Executive Stock Options and Incentive Effects due to Systematic Risk

Transcription

1 Executive Stock Options and Incentive Effects due to Systematic Risk Jin-Chuan Duan and Jason Wei Joseph L. Rotman School of Management University of Toronto 105 St. George Street Toronto, Ontario Canada, M5S 3E6 First version: March 2002 Current version: July 2003 Abstract Previous research on executive stock options mainly focuses on total risk when studying risk incentives. In this study, we use a GARCH option pricing framework to show that executive stock options have important incentive effects on the composition of risk. The value of executive stock options increases with systematic risk and this effect is stronger when the total risk is low. Thus, when firms grant standard or non-indexed options, CEOs will have incentives to increase systematic risk even when the total risk remains constant. By contrast, granting indexed options provides CEOs with incentives to reduce systematic risk since the value of indexed options decreases with systematic risk. An implication of our findings is that an optimal mix of indexed and non-indexed option grants will provide CEOs with incentives to take the desired level of systematic risk. Both authors gratefully acknowledge the financial support from the Social Sciences and Humanities Research Council of Canada. We thank Melanie Cao, Vidhan Goyal, Jian Wu, workshop participants at the University of Arizona, Case Wester Reserve University and Hong Kong University of Science and Technology, and conference participants at the 2002 annual meeting of the Northern Finance Association for useful comments and suggestions.

2 Executive Stock Options and Incentive Effects due to Systematic Risk Abstract Previous research on executive stock options mainly focuses on total risk when studying risk incentives. In this study, we use a GARCH option pricing framework to show that executive stock options have important incentive effects on the composition of risk. The value of executive stock options increases with systematic risk and this effect is stronger when the total risk is low. Thus, when firms grant standard or non-indexed options, CEOs will have incentives to increase systematic risk even when the total risk remains constant. By contrast, granting indexed options provides CEOs with incentives to reduce systematic risk since the value of indexed options decreases with systematic risk. An implication of our findings is that an optimal mix of indexed and non-indexed option grants will provide CEOs with incentives to take the desired level of systematic risk. 0

3 1. Introduction Executive stock options have become an important compensation vehicle for most companies. Since the beginning of the last decade, the value of stock option grants as a percentage of total executive compensation has been growing steadily. According to Hall and Murphy (2001), in the fiscal year of 1999, 94% of the S&P500 companies granted stock options to their top executives, while in 1992, only 82% did so. In 1999, for S&P500 companies, the value of stock options accounted for 47% of the total pay, compared with 21% in Executive stock options are call options written on the company s own stock, and are granted to the company s CEO and senior managers. 1 Typically, they are issued at-themoney with a time to maturity of ten years, and a certain vesting period. The vesting period can last anywhere between six months and five years, during which time the options cannot be exercised. A single grant for a year may be divided into bundles with different exercise prices, time to maturities, and vesting periods. For example, as shown in Table 1, for each year from 1994 to 1999, Hewlett Packard granted four equal-sized bundles of options to its chairman of the board, president and chief executive officer, Lewis E. Platt. The time to maturity was ten years and the vesting period ranged from one year to four years. In contrast, as shown in Table 2, some of the stock options granted to the CEO of Philip Morris, Geoffrey C. Bible, carried a much shorter time to maturity and vesting period. In general, after the vesting period, most executive stock options become American options in that they can be exercised anytime before maturity. 2 CEOs are prohibited from selling their holdings of executive stock options and are not allowed to short their own company s 1 Stock options are sometimes also granted to other employees. In this paper, we only examine executive stock options since our focus is on risk incentives as opposed to performance boosting or talent retention. 2 It is well known that CEOs tend to exercise a significant portion of their stock options once the vesting period is over, which seems to be a sub-optimal strategy in light of the standard option valuation theory. Researchers have proposed various reasons to explain this behavior. Please see Hall and Murphy (2001) for the latest explanations. 1

4 stocks. This inability of perfectly hedging the option can potentially make the option s value to the CEO deviate from its value to persons without such a trading constraint. The fundamental objective of granting stock options is to align the CEO s interest with shareholders, i.e. to alleviate the agency problem in the sense of Jensen and Meckling (1976). Insofar as shareholders wealth is reflected in the stock price, choosing the stock price as the ultimate performance measurement and granting call options to CEOs are a logical device. However, typical stock options are also designed to achieve other related objectives. For instance, a vesting period and a longer option maturity can both help retain capable CEOs. Empirical evidence seems to support the general rationale of stock options. For example, Hall and Liebman (1998) found a strong link between a firm s performance and its CEO s compensation. Broadly speaking, the existing research on executive stock options focuses on four major issues: 1) valuation and incentive effects from the CEO s perspective in light of the CEO s risk-aversion and hedging restrictions (e.g., Lambert, Larcker and Verrecchia (1991), Hall and Murphy (2001), Meulbroek (2001a&b), Tian (2001&2002), and Jin (2002)); 2) repricing of stock options when the stock price has declined substantially from the granting date (e.g., Brenner, Sundaram and Yermack (2000), Chance, Kumar and Todd (2000), and Emm and Ince (2000)); 3) alternative design of stock options for the purpose of better aligning performance and pay (e.g., Johnson and Tian (2000), Meulbroek (2001b), and Tian (2001)); and 4) empirical efficacy of stock options (e.g., Hall and Liebman (1998), and Williams and Rao (2000)). When considering CEOs risk-aversion, researchers typically assume a certain form of utility function and then find the so-called certainty equivalent value of the stock option. Otherwise, a Black-Scholes setup is generally assumed to study valuation, incentive effects and repricing. For instance, DeFusco, Johnson and Zorn (1990), 2

5 and Williams and Rao (2000) found that stock options induce CEOs to take on risky projects that they otherwise would not have taken. Risky projects increase the overall variance of the stock returns and hence, in the Black-Scholes valuation context, the stock options value is enhanced. When studying risk incentives, most researchers only focus on the total risk or the stock return s standard deviation. Very little attention has been paid to the breakdown of the total risk. Among few exceptions is the study by Meulbroek (2001a). From a diversification perspective, Meulbroek (2001a) argued and demonstrated that managers would value the granted stocks and options much less than the company or the market would, simply due to the fact that a major portion of their wealth is tied to the company s fortune. While investors at large could easily diversify away the company specific risk and demand a return commensurate with only the systematic risk, the managers must demand a higher return on the company s stock since they cannot fully diversify their holdings. The same logic applies to stock options. Through a Sharpe ratio analysis, Meulbroek (2001a) was able to demonstrate precisely how much a CEO would under-value his or her holdings of stocks and stock options. From the company s perspective, granting stocks and stock options involves what Meulbroek (2001a) called a dead-weight loss, since the CEO values them less than the cost to the company; from the CEO s perspective, he or she will have an incentive to increase the systematic risk of the company s stock, so that the exposure to the company specific risk is reduced. 3 Another exception is Tian (2002). Using a utility and certainty equivalent framework, he showed that stock options create incentives for CEOs to reduce idiosyncratic risk and, under certain conditions, the systematic risk as well. The direction of risk incentive concerning 3 The discussions of systematic risk versus non-systematic risk also appear in Meulbroek (2001b), with a focus on index-options. Jin (2002) also studied the relation between CEOs incentives and their firms risk characteristics. 3

6 the systematic risk is not always clear. The ambiguity stems from the trade-off between risk aversion and higher option payoffs. When the systematic risk is in the modest range, the benefit of potential higher payoffs due to higher expected returns more than offsets the cost of higher risk; the trade-off reverses when the systematic risk is very high. Therefore, due to different frameworks, Meulbroek (2001a) and Tian (2002) did not arrive at the same conclusion as far as CEOs desired systematic risk level is concerned. Both papers rely on the CEO s trading constraint to arrive at their respective conclusions which in essence amount to the CEO s shadow price differing from the market value. In the current paper, we study the risk incentive effects of executive stock options by assuming a GARCH process for the stock returns. We employ the GARCH option pricing theory to break down the total risk into systematic and nonsystematic risks, and show explicitly how option values depend on the amount of systematic risk. We then study the risk incentive effects of stock options within this framework. The key in linking the systematic risk to option values is to recast the risk premium term in the GARCH process as parameters pertaining to the stock s systematic risk and the overall market conditions. In the Black-Scholes framework, only the total risk enters into the valuation, and it is impossible to study the separate impacts of its components. This is perhaps why the current literature is lacking in studying risk incentives from the perspective of systematic versus nonsystematic risks. It should be noted up-front that, in this study, we do not attempt to distinguish a stock option s value to the CEO from its cost to the firm as many authors have done (e.g., Lambert, Larcker and Verrecchia (1991), Hall and Murphy (2001), Meulbroek (2001a), Meulbroek (2001b), Tian (2001), and Tian (2002)). In the current literature, it is typically assumed that the cost of a stock option to the firm is the Black-Scholes formula price, but 4

7 the option s value to the CEO can be different from this formula price. In general, it is lower than the cost due to hedging restrictions and CEOs risk-aversion. In this paper, we utilize the role of risk premium in the GARCH option pricing theory but do not explore the asymmetry between the CEO s private value and the firm s cost. In fact, we argue that the qualitative conclusions would remain valid when trading restrictions are explicitly considered. Thus, with respect to systematic risk, our model is complementary to the line of work by Meulbroek (2001a) and Tian (2001). We find that, with non-indexed options, CEOs have an incentive to increase the systematic risk in their stocks, consistent with the findings by Meulbroek (2001a) but for a different reason. A higher systematic risk will enhance the value of the stock option, even when the total risk remains constant. The incentive to increase the systematic risk is stronger when the total risk is lower. The Black-Scholes model tends to under-estimate the value of the stock option when the systematic risk is high, since it is incapable of distinguishing the separate effects of systematic and nonsystematic risks. With indexed options, CEOs generally have an incentive to reduce systematic risk. The opposite incentives induced by the two types of options imply that firms could grant a mixture of indexed and non-indexed options to target a desirable composition of systematic and nonsystematic risks. The rest of the paper is organized as follows. Section 2 presents the valuation framework which allows the systematic risk to be an explicit factor. Section 3 reports numerical results pertaining to valuation and risk incentives for both non-indexed and indexed stock options. Section 4 provides a brief summary and some concluding remarks. 5

8 2. The Theoretical Framework The seminal work of Black and Scholes (1973) on option pricing lays the foundation for the modern option pricing theory. One of the most significant theoretical implications of the Black-Scholes theory is the irrelevancy of the risk premium of the underlying asset. Specifically, it stipulates that an option s value is a direct function of the underlying asset s volatility but is not a function of the expected return. This implies that an option s value is independent of the relative proportions of systematic and nonsystematic risks as long as the total asset risk remains the same. Although it logically follows from a continuous hedging argument within the Black-Scholes framework, this result seems at odds with our simple economic intuition: a higher expected return (due to a bigger systematic risk) should increase the chance for a call option to finish in the money keeping other variables fixed. 4 In general, it is indeed meaningful to ask the following question, Will two call options have the same value if all aspects of the contracts and the underlying assets are exactly the same, except for the systematic risk? As discussed above, within the Black-Scholes framework, the answer is a simple yes. This means that, when applied to executive stock options, the Black-Scholes framework will lead to specific but likely erroneous implications for risk incentives. CEOs have control over not only the overall risk level but also the composition of the total risk. Activities such as streamlining product lines will substantially reduce systematic risk and increase idiosyncratic risk, while perhaps leaving the total risk unchanged. Within the Black-Scholes framework, CEOs will not have any incentive to engage in such activities since the value of the compensation (i.e., call option value) depends only on the total risk level. But such activities will surely affect the fortune of the company 4 A higher expected return can be a result of (1) a lower current asset price in conjunction with the same level of future cash flows, or (2) the same current asset price in conjunction with a higher level of future cash flows, or both. Our discussion on the effect of the expected return is concerned with the second case. 6

9 and hence the compensation for the CEO. Is it then possible to have a theoretical framework which allows the breakdown of the total risk to play a role in option valuation? The answer is a positive one due to the recent development in GARCH option pricing theory. In what follows, we will lay out the valuation framework, with a particular focus on the role of systematic risk. To this end, we consider a specific version of the GARCH process, the exponential GARCH(1, 1)-inmean or EGARCH(1, 1)-in-mean process for the stock return with respect to the physical probability measure P : 5 ln S t+1 S t = r t+1 + λ t+1 σ t σ2 t+1 + σ t+1ε t+1 ln σ 2 t+1 = α 0 + α 1 ln σ 2 t + α 2 ( ε t θε t ) (2.1) ε t+1 φ t P v N(0, 1) where φ t is the information set containing all information up to and including time t; r t+1 is the risk-free interest rate; λ t+1 is the risk premium per unit of standard deviation; α 0, α 1, α 2,andθ are the EGARCH parameters governing the variance process; and N(0, 1) denotes a standard normal distribution. Note that the term λ t+1 σ t+1 captures the total risk premium in the expected return. In the above system, we use time subscript for the interest rate and the risk premium so as to allow them to be potentially stochastic. As stated in Duan (1995), making them stochastic but predictable (i.e., r t+1 and λ t+1 are measurable with respect to φ t ) does not alter the GARCH option pricing results. For the return process to be variance stationary under P, the EGARCH parameter α 1 must be strictly between one and minus one. Although the stationary (or unconditional) variance of the one-period stock return is finite, there is no convenient formula because of the logarithmic transformation. However, it can be easily computed by simulation and we will delineate this point in the 5 Please note that our general conclusions remain valid under other GARCH specifications. We use EGARCH to ensure a better convergence in simulations. 7

10 numerical analysis section. Duan (1995) showed by an equilibrium argument that, for the purpose of option valuation, the pricing system corresponding to the physical system in (2.1) becomes ln S t+1 S t = r t σ2 t+1 + σ t+1ξ t+1 ln σ 2 t+1 = α 0 + α 1 ln σ 2 t + α 2 [ ξ t λ t θ (ξ t λ t )] (2.2) Q ξ t+1 φ t v N(0, 1) where Q is the locally risk-neutralized pricing measure and ξ t = ε t + λ t. It is seen that the risk premium, λ t, plays a critical role in the GARCH option pricing model. In contrast to the Black-Scholes framework, an option s value is a direct function of the stock s expected return (via the risk premium term, λ t ). According to the standard asset pricing theory, the total risk premium is proportional to systematic risk. Therefore the next step is to explicitly characterize the risk premium, λ t, in terms of systematic risk. Once this is done, we will be able to analyze in a precise manner the impact of systematic risk on option pricing. To this end, we define Y t = κ ln (U 0 (C t )/U 0 (C t 1 )) with U(C t ) being the utility function of time-t consumption C t and κ the impatience parameter for exponential discounting over time. In words, Y t is the negative of the logarithmic one-period marginal rate of substitution (or the logarithmic one-period stochastic discount factor). As shown in the appendix, λ t = q t δ t (2.3) where q t = Corr P ³ln S t+1 S t,y t+1 φ t and δ 2 t = Var P (Y t+1 φ t ). In order to relate to the notion of systematic risk in terms of the market portfolio, we adopt the following one-factor model for the stochastic discount factor: Y t = a + b ln 8 I t I t 1 + ² t (2.4)

11 where I t is the value of the market portfolio at time t and ² t is the regression residual. We further assume that the residual ² t has a constant variance and is independent of any asset return. The market portfolio return is allowed to have a time-varying volatility. The specific model for the market portfolio return process is not critical to the theoretical discussion but is important for actual implementation. We will later adopt an EGARCH process for the market portfolio return. With the assumption of (2.4), δ t = q b 2 σ 2 I,t + c (2.5) where σ 2 I,t = VarP ³ln I t+1 I t φ t and c = Var P (² t ). Moreover, it is straightforward to derive λ t = bβtσ2 I,t σ t q b 2 σ 2 I,t +cδ t (2.6) ³ where β t = CovP ln S t+1,ln I t+1 S t σ 2 I,t I t φ t = bβ tσ 2 I,t σ t. The risk premium per unit of total asset risk (λ t )isthus completely characterized relative to the market portfolio. Note that b is constant across different assets and can be identified using the market portfolio because its own systematic risk equals one by definition. The difference across assets only comes from the difference in the systematic risk as measured relative to the market portfolio. Since the total risk premium is λ t σ t = bβ t σi,t 2, it is seen that an asset with a larger systematic risk will have a higher total risk premium, a result that is quite intuitive. To actually implement the model, we need to specify a model for the market portfolio return. Specifically, we assume an EGARCH(1, 1) model with respect to the physical probability measure P as follows: ln I t+1 I t = r t+1 d I,t+1 + bσ 2 I,t σ2 I,t+1 + σ I,t+1ε I,t+1 ln σ 2 I,t+1 = α I,0 + α I,1 ln σ 2 I,t + α I,2( ε I,t θ I ε I,t ) (2.7) ε I,t+1 φ t P v N(0, 1) 9

12 where d I,t+1 is the dividend yield on the market portfolio and Corr P (ε t+1, ε I,t+1 φ t )=ρ. 6 Note that the risk premium term above is not arbitrary. It follows directly from (2.1) and (2.6) by using the fact that the systematic risk of the market portfolio equals one. The pricing system under the locally risk-neutralized pricing measure Q again follows directly from Duan (1995) and in this case can be written as ln I t+1 I t = r t+1 d I,t σ2 I,t+1 + σ I,t+1ξ I,t+1 ln σ 2 I,t+1 = α I,0 + α I,1 ln σ 2 I,t + α I,2 [ ξ I,t bσ I,t θ I (ξ I,t bσ I,t )] (2.8) Q ξ I,t+1 φ t v N(0, 1) Note that the correlation coefficient is not affected by the measure transformation. In other words, Corr Q (ξ t+1, ξ I,t+1 φ t )=ρ. In addition, it can be seen that q t = ρ and λ t in (2.6) can be simplified to ρbσ I,t. With the system in (2.2) and (2.8), any options on the stock and the market portfolio can be valued. For instance, with a constant risk-free interest rate r, anexecutivestock option of European style with an exercise price X and maturity T canbevaluedas C t = e r(t t) E Q [max (S T X, 0) φ t ] (2.9) where S T is the stock price at the option s maturity, and E Q ( ) is the expectation operator with respect to the risk-neutral measure Q. An indexed option as proposed by Johnson and Tian (2000) can be valued in a similar fashion: C indexed t = e r(t t) E Q [max (S T H(S 0,I 0,I T ), 0) φ t ] (2.10) where H(S 0,I 0,I T ) in general is a function that sets the exercise price in relation to the future index level, the current index level and the current stock price. There is no closed-form 6 Adividendyieldtermhasbeenincorporatedin(2.7) to account for the fact that many component stocks pay cash dividends. The situation can be likened to a stream of small dividends and approximately modeled as a rate. For individual stocks, we did not incorporate a similar term in (2.1) because periodic cash payments are typically treated by an adjustment to the stock price. 10

13 solution to either (2.9) or (2.10) but numerical procedures such as Monte Carlo simulation, the lattice method of Ritchken and Trevor (1999) and the Markov chain method of Duan and Simonato (2001) can be used to price European executive options or American style options subject to a vesting period restriction. 3. Valuation and Risk Incentives of Executive Stock Options In this section, we study the value and risk incentives of executive stock options. Following pervious authors, for simplicity, we assume away the complications associated with vesting and the early exercise feature. We treat the stock option as a European call option. Monte Carlo simulations are used to value the options. In what follows, we will first examine non-indexed executive stock options with fixed exercise prices, and then examine indexed stock options with stochastic exercise prices Non-Indexed CEO Stock Options Parameter values and numerical analysis design To obtain the value of a particular stock option, we will simulate the systems in (2.2) and (2.8), and evaluate (2.9) numerically. 7 Specifically, we generate 10,000 correlated paths of the stock price and the index value via (2.2) and (2.8) to obtain 10,000 stock prices at the option s maturity. We then obtain the option s value by calculating the average of the 10,000 discounted payoffs via (2.9). Recall that λ t = ρbσ It, which is the reason for simulating both the index value and stock price processes. To obtain the GARCH process parameters, we empirically estimate (2.1) and (2.7) 7 Although the option s payoff is independent of the market index, we still need to simulate the index return process since the risk premium term, λ t is a function of the market return s variance. 11

14 using the daily closing prices for Philip Morris stock and the S&P500 index for the period of January 2, 1992 to December 29, To obtain some guidance for choosing parameter values for comparative analysis, we also estimate (2.1) for the other 29 component stocks of the Dow Jones Industrial Average (DJIA). (Philip Morris is a component stock of the index.) Table 3 contains the estimation results for Philip Morris and the S&P500 index. For all cases, we keep the S&P500 parameters constant. For the stock, we will vary the volatility asymmetry parameter θ andthecorrelationcoefficient ρ to obtain comparative static results. For the 30 DJIA component stocks, θ ranges from to 1.07, with an average close to 0.6. The parameter takes a negative value for only two stocks. We will examine three values of θ: 0.2, 0.6, and 1.0, representing respectively low, average and high levels of θ. Notice that the volatility asymmetry parameter, θ is crucial in determining the return distribution properties and it is directly linked to the so-called leverage effect. It is also related to the skew of the implied volatility smile. Roughly speaking, a positive θ induces a negative skewness in the unconditional stock return. The range for the correlation coefficient, ρ for the 30 DJIA component stocks is from 0.20 to To ensure generality of the numerical results, we will examine three values of ρ: 0.0, 0.35, and Notice that given the market volatility and an overall level of the total volatility for the stock, the amount of systematic risk is purely determined by the correlation coefficient. The annualized standard deviation for the 30 DJIA component stocks ranges from 0.27 to 0.51, and that for the S&P500 is Unless otherwise specified, we set the stock s standard deviation at 0.40, and the market s at 0.2, both under measure P.Atthispoint, we need to discuss how to ensure comparability between the Black-Scholes and GARCH option values. Traditionally, within the Black-Scholes framework, we estimate the constant 12

15 volatility using total returns. Once we stipulate that GARCH properly describes the real world, we should use the total return generated via (2.1) to estimate the constant volatility. Although we typically refer to the average standard deviation of the residual in (2.1) as the stationary volatility for a GARCH process, in the current setting, we will loosely call the constant standard deviation of the total return the stationary volatility. With the above understanding, we then back out the value for α I,0 in (2.7) for the given level of the market index volatility, 0.2. The constant volatility is estimated by calculating the standard deviation of total returns in (2.7) for a very long time series (i.e. 50,000,000 days). The search algorithm will pinpoint a value of α I,0 which leads to the constant volatility of 0.2. Similar procedures can be followed to back out the value of α 0 in (2.1) for a given level of the stock s constant volatility. Since changes in θ will lead to changes in the overall standard deviation under measure P, whenever we vary this parameter, we must re-adjust α 0 so that the total volatility remains at 0.4. Similarly, changes in the correlation coefficient ρ will also affect the constant volatility of the total return in (2.1), although this impact is very minor. Nonetheless, to ensure comparability, we will also re-adjust the value of α 0 in (2.1) so that the constant volatility remains at 0.4. Although changes in ρ only slightly affect the total return standard deviation under measure P, they will greatly affect the corresponding standard deviation under measure Q. This is essentially how the impact of systematic risk is manifested in the GARCH framework. All else being equal, varying ρ amounts to varying the amount of systematic risk while keeping the total risk constant. Given the unconditional volatilities, ρ =0.0, 0.35 and 0.70 correspond to a stationary beta of 0.0, 0.7, and 1.4. Without loss of generality, we set the risk-free interest rate to 5%, and the current stock price to $100. We examine nine moneyness situations: PV(X)/S = 0.80, 0.85, 0.90, 0.95, 13

16 1.00, 1.05, 1.10, 1.15, and 1.20, where PV(X) stands for the present value of the exercise price, i.e., the exercise price discounted at the risk-free rate. For brevity, we only report five moneyness situations: deep-in-the-money (DITM), in-the-money (ITM), at-the-money (ATM), out-of-the-money (OTM), and deep out-of-the-money (DOTM). For the ITM case, we average the two prices for PV(X)/S = 0.95 and 0.90; for the DITM case, we average the two prices for PV(X)/S = 0.85 and Similar calculations and reporting apply to the OTM and DOTM cases. 8 The option s maturity ranges from 182 days (i.e., 0.5 years) to 3650 days (i.e., 10 years), assuming 365 days in a year. For brevity, we report only eight maturities: 182 days (0.5 years), 365 days (1 year), 730 days (2 years), 1460 days (4 years), 2190 days (6 years), 2920 days (8 years), and 3650 days (10 years). As mentioned before, most of the executive stock options are granted with a 10-year maturity, so the longest maturity corresponds to the case of newly granted stock options. To reduce simulation errors, we employ three variance reduction techniques: antithetic variable, control variate and empirical martingale correction. The first two adjustments are well known and the third was proposed by Duan and Simonato (1998). The control variate is the Black-Scholes value with the aforementioned stationary standard deviation under measure P as the volatility input. The empirical martingale correction is important for long-term options because the stock price is typically modeled as a stochastic exponential. Without such a correction, the Monte Carlo option value tends to have a serious downward bias. 8 Note that our definition of moneyness deviates from the normal practice in the CEO stock option literature. Almost all stock options are granted with an exercise price equal to the stock price at the time of granting, and they are typically considered as at-the-money options. When studying long term options, with a non-zero interest rate, this type of at-the-money options effectively becomes in-the-money options. The advantage of our definition is that the option values are actually independent of the risk-free interest rate. In other words, it is equivalent to using the usual moneyness definition while setting the interest rate to zero. 14

17 Risk Incentive Analysis Since most previous researchers used the Black-Scholes framework to analyze executive stock options, we will use the Black-Scholes value as the benchmark value. The volatility input is fixed at 0.4. We consider the GARCH option value as the true value for the executive stock option. Risk incentives are also derived from the GARCH option prices. Table 4 reports the numerical results. Here, for each parameter combination, we calculate and report the percentage difference between the GARCH option price and the Black- Scholes value. In addition, for the GARCH process, we set the initial volatility equal to the stationary volatility. Again, it should be kept in mind that the stationary volatility under measure P is kept constant at 0.4 for all cases. Several observations are in order. First, for all parameter combinations, the true value of executive stock options increases as the systematic risk increases (i.e., as ρ increases). An immediate implication is that CEOs will have an incentive to increase their companies systematic risk in order to enhance their stock options value. This can be accomplished by acquiring companies in different business lines. Without proper monitoring, the company s systematic risk level may be above the desired level by moving away from its core competency and becomes detrimental to the company s performance. Conceivably, the CEO could achieve a higher systematic risk by keeping the total risk constant. Within the Black- Scholes framework, this would have no effect on the value of their stock options. Therefore, using the Black-Scholes framework to study stock options risk incentives will overlook the incentive for the CEO to take on excessive systematic risk. It is interesting to note that our conclusion is in agreement with that of Meulbroek (2001a) who studies the value of a stock option from a diversification perspective. Second, with several minor exceptions, the true stock option value is always lower than 15

18 the Black-Scholes counterpart when the systematic risk is zero (i.e., when ρ = 0.0), and higher when the systematic risk is high (i.e., when ρ = 0.7 corresponding to a stationary beta of 1.4). The difference in value can be substantial in many cases. For instance, with a zero systematic risk and a volatility asymmetry parameter of 1.0, the true stock option value is about 10% lower than its Black-Scholes counterpart when the maturity is four years. The percentage difference is even larger for options with shorter maturities. With a medium level of systematic risk (i.e., when ρ =0.35 corresponding to a stationary beta of 0.7), it appears that the true option value is lower (higher) than its Black-Scholes counterpart for out-of-the-money (in-the-money) options. Overall, the results imply that using the Black- Scholes formula to estimate the cost of executive stock options to the firm can lead to substantial errors. Whether the Black-Scholes formula under- or over-estimates the true stock option s value (when the stock return indeed follows the EGARCH process) depends on the stock s systematic risk and moneyness of the option. For instance, if the stock has a zero systematic risk, most of the time, the Black-Scholes formula will be upward biased. But when the systematic risk is high, the Black-Scholes formula will be downward biased. 9 Third, in order to see whether the impact of changing systematic risk is different across maturities, for each moneyness situation and under a particular level of the volatility asymmetry parameter, θ, we examine the difference between the percentage value differences, 9 Note that there is a subtle reason for the GARCH option values to be lower than their Black-Scholes counterparts when the correlation coefficient is zero (i.e., no systematic risk). The standard way of estimating the constant volatility is based on one sample path, which is what one does in practice. Implicit in such an estimate is the i.i.d. return assumption. For a time series model such as GARCH, it is equivalent to computing the average variance of one-period return. By the Central Limit Theorem, the standardized cumulative stock return (continuously compounded) will converge to a normal distribution, but the normalizing constant should be the standard deviation of the standardized cumulative return. Once a correlation exists among one-period returns such as the case of GARCH, the variance of the standardized cumulative return no longer equals the average variance of one-period return. In other words, although the Black-Scholes formula is applicable for long-term options due to the Central Limit Theorem, it must be evaluated at a volatility level different from the standard constant volatility estimate. We performed a separate investigation on this issue, and it turns out that the average variance of one-period return is always higher than the variance of the standardized cumulative return. As a result, one would expect the Black-Scholes value to be higher than the GARCH option value for long-term options when the systematic risk is zero. 16

19 since the latter are both with respect to the same Black-Scholes value. For example, when the volatility asymmetry parameter θ is equal to 1.0, for deep-in-the-money options with a maturity of 1460 days (i.e. four years), the option value would go up by [7.27 ( 1.30)]/( ) = 8.68% when the correlation coefficient goes from 0.0 to 0.7. This is approximately equal to 7.27 ( 1.30) = 8.57%. 10 A casual examination reveals no apparent pattern of the value change across different maturities for the same moneyness. This means that the CEO s incentive to increase systematic risk is uniform across options with different maturities, as long as the options have similar exercise prices. Insofar as different maturities means different granting dates, the above implies that CEOs will try to increase the systematic risk to equally benefit all existing stock options in his/her compensation portfolio. Fourth, by and large, the percentage difference in price is bigger, the higher the volatility asymmetry parameter, θ. As mentioned before, this parameter determines the distribution property of the stock return. Specifically, it determines the skewness of the distribution. 11 The results imply that, other things being equal, CEOs of those companies whose stock returns exhibit more skewness will have more incentive to increase the systematic risk. So far, all the results are based on the assumption that the current volatility of the stock is equal to the stationary volatility. However, the model allows the volatility to evolve, and at a particular point in time, the actual volatility can be above or below the long run average. It is of interest to know if the previous observations still obtain under those circumstances. To this end, we re-run the simulations for two scenarios by setting the initial standard deviation to, respectively, 50% and 150% of the long run average. The 10 Although not shown here, we have directly calculated the percentage change in option values when the correlation coefficientchangesfrom0.0to0.35and It is similar to the volatility asymmetry parameter in the non-linear GARCH model. For a more in-depth discussion of its effects, see Duan and Wei (1999). 17

20 results are reported in Table 5 and Table 6. Understandably, the true stock option prices are all lower when the initial volatility is lower than the long run average, and vice versa. More important, it remains true that a higher systematic risk (i.e., higher ρ) leads to a higher option value. Therefore, the incentivetoincreasethesystematicriskispresent regardless of the current level of the volatility. It is interesting to note that the difference between the true stock option value and the Black-Scholes value is much more pronounced with shorter term options. This is due to a dampening effect. In a GARCH framework, the stochastic volatility reverts to the long-run stationary level when the time horizon is long. Over a shorter horizon, the current level of volatility dominates valuation. That is why we observe bigger percentage differences between the GARCH value and the Black-Scholes value in Tables 5 and 6 when the maturities are relatively short. This again reveals the limitations of the Black-Scholes model in estimating the value of seasoned CEO stock options. When the stock s volatility changes, the value of the option can change significantly. In the Black-Scholes setting, when the long-run volatility is estimated using historical data, the option value will not respond to current fluctuations of the return volatility. The above analyses focus on the varying levels of the current volatility relative to the long run volatility. Stocks can have inherently different levels of long run or average volatilities due to different business lines. For instance, high tech stocks will have a much higher average volatility than utility stocks. It is useful to know if the risk incentives are different under different levels of long run volatilities. To this end, we repeat calculations similar to those in Table 4 by setting the initial volatility to the long run average. However, rather than varying θ and keeping the long run volatility constant, we vary the systematic risk parameter ρ under different levels of long run volatilities. Specifically, we examine three scenarios: the 18

21 annualized stationary standard deviation being 0.2, 0.4, and 0.6. The volatility asymmetry parameter θ is kept at the base level of 0.6. Table 7 contains the results. 12 There are several interesting observations. First, it still remains true that the higher the systematic risk, the higher the stock option s value. This implies that the CEO always has the incentive to increase the proportion of systematic risk, regardless of the total risk level. To this point, we have basically established that if the stock returns are accurately described by the GARCH process, then we can infer that CEOs will always have the incentive to increase their stocks systematic risk in order to enhance their own wealth. Second, the results in Table 7 reveal that increasing the systematic risk will have a bigger impact on the true option value when the long run volatility is low. This result is quite intuitive. On the one hand, the stock option s value is always higher when the long run volatility or total risk is higher. On the other hand, when the total risk is low, any change in the systematic risk will likely bring about a bigger impact on the option price. This is what we see in Table 7. The implication is that for companies with a lower total risk, the board should be extra vigilant about their CEOs actions. Taken together, the above results seem to suggest the following: 1) CEOs will have an incentive to increase their stocks systematic risk to enhance their own wealth; 2) the risk incentive is stronger when the total risk level is lower; 3) when the systematic risk is high, the Black-Scholes formula tends to under-estimate the true option value because of its inability to incorporate the systematic risk into pricing; when the systematic risk is very low, the Black-Scholes formula tends to over-estimate the true option value due to the failure to properly account for return dependency in computing the constant volatility 12 Please note that the Black-Scholes prices are different for different stationary volatilities. The percentage differences are calculated using the proper Black-Scholes price under each stationary volatility. For brevity, we omit the Black-Scholes prices in the table. In addition, we have also done the calculations for lower and higher initial variance cases. Since they do not offer additional insights, we omit the tables for brevity. 19

22 estimate Indexed CEO Stock Options Johnson and Tian (2000) proposed that the exercise price of a stock option should be indexed, since part of the stock price movements is affected by the overall market and is beyond the CEO s control. In other words, CEOs should be rewarded only for his / her contribution to the idiosyncratic stock price appreciations, and should not be penalized for stock price declines caused by the overall market downturn. Thus indexing will not only better align incentives with performance, but also save compensating cost for the company. In the Black-Scholes environment, assuming constant returns and constant volatilities for the stock and the market index, Johnson and Tian (2000) proposed an indexed exercise price which is essentially the expected stock price conditional on 1) the level of the realized market index, and 2) zero excess return of the stock with respect to the market. Using their notation and ignoring dividend yield, for an option granted at time 0, the indexed exercise price at time t is defined as H t = S 0 (I t /I 0 ) β e [r(1 β)+0.5ρσsσ I(1 β)]t (3.1) where S 0,I 0 are the stock price and the market index at time 0, I t is the market index at time t, σ s and σ I are the volatilities for the stock and the market index, ρ is the correlation between the stock and the market index returns, r is the risk-free rate, and β = ρσ s /σ I. Clearly, at the time of granting, the exercise price is equal to the prevailing stock price. As shown by Johnson and Tian (2000), one can easily re-scale the exercise price by a constant to make it in-the-money or out-of-the-money at the time of granting. With the above setup, 20

23 the option s value using the Black-Scholes pricing theory can be expressed as C indexed t = S t N(d 1 ) H t N(d 2 ) (3.2) where d 1 = ln(s t/h t )+0.5σ 2 a(t t) σ a T t, d 2 = d 1 σ a T t, p σ a = σ s 1 ρ 2, T t is the option s time to maturity, and N( ) is the cumulative probability function for a standard normal distribution. 13 To study the value and risk incentive effects for indexed CEO stock options within the GARCH framework, we must slightly modify the numerical analysis framework in the previous section and make some simplifying assumptions. Specifically, 1) we index the exercise price according to (3.1), 2) we use the long run or stationary volatilities in place of σ s and σ I, 3) we assume that the current stock and index levels are the same as those at the time of granting, and 4) we scale the overall exercise price to make it in-the-money or outof-the-money. In this environment, the Black-Scholes value will be given in (3.2), and it is also used as the control variate in the simulations. Similar to the case of non-indexed stock options, the difference between the Black-Scholes and GARCH option values stems from the stochastic volatilities within the GARCH framework. When the option s exercise price is indexed, the systematic risk becomes important even within the Black-Scholes framework. But its effect is solely through indexing. Within the GARCH framework, the systematic risk affects both indexing and the return dynamics. 13 Meulbroek (2001b) made a critique of such indexing schemes by pointing out that the option s value would increase by the same percentage in which both the stock price and the market index have increased (assumingaunitbeta). Thisissobecausetheoption value is homogenous of degree one with respect to the stock price and the exercise price. She then offered an alternative indexing scheme which takes the index-hedged stock price as the monitoring variable. In this paper, we will not attempt to delineate on the merit of indexing per se. Instead, we will focus on the risk incentives regarding the systematic risk. 21

24 With the above setup, we calculate option values under different systematic risk levels by varying the correlation coefficient, ρ. Similar to the study of non-indexed stock options, we perform the calculations under different levels of the volatility asymmetry parameter and the initial volatility for the stock. Since the results under various parameter combinations do not offer additional insights, for brevity, we only report in Table 8 the numerical results for the base case where the initial volatility is equal to the long run volatility at 0.4, and the volatility asymmetry parameter is at 0.6. Several observations are in order. First, unlike the non-indexed options whose value is totally independent of the systematic risk in the Black-Scholes environment, the Black-Scholes value for an indexed option is negatively affected by the systematic risk. As already established by Johnson and Tian (2000), a higher systematic risk means a smaller contribution of the idiosyncratic risk; most of the stock movements is attributable to the market movements, and the CEO is rewarded much less than if the exercise price is not indexed. Incidentally, comparing Panel A of Table 8 with Panel B of Table 4, we notice that the indexed and non-indexed options have the same value (both Black-Scholes and the GARCH) when the correlation is zero. This is expected since the indexed exercise price in (3.1) reduces to S 0 e r(t t) at maturity, which is identical to the present-value-adjusted exercise price for the non-indexing case. Second, the GARCH option values appear to be negatively affected by the systematic risk. This is in sharp contrast with the results for non-indexed options. Increasing systematic risk is no longer in the CEO s interest. It should be noted, however, that the table has only presented the results for three values of ρ. Once we consider the whole spectrum of possible values, the GARCH option value for the indexed contract actually increases withthesystematicriskforasmallrangeofvaluesforρ andthenstartstodecline,asreflected in Figure 1. This result is due to the interplay between the increasing force from the 22

25 GARCH dynamic and the dampening effect of indexing the contract. Once the systematic risk reaches a certain level, the dampening effect becomes the dominant force. Third, as long as the systematic risk is not close to zero, GARCH option prices are higher than their Black-Scholes counterparts. Thus, for indexed options, using Black-Scholes model to estimate the compensation cost can lead to an under-estimation. The higher the systematic risk, the bigger the under-estimation. When the return correlation is zero, the Black-Scholes model tends to over-estimate the true option value, similar to the non-indexed option case. Fourth, comparing option values across panels (both Black-Scholes and GARCH values), we notice that, relatively speaking or percentage-wise, systematic risk exerts a bigger impact on out-of-the-money options than on in-the-money options. Intuitively, out-of-the-money options have less cushion against exercise price fluctuations than in-the-money options. The above analysis indicates that whether the stock options are indexed or not may have completely opposite impacts on the CEO s risk incentives as far as systematic risk is concerned. With non-indexed options, the CEO has an incentive to increase the systematic risk; with indexed options, it will be in the CEO s interest to reduce the systematic risk for a wide range of systematic risk level. Neither of the two alone can induce the CEO to opt for a level of systematic risk that is best for the company. The CEO is likely to pursue a corner solution by increasing the systematic risk to its fullest extent when granted with non-indexed options. If indexed contracts are awarded, the systematic risk will be set at a low level for which the option value is the highest. Using a basket of the two options has the potential of inducing the CEO to choose a level of systematic risk deemed by the board of directors to be desirable for the company. In other words, an optimal compensation scheme can be devised by partially indexing the CEO stock options. Partial can be 23