is obviously a simplistic version of reality, assuming only two outcomes. The model can be improved and result in better estimates of intrinsic value by breaking the single period into sub-periods. For example, if the one-year period is divided into two six-month periods, a price tree for the underlying stock can be obtained as depicted in Figure 36.7, assuming prices increase by 5% or decrease by 2.5% during each six-month period. Note that this is equivalent to an annual increase of 10.25% and decrease of 4.938%.

[FIGURE 36.7 OMITTED]

There are now three possible ending values. As the number of sub-periods increases, the number of potential ending values increases as well. The process for determining the value of the call option at time 0 (today) is the same as before. First, the intrinsic value at maturity is determined at the far right of the binomial tree. Then each sub-period option value is determined working right to left until the current option value is determined. The resulting binomial option tree is presented in Figure 36.8.

[FIGURE 36.8 OMITTED]

The end result is an option value today of $1.60 versus $1.93 obtained in the single period model. This new value is a better one to use. The more periods that are used in the model, the better the estimate of value is.

Although this demonstration was for a European style option, it can easily be extended for an American style option. The only difference is that an American style option can be exercised during any of the sub-periods. Therefore, the computed option value must be compared to the intrinsic value at each node. If the intrinsic value exceeds the computed value, the intrinsic value is used instead. For example, the computed value of $2.23 in the first sub-period is compared to the intrinsic value of $1.50. In this case, the computed value is greater, and there is no need for an adjustment.

Black-Scholes Model

The binary option pricing model assumes two outcomes and discrete time periods. A mathematical extension of this process to a continuous time period with a wider distribution of outcomes is obtained with the Black-Scholes option pricing model. While mathematically complex, the latter model is useful to examine what factors influence option value. Without going into the details of how it was derived, Fischer Black, Myron Scholes, and Robert Merton developed a formula for pricing call options (Scholes and Merton subsequently won the Nobel Prize in Economics for this work). The theoretical options price is:

Equation 36-6

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here, S is the current stock price, N([d.sub.1]) and N([d.sub.2]) represent cumulative probability distributions, K is the option strike price, r is the risk free rate of return, t is the fraction of the year remaining until expiration and a is the standard deviation of the annual rate of return on the underlying security.

Note that earlier the intrinsic value of a call option was defined as the stock price (S in the formula above) minus the strike price (K in the formula above) but not less than zero. The two components in the Black Scholes formula above are similar, but reflect the theoretical total value of the call option (intrinsic value plus time value). The probability distributions are statistical concepts that consider the range of potential outcomes.

From that model, several things can be derived about the value of the option as it relates to different factors:

* As the value of the underlying stock increases, so does the value of the call option.

* As the time to expiration increases, so does the value of the option.

* As the volatility of the underlying stock increases, so does the value of the option.

* As the strike price increases, the value of the call option decreases.

* As the risk free rate of return increases, the value of the call option increases.

The Black-Scholes formula is based on a European style option and is best used in valuing those types of options.

Put Option Pricing

As noted earlier, the holder of a put option has the right to sell the underlying security at a certain price. Selected put options for a hypothetical XYZ Computers as of early July are presented in Table 36-5.

Intrinsic value for a put option only exists where the strike price exceeds the current market price, and it is measured as the strike price less the current market price (K-S, but not below zero) (see Table 36-6). For the options with a strike price of $120, there would be an intrinsic value of $7 ($120 - $113). With a current price of $113, the options with strike prices of $100, $105, and $110 would have no intrinsic value. The holder of the option would not want to exercise his right to sell something at a price lower than the current market price. The intrinsic value for each strike price is shown in Table 36-6.

Put options with a strike price below the current market price of the underlying security are referred to as "out of the money." They have no intrinsic value. Put options with a strike price above the current market price of the underlying security are known as "in the money." Put options with a strike price equal to the current market price are known as "at the money."

The time value equals the difference between the current option price (premium) and the intrinsic value. For example, an $8.00 option price minus a $7.00 intrinsic value equals a time value of $1.00 for options above with a strike price of $120 and an expiration month of July. As with a call option, a longer time to maturity equates to greater time value. The observed time values for a strike price of $120 are summarized in Table 36-7.

There are several methods available to determine the value of a put option, including variations of the binomial and Black-Scholes models. They can also be valued with reference to the value of call options.

The binomial pricing model for put options is applied in the same manner as for call options (see above). The intrinsic value is determined at maturity based on the excess of the strike price over the value of the underlying security. The put value at each node is computed based on the weighted average of the values for the put option in the upside and downside cases.

The formula for the value of a European put option under the Black-Scholes option pricing model (see above) is:

Equation 36-7

P = K ([e.sup.-rt]) [1 - N([d.sub.2])] - S [1 - N([d.sub.1])]

Intuitively, this is the excess of the strike price over the current stock price, adjusting for the time value of money and a normal distribution of outcomes.

Put options can also be theoretically valued using a concept known as Put-Call Parity. Put-Call Parity is derived from the assumption that puts and calls should be priced relative to the underlying security such that no arbitrage opportunity exists. Theoretically, purchasing a stock and a put option should be an equivalent position to holding cash equal to the exercise price and purchasing a call option (where the put and call options have the same exercise price):

S + P = K/[e.sup.rt] + C

Here, S is the security's current market price, P is the price of a put option, K is the exercise price (which is being discounted using the risk fee rate) and C is the call price.

Using the example above, if an investor purchases XYZ stock for $113 and a July 110 put option, the investor will profit if the price of IBM rises but is protected with the put option on the downside (if the price falls below $110, the investor will exercise the option and sell for $110). No matter what happens, the investor is ensured of having a value of $110 at expiration. Note that the total investment is $115.25 ($113 plus $2.25).

The investor would be in a similar position if the investor took the present value of the exercise price (PV of $110 over about two weeks at 5% would be about $109.75), invested it at the risk free rate of interest until option expiration and simultaneously purchased a July 110 call option for $5.50. If the underlying security rises in value, the call price will rise accordingly. If the price falls and the call option expires worthless, the investor will be left with $110. Note that the total investment is $115.25 ($109.75 plus $5.50). Therefore Put-Call parity holds for this example.

This is not always the case for real data due to bid-ask spreads and the timing of the quotations. Rearranging the above equation, the theoretical price for a put option is:

P = K/[e.sup.rt] + C - S

Holding a put option is equivalent to investing the present value of the strike price, purchasing a call option and selling the stock short.

WHERE CAN I FIND OUT MORE ABOUT IT?

1. John Stowe, Thomas Robinson, Jerald Pinto, and Dennis McLeavey, Analysis of Equity Investments: Valuation (Association for Investment Management and Research, 2002).

2. A Practitioner's Guide to Factor Models, The Research Foundation of The Institute of Chartered Financial Analysts, Association for Investment Management and Research.

3. Harry Markowitz, "Portfolio Selection," Journal of Finance, 7, No. 1, March 1952.

3. William F. Sharpe, "Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk," Journal of Finance, 19, No. 3, September 1964, pp. 425-442.

4. John Lintner, "Security Prices, Risk and Maximal Gains from Diversification," Journal of Finance, 20, No. 4, December 1965, pp. 587-615.

5. Stephen Ross, "The Arbitrage Theory of Capital Asset Pricing," Journal of Economic Theory, 13, No. 2, December 1976, pp. 341-360.

6. Don M. Chance, Analysis of Derivatives for the CFA Program (Charlottesville, VA: Association for Investment Management and Research, 2003).

7. Lawrence G. McMillan, Options as a Strategic Investment, 4th Edition (New York, NY: New York Institute of Finance, 2002).

QUESTIONS AND ANSWERS

Question--Under what conditions would an investor choose a portfolio that lies below the efficient frontier?

Answer--None. For any portfolio below the efficient frontier, alternative portfolios are available offering higher returns, lower risk or both.

Question--Given a stock with the following characteristics, use the CAPM and a constant growth dividend discount model to estimate the value of the stock:
Annual dividend per share        $1.00
Expected growth rate                4%
Risk-free rate                      5%
Expected market return             11%
Beta                               1.5


Answer

[E.sub.r] = [r.sub.f] + B ([r.sub.m] - [r.sub.f]) = 0.05 + 1.5 (0.11 - 0.05) = 0.14 = 14%

[V.sub.0] = [D.sub.0] (1 + g)/r-g = 1.00 (1.04)/ (0.14 - 0.04) = 1.04/0.10 = $10.40

Question--On July 14, 2006, DELL was trading at $21.82 per share and the option prices below were observed. Why is the July $20.00 put priced so cheaply?
Expiration       Strike        Call         Put

July 06          20.00         1.85         0.05
July 06          22.50         0.15         0.85
July 06          25.00         0.05         3.20
August 06        20.00         2.40         0.40
August 06        22.50         0.80         1.45
August 06        25.00         0.20         3.40


Answer--Because on July 14, 2005 there was only one week to expiration and the option was out of the money. The stock would have to fall more than 8.6% to below 19.95 in that week for the option to have value at expiration.

Question--Which of the options for Dell, above, has the greatest time value?

Answer--The total value of an option is comprised of intrinsic value and time value. Time value is the total value less the intrinsic value. For a call option the intrinsic value is the greater of zero and the stock price minus the exercise price. So the intrinsic values for the call options are zero ($22.50 and $25.00 strike) and 1.82 ($20.00 strike). The highest time value for a call is $0.80 for the August $22.50's. For the puts, the intrinsic value is the greater of zero and the exercise price less the stock price. The 20's have no intrinsic value, the $22.50's have $0.68 intrinsic value and the $25.00's have $3.18. The greatest time value is again for the August $22.50's, for which it is $0.77. This also exceeds the highest call option time value and is thus the highest overall.

Question--On the same date, Microsoft traded at $22.36 and the prices below were observed for Microsoft options. Although the prices of both stocks (Dell and Microsoft) were closest to the $22.50 stock price, it was cheaper to buy options on Microsoft than on DELL for that exercise price. This phenomenon was true of both the call and put options. Why?
Expiration       Strike        Call         Put

August 06        20.00         2.50         0.10
August 06        22.50         0.65         0.75
August 06        25.00         0.10         2.65


Answer--From the Black-Scholes model, it can be seen that:

* As the value of the underlying stock increases, so does the value of the call option. Microsoft had a higher stock price, so this cannot be the explanation.

* As the time to expiration increases, so does the value of the option. They both have the same time to expiration, so this cannot be the explanation.

* As the strike price increases, the value of the call option decreases. They both have

the same strike price, so this is not the explanation. 1.

* As the risk free rate of return increases, the value of the call option increases. The risk free rate is the same for both securities, so 2. this is not the explanation.

* As the volatility of the underlying stock increases, so does the value of the option. DELL must be a more volatile stock than 3. Microsoft.

CHAPTER ENDNOTES

(1) Note, small b is used for the coefficients here to distinguish them from Beta in the CAPM. Essentially, these small b's represent individual Beta coefficients for each risk Factor.

(2) Edwin Burmeister, Richard Roll and Stephen A. Ross, "A Practitioners' Guide to Arbitrage Pricing Theory," in A Practitioner's Guide to Factor Models, The Research Foundation of The Institute of Chartered Financial Analysts (Association for Investment Management and Research).

(3) Since each options contract denotes 100 shares, the contract has an intrinsic value of $1,300.
Table 36-1

                Option Writer (Seller)       Option Holder (Buyer)

Call Option    Must sell the underlying    May purchase underlying
               security at the agreed      security at agreed upon
               upon price if the option    price if doing so is
               is exercised                advantageous

Put Option     Must buy the underlying     May sell underlying
               security at the agreed      security at agreed upon
               upon price if the option    price if doing so
               is exercised

Table 36-2
XYZ Call Option Premium

Expiration                       Strike Price
  Month
              100      105      110      115      120      125

July         14.30    11.20     5.50     2.55     1.00      .30
August       14.90    11.50     7.50     4.70     2.50     1.25
October      19.00    13.00    11.70     8.70     5.80     3.80
January      21.60    19.00    13.90    11.60     9.50     7.00

Table 36-3

    Strike Price         Intrinsic Value

100                           13.00
105                            8.00
110                            3.00
115                            None
120                            None
125                            None

Note that strike prices above $113 have no intrinsic
value.

Table 36-4

  Expiration Month          Time Value

July                          $1.30
August                        $1.90
October                       $6.00
January                       $8.60

Table 36-5
XYZ Put Option Premium

 Expiration                        Strike Price
   Month
                100      105      110      115      120      125

July            0.55     1.10     2.25     4.50     8.00     9.50
August          1.60     2.55     4.10     6.20     9.00    12.00
October         3.60     4.70     5.90     8.70    11.60    13.10
January         5.50     7.30     9.20    11.00    13.60    15.50

Table 36-6

    Strike Price         Intrinsic Value

100                            None
105                            None
110                            None
115                           $2.00
120                           $7.00
125                           $12.00

Table 36-7

  Expiration Month          Time Value

July                          $1.00
August                        $2.00
October                       $4.60
January                       $6.60
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Title Annotation:Techniques of Investment Planning
Publication:Tools & Techniques of Investment Planning, 2nd ed.
Date:Jan 1, 2006
Words:5913
Previous Article:Chapter 35 Security valuation.
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