NEW PERFORMANCE-VESTED STOCK OPTION SCHEMES AN CHEN∗ , MARKUS PELGER‡ , AND KLAUS SANDMANN§ Abstract. In the present paper, we analyze two effective non-traditional performancebased stock option schemes which we call Parisian and constrained Asian executives’ stock option plans. Both options have a criterion on the terminal value similar to a call option, but in addition impose a restriction on the path of the firm’s assets process. Under a Parisian option scheme, the bonus of the executives becomes effective when the stock price has outperformed a certain threshold for a fixed length of time. Under the constrained Asian scheme, the executives’ compensation is coupled with the average performance of the stock price. We show that the value of both ESO schemes are less sensitive to changes in risk than plain vanilla options and hence represent an alternative compensation scheme that could make exaggerated risk taking through the executives less likely.
Keywords: Executive Stock Options, Asian Options, Parisian Options JEL: G12, G13, G30
Date. January 22, 2013. ∗
(Corresponding author) Faculty of Mathematics and Economics, University of Ulm, Helmholtz 20, 89081
Ulm, Germany. Tel: 0049-731-5031183, Fax: 0049-731-5031188, E–Mail:
[email protected]. ‡
Department of Economics, University of California, Berkeley, 508-1 Evans Hall, Berkeley, California
94720-3880, USA, E–Mail:
[email protected]. §
Department of Economics, University of Bonn, Adenauerallee 24-42, 53113 Bonn, Germany. Tel: 0049-
228-739227, Fax: 0049-228-735048, E–Mail:
[email protected]. 1
New performance-vested stock option schemes Abstract. In the present paper, we analyze two effective non-traditional performancebased stock option schemes which we call Parisian and constrained Asian executives’ stock option plans. Both options have a criterion on the terminal value similar to a call option, but in addition impose a restriction on the path of the firm’s assets process. Under a Parisian option scheme, the bonus of the executives becomes effective when the stock price has outperformed a certain threshold for a fixed length of time. Under the constrained Asian scheme, the executives’ compensation is coupled with the average performance of the stock price. We show that the value of both ESO schemes are less sensitive to changes in risk than plain vanilla options and hence represent an alternative compensation scheme that could make exaggerated risk taking through the executives less likely.
1. Introduction Executives’ stock options (ESOs) were introduced in the managers’ compensation plans in the US in 1950s (see e.g. Lambert et. al. (1989)). Nowadays ESOs are included in most of the executives’ compensations in the US, UK and other developed countries. As stated in Jensen and Meckling (1976), one of the original purposes of ESOs is to alleviate the principal and agent problem and to better align the interests of shareholders and management. They are designed to increase the risk-taking of risk-averse managers. The incentive alignment argument is due to the belief that ESOs are positively related to the firm performance. However, there is very mixed empirical evidence for this alignment argument. The recent financial crisis has caused more doubt about ESOs and raised the question if some managers have taken too much risk. There are many different forms of ESOs. A European style call option is the simplest form of an ESO. It is a performance-vested option with a fixed absolute performance target. ESOs are based on the performance of 2
the firm’s assets directly and understood as European call options granted with a fixed strike price equaling the stock price at the granting date. Most of stock option plans in the US and some of German ESOs like premium options belong to this category. Bettis et. al (2005) show that the typical life of ESOs is 10 years, and the vesting period ranges from 2 to 4 years. In Germany, the life of ESOs varies from 5 up to 10 years. The choice of a specific maturity date depends on some firm- and industry-specific factors. For instance, the ESOs of Pharmacy companies and machine building industry own a longer maturity than in food industry (see e.g. Korn at. al (2012)).
The use of these simple ESOs has caused some concern because these options can be easily manipulated by the executives. The manipulation is done by influencing either the timing of granting or the stock price at the maturing date. Since the executives’ bonuses are solely dependent on the terminal stock price, the executives will try to increase the terminal value as much as possible. This could be achieved for example by taking higher risks. It might have the consequence that high bonuses will be paid to the mangers, although the firm has performed very averagely or poorly during the granting period. Apart from volatility, the manipulation can also occur through influencing the strike or granting time of the simple ESOs. Since the simple ESO can be considered as a European call option, a higher option value can be achieved by adjusting the strike to a lower level. If the executives have impact on the time of granting, they might grant the options after an anticipated future stock price decrease (i.e. strike is anticipated to be lower).1
Lie
(2005) provides a discussion on this issue. In order to make the manipulation less likely 1
Usually, manipulating the firm’s volatility is legal, in particular when the manipulation takes place in
the beginning, whereas manipulation other parameters might lead to legal problems. In this paper, we focus on the effect caused by volatility change. 3
or very risky strategies less attractive for the management, more complicated forms have been introduced. For instance, the bonuses are sometimes coupled with a reference index (instead of the firm’s stock). Sometimes ESOs are linked to some performance hurdles. Johnson and Tian (2000) introduce a performance-vested stock option with an absolute performance hurdle, i.e. the ESOs cannot be exercised unless the firm’s stock price hits a prescribed barrier (larger than the stock price at the grant date) during the option life. However, it should be pointed out that the executives can still manipulate the exercise of the option quite easily in their favor, because the barrier trigger only depends on a single touch of the barrier by the underlying stock price.
With the purpose of better avoiding manipulation of the management, the present paper introduces two types of performance-vested stock option plans which have not been studied in the ESO literature. We focus on a particular kind of manipulation, where the manager increases the risk of the company, measured by the volatility of the stock price process, in order to increase the payoff of his ESOs. In the first type, the bonus is coupled with a performance hurdle combined with a reference period. This feature can be characterized as a Parisian up-and-in call option.2 Under an ESO scheme with Parisian feature, the manager will be rewarded only when the rate of return of the underlying asset has stayed above a certain rate for at least a fixed time window d. An example for this ESO is the Group Incentive Plan of Allianz AG, in which the option is only valuable when an benchmark index is outperformed for at least a period of five consecutive trading days. Since out-performance for a time period is required to trigger the knock-in condition of the option, manipulation becomes more difficult. The longer the time period, the more 2
The formal formulation of Parisian options is introduced in Section 3. 4
difficult it becomes. In the second type, we let the bonuses coupled with the average performance of the stock price and characterize it as an Asian ESO.3 If it is required that the average rate of return of the stock exceeds a fixed rate of return, it is very unlikely that the executives can benefit from a one-time manipulation of the stock price. We also consider a constrained Asian scheme which requires additionally that the total return shall lie above a reference level.
The focus of the paper is twofold. First, ignoring some special characteristics of ESOs and particularly the non-tradable restriction, we determine the market value of the above two new ESOs. We use the inverse Laplace transform approach introduced by Chesney et. al (1997) to value the Parisian ESOs. Our evaluation of the constrained Asian ESO is novel and is based on the conditioning approach presented in Curran (1994) and Rogers and Shi (1995) and extended by Nielsen and Sandmann (2003).4 The second contribution is to analyze the properties of these new ESOs with respect to their risk-sensitivity.5 For this purpose we compare these two ESOs schemes with European style ESO to find out whether compensation packages based on these two schemes are less sensitive to changes in the risk as measured by the volatility. To make different schemes comparable, we first choose the number of new ESOs so that the price of the portfolio equals the value of a plain vanilla call option, and fix an initial volatility level with which the ESOs are granted. Furthermore, we assume that the managers can immediately adjust (only once) their risk 3The 4In
formal formulation of Asian options is introduced in Section 4.
our valuation framework, we ignore the following features incorporated in these options: early-
exercise feature (see. e.g. Bettis et. al. (2005) and Sircar and Xiong (2007)), non-tradable restriction (see e.g. Carpenter (2000)), and reloading or resetting feature (see e.g. Dybvig and Loewenstein (2003)). 5We
do not model manipulation dynamically. 5
management strategy which has a higher or lower volatility than the initial one. We particularly examine how the market value of the ESOs changes when the manager decides to invest in a more risky (higher volatility) strategy.6 We observe that under both new schemes, the market value of the ESOs falls below the vanilla call when the manager deviates to a more risky strategy. Hence, the incentives for risk-taking behaviors are hampered. Moreover, we formally study the risk sensitivity of the ESOs. One might argue that by taking a plain vanilla call option and imposing some additional restrictions on the path one would automatically reduce the sensitivity to the volatility. One might also think that increasing the excursion periods of a Parisian option or increasing the strike for the average value of an Asian option are simple ways to make these options less volatile to risk. These argumentations are only partly true as what really matters is not the behavior of a single option but the price of a normalized option which takes into consideration the number of options which makes the price of the portfolio equal some baseline value. By imposing more restrictions on the option, one typically reduces the sensitivity to the volatility of a single option but at the same time also reduces the price. Taking account of the number of options, we show that the price of a constrained Asian and a Parisian stock option is less sensitive to changes in the volatility than the corresponding call option. Hence, the new stock options provide less incentives for a manager to increase the risk at the cost of the company.
6Although
the market value of ESOs is different from the managers’ private value, we do not distinguish
these two values. 6
The existing literature (e.g. Hall and Murphy (2003)) has found empirical evidence that the option values in the remuneration of managers do not always increase in volatility, particularly when the volatility level is high. The use of European style ESOs disputes this empirical observation, because the value of a European call always increases in volatility. Our new performance-vested ESOs illustrate a good example for this argument and deliver similar results.
The remainder of the article is organized as follows. In Section 2, assumptions about the underlying firm’s asset process are made and European style ESOs are reviewed. Section 3 is dedicated to introducing performance-vested stock options with Parisian feature. Some numerical results (particularly concerning the length of excursion and the riskiness (volatility) of a portfolio) are conducted. Section 4 focuses on the introduction and valuation of the performance-vested stock options with Asian option schemes. In Section 5, different ESO schemes are compared to investigate the question of whether the new schemes reduce the mangers’ incentives for excessive risk-taking. Finally, the results of this article and possible further research are summarized in the conclusion.
2. Base contract: European stock option plan This section reviews the simple valuation framework for the simplest ESO form, the European-style ESO. Throughout the paper, we use relative stock price performance as the S(t) underlyings of ESOs, i.e. the ratio process { S(0) }t∈[0,T ] . To specify the underlying assets
process, we start immediately with the risk-neutral probability (also called equivalent martingale) measure Q. It is assumed that under Q the price process of the firm’s assets 7
{S(t)}t∈[0,T ] follows a geometric Brownian motion dS(t) = S(t)(rdt + σdWt )
(1)
in which σ denotes the deterministic volatility of the asset price process {S(t)}t∈[0,T ] . We assume that the firm continuously rebalances its investment portfolios such that the asset return volatility remains the same over time. Furthermore, {Wt }t∈[0,T ] in (1) is the unique risk–neutral Q–martingale. Solving this differential equation, we obtain S(t) = S(0) exp
1 2 r − σ t + σWt . 2
The benchmark contract specification is formulated immediately as a standard European call option: max
+ S(T ) S(T ) − K, 0 =: −K S(0) S(0)
(2)
in which K is the fixed strike of the call option. For instance, if ESOs are granted at the money, K = 1. The arbitrage-free price of the contract payoff in (2) results from the Black-Scholes formula: " E e−rT
d1/2
S(T ) −K S(0)
+ #
=Φ(d1 ) − Ke−rT Φ(d2 )
ln 1−rT ± 1 σ 2 T = Ke √ 2 , Φ(t) = σ T
Z
t
−∞
(3)
x2 1 √ e− 2 dx. 2π
Note under such a contract specification the bonus participation of the executives is unconstrained. The value of a plain vanilla call increases in the volatility level. 3. Parisian performance-based stock option scheme This section mainly introduces Parisian-type performance-vested executives’ stock options, i.e. ESOs incorporating a performance hurdle combined with a reference period. We 8
focus on introducing the payoffs of ESOs under the Parisian scheme and their valuation. Furthermore, comparisons are made between these new performance-vested ESOs and the base contract specification introduced in Section 2. Parisian options do not have a long history in the literature on exotic options. They are introduced by Chesney et al. (1997) and subsequently developed by Moraux (2002) and Anderluh and van der Weide (2004) etc.7
Under a Parisian ESO scheme, the bonus will be paid out only when the rate of return of the underlying asset has stayed above a certain rate sufficiently long. In other words, the bonus payment corresponds to a Parisian up–and–in option. Letting d denote a prespecified time window d, we assume the bonus payment follows when the underlying asset value shall stay consecutively above a certain boundary for a time longer than d before the maturity date. The exercise condition for the executives’ stock options is the following condition: S TB+ := inf t > 0 t − gB,t 1{S(t)>B(t)} > d < T
(4)
with S gB,t := sup{s ≤ t|S(s) = B(s)},
S where gB,t denotes the last time before t at which the value of the assets hits the barrier.
TB+ gives the first time at which an excursion above Bt lasts more than d units of time. In the following we will assume an exponential barrier B(t) = B(0)egt , i.e. the executives shall aim to increase the rate of return of the firm with a constant rate g. Under a Parisian 7Parisian
options distinguish themselves between standard and cumulative Parisian options (see e.g.
Chen and Suchanecki (2007)). Only standard Parisian options are considered here. 9
scheme, the manager owns the following option: h(S(T )) = max
+ S(T ) S(T ) − K, 0 1{T + 0 but the total return is 0! In this case it would be economically doubtful to pay the manager a bonus. With respect to option pricing theory, a novel contribution of this paper is to derive closed form solutions for the lower and upper bounds of the following payoff scheme which we call constrained average return contract. By the assumption that the contract is traded (or can at least be perfectly hedged by traded securities) the price equals the expected discounted payoff under the equivalent martingale measure Q: " #+ N −1 X S(ti+1 ) CA2 (K1 , K2 , N ) = e−rT E − K1 1n S(T ) ≥K o 2 S(t0 ) S(t ) i i=0
(7)
with T = tN . Appendixes D and E provide a lower and an upper pricing bound for (7).
4.3. Comparative Statics. The subsection illustrates some numerical results for the unconstrained and constrained average return ESO contracts, which we label contract 3 and 4. We start with contract 3. With the incorporation of the arithmetic weighting feature, the value of the ESOs (expected rate of return of the manager) decreases substantially. As 14
illustrated in Figure 3, the more time periods are considered, the lower the value of the unconstrained average return ESOs.
Figure 3. Price of
1 N
Asian calls as a function of N with parameters σ =
0.2, T = 10, r = 0.05 and K = N .
Figure 4. Relative difference between lower and upper bound of
1 N
Asian
calls as a function of σ and relative error of the lower bound compared to the true Monte Carlo value with parameters N = 20, T = 10, r = 0.05 and K = N . We use 100,000 Monte Carlo simulations. 15
Price as a function of !
0.035
Lower bound Upper bound Monte Carlo
0.03
Price
0.025
0.02
0.015
0.01
0
0.05
Figure 5. Price of
0.1
1 N
0.15
0.2
0.25 !
0.3
0.35
0.4
0.45
0.5
Asian calls as a function of σ with parameters N = 12,
T = 3, r = 0.05 and K = N . We use 100,000 Monte Carlo simulations.
The error made by using the lower bound as an approximation for the price of an Asian call seems to be negligibly small. Figure 4 plots the maximal error made by using the lower bound. In relative terms the error is quite small compared to the price of the contract. In Figure 5 we use a Monte Carlo simulation to approximate the true value and compare it with the bounds. In line with the results reported by Nielsen and Sandmann (2003), the lower bound performs better than the upper bound. As observed in Figure 4, the percentage error of the lower bound compared to the true Monte Carlo value is small.
Now we come to contract 4. In Figures 6, 7, 8 and 9 we plot the upper and lower bounds for contracts 3 and 4 for T = 3 and T = 10. Note, that in Figures 8 and 9 we know the exact value for contract 4. For a better comparison we also calculate the price for the two contracts with a Monte Carlo simulation. As expected introducing the additional restriction of contract 4 lowers the price. Our lower and upper bounds for contract 4 perform very well - even better than for contract 3! Surprisingly, we can show that for 16
0.055
0.03
Contract 3 upper bound Contract 3 lower bound Contract 4 upper bound Contract 4 lower bound Contract 4 Monte Carlo Contract 3 Monte Carlo
Contract 3 upper bound Contract 3 lower bound Contract 4 upper bound Contract 4 lower bound Contract 4 Monte Carlo Contract 3 Monte Carlo
0.05
0.045
Price
Price
0.025
0.04
0.02
0.035
0.015 0.03
0.01 0.1
0.15
0.2
0.25
0.3 !
0.35
Figure 6. Price of
.
0.4
1 N
0.45
0.025
0.5
Asian
0
0.05
0.1
0.15
0.2
0.25 !
0.3
Figure 7. Price of
0.35
1 N
0.4
0.45
0.5
Asian
calls of contract 3 and contract
calls of contract 3 and contract
4 as a function of σ with param-
4 as a function of σ with param-
eters N = 12, T = 3, r = 0.05,
eters N = 12, T = 10, r = 0.05,
K1 = N and K2 = 0.9. We use
K1 = N and K2 = 0.9. We use
250,000 Monte Carlo simulations.
250,000 Monte Carlo simulations. .
certain parameter values the price of the new contract is not monotonically increasing in σ any more. For low values of σ an increase in the volatility can actually decrease the price. In contract 3 the payoff was increasing in σ as only the positive effect of a higher risk was taken into consideration. On the contrary a higher σ in contract 4 has two effects. On the one hand there is the same positive effect on the payoff as in contract 3. On the other hand for a certain range of σ a higher volatility makes it also more likely that the final return is below K2 and hence the price becomes lower. In addition, longer maturities do not lead to too much changes in results concerning the quality of our approximation. 17
0.055
0.035 Contrac 3 upper boundt Contract 3 lower bound Contract 4 price Contract 4 Monte Carlo Contract 3 Monte Carlo
0.03
0.05
0.045
Price
Price
0.025
0.02
0.04
0.015
0.035
0.01
0.03
0.005
Contract 3 upper bound Contract 3 lower bound Contract 4 price Contract 4 Monte Carlo Contract 3 Monte Carlo
0.025 0
0.05
0.1
0.15
0.2
0.25 !
0.3
Figure 8. Price of
0.35
1 N
0.4
0.45
0.5
Asian
0
0.05
0.1
0.15
0.2
0.25 !
0.3
Figure 9. Price of
0.35
1 N
0.4
0.45
0.5
Asian
calls of contract 3 and contract 4
calls of contract 3 and contract 4
as a function of σ with the same
as a function of σ with the same
parameters as in Figure 6 except
parameters as in Figure 7 except
for K2 = 1.1.
for K2 = 1.1.
5. Comparing the different stock option schemes 5.1. The setup. The most interesting question is how “comparable” option schemes influence the manager’s risk-taking decisions. Imagine the following baseline scenario: The manager is granted a stock option compensation scheme. Typically such a compensation has a duration of 8 to 10 years but the performance-based components will not have a maturity longer than 3 years. We can think of such a scheme as a portfolio of performancevested stock options, each of which has a maturity of 3 years but with a different starting date. Hence, without loss of generality it suffices to consider the volatility effect on a single stock option with a maturity of 3 years. When the stock options are granted the company’s firm’s value process has a fixed volatility, e.g. σ = 0.2. Now the manager can decide about different investment opportunities or new strategies for the company. This will affect the volatility of the company and thus the payoff of his stock options. A simple example would 18
be the entry into a new market which offers potentially higher returns but also the possibility of failure. Here we assume that the manager can make his decision at time t = 0, which changes the volatility to a new constant value. Typically, these investment decisions would mean a tradeoff between risk and return, e.g. a higher expected rate of return is usually accompanied by a higher volatility. As prices are calculated under the risk-neutral measure, the expected rate of return µ does not directly enter the optimization problem. Hence, we assume here that the manager’s decision is based on the market value of his stock options. If the market value of the ESOs increases substantially by an increase in the volatility, this implies that the manager would more eagerly accept a new more risky asset.
The major challenge in comparing the different payoff schemes is to make the options as “equal” as possible. First, the payoff schemes must have the same costs for the company at time t = 0, i.e. we adjust the number of options in a portfolio of Asian or Parisian options in such a way that the price of the portfolio is equal to one plain vanilla call option. The next problem is that each option has certain parameters which do not have a counterpart in the other options. In the following we will fix the strike of a call option with maturity T equal to three years to be K = 1.2. This means that for example an average yearly return of 6% is sufficient to be in the money, which seems to be a realistic value. The Parisian option and the constrained Asian option which we labeled contract 4 are both call options with an additional condition on the path of the stock market value. Hence, it makes sense to fix the strike on the total return (i.e. K in the Parisian case and K2 for contract 4) to the value 1.2 as well. The parameter g and K1 still need to be chosen. The intuition should be that if the manager runs the company in such a way, that it has an average yearly return of 6 %, his stock options should be in-the-money. Therefore, we can choose
19
for example g = 0.02 and K1 = 1.0 which satisfy this condition. Usually companies publish information quarterly and natural choice for N would be N = 4T = 12. A realistic value for the excursion time of the Parisian option is a quarter or a half year (d = 0.25 or d = 0.5).
The plots in Figure 10 show the sensitivity of the normalized prices to the volatility8. We have successfully created two new contracts which encourage the manager to achieve the same final stock market value but lower his risk-taking incentives. Hence, these contracts provide shareholders with an effective instrument to control for the riskiness of CEO’s decisions. In comparison, the effect of Parisian ESO scheme is less substantial. We observe that the constrained Asian option (contract 4) is best suited to discourage managers from increasing the risk to higher values than the normalization point σ = 0.2. For lower values than σ = 0.2 the baseline Asian case seems to do a slightly better job. Note, that for values of σ between 0 and 0.1 the price function of contract 4 can have the undesirable feature of being humped-shaped (e.g. for K2 = 1.1), i.e. for very low values of σ there may be an incentive for the manager to further reduce the risk. But usually, a realistic range of volatility values would be between 0.1 and 0.3 such that this phenomenon is a minor problem.
Our findings are by no means trivial. One might argue that by taking a plain vanilla call option and imposing some additional restrictions on the path one would automatically reduce the sensitivity to the volatility. One might also think that increasing the excursion periods of a Parisian option or increasing the strike for the average value of an Asian
8In
the following we use the lower bounds as an approximation of the prices for Asian and constrained
Asian contracts. 20
option are simple ways to make these options less volatile to risk. These argumentations are only partly true as what really matters is not the behavior of a single option but the price of a normalized option which takes into consideration the number of options which makes the price of the portfolio equal some baseline value. By imposing more restrictions on the option, one typically reduces the sensitivity to the volatility of a single option but at the same time also reduces the price. Hence, in order to compare this option with an alternative, one has to increase the number of options in a portfolio which can counteract the initial effect. Therefore, imposing new restrictions has the desired effect (at least locally) only if the slope of the option around a reference value for σ decreases more than the price. Comparing the normalized price as a function of !
0.35
Contract 3 Contract 4 Plain vanilla call Parisian call exponential barrier Parisian call constant barrier
0.3
normalized price
0.25
0.2
0.15
0.1
0.05
0
0
0.05
0.1
0.15
0.2
0.25 !
0.3
Figure 10. Normalized prices of portfolios of
0.35
1 N
0.4
0.45
0.5
Asian calls of contract
3 and contract 4 and a single plain vanilla call option as a function of σ with the parameters N = 12, T = 3, r = 0.05, K1 = N , K2 = 1.2 and K = 1.2. The Parisian call with exponential barrier has the parameters S0 = 100, B0 = 100, g = 0.02, K = 1.2 and d = 0.5 while the Parisian call with constant barrier has S0 = 100, B0 = 100, g = 0, K = 1.2 and d = 0.25. 21
5.2. Mathematical formulation for risk-sensitivity. In the last subsection we observed that the price of a constrained Asian and a Parisian stock option is less sensitive to changes in the volatility than the corresponding call option. The crucial assumption is to make the different options comparable to each other by normalizing the number of options in a portfolio such that the total value of every portfolio is the same for a given reference level of σ. In this subsection, we formally derive the results exhibited in the above subsection.
Denote by f (σ) the price of a single option depending on the volatility. Recall, that in the last subsection we have chosen the number α of options of a certain kind (at the ESO-grating time) such that αf (σ0 ) = c, where c > 0 is a constant and in the above example it equals the value of a single plain vanilla call option. The slope of the portfolio is the decision criterion for the sensitivity of the options: the flatter the price curve of the portfolio at the reference level the less the price is influenced by changes in the volatility. The slope is given by αf 0 (σ)|σ=σ0 := αf 0 (σ0 ) =
cf 0 (σ0 ) . f (σ0 )
(8)
Table 1 delivers some number for this measure under different ESO schemes. First, both constrained Asian and Parisian scheme deliver smaller αf 0 (σ0 ) values than the benchmark case, i.e. the risk sensitivity is reduced by introducing the new schemes. Second, the higher the starting value of the volatility, the smaller this value. Third, the constrained Asian scheme does a better job than the Parisian scheme. In particular, the constrained Asian case with a high strike K2 for the absolute return seems to dominate all other payoff 22
schemes. Fourth, under the Parisian scheme, when the length of excursion becomes longer, the option becomes less sensitive. Furthermore, the last two columns for the plain vanilla call option indicate that imposing a stricter restriction does not necessarily mean that the sensitivity is reduced.
constrained Asian ρ
Parisian
Standard call
K2 = 1.2 K2 = 0.9 d = 0.25 d = 0.5 K = 1.2 K = 1.3
αf 0 (σ)|σ=0.2
3.2963
2.7842
4.8925
4.7564
5.5572
7.2878
αf 0 (σ)|σ=0.3
2.3174
2.2417
3.2832
3.2631
3.5234
4.2442
αf 00 (σ)|σ=0.2
-2.8393
2.6072
0.7353
1.6855
−0.5917
2.7407
αf 00 (σ)|σ=0.3
-1.7644
-1.0214
−0.3018
0.1424
−0.7473 −0.2934
Table 1. ρ(f, σ1 , σ2 ) for different ESOs. All ESOs have in common: T = 3, r = 0.05 and c = 1. For the Parisian case we have S0 = 100, B0 = 100, T = 3, r = 0.05, g = 0.02, K = 1.2 and for the constrained Asian case N = 12, K1 = 1.0.
We have argued that a low value of αf (σ0 ) is desirable, as it means that the value of the portfolio does not change locally for changes in the volatility. However, if αf (σ0 ) is only very low at σ0 it does not say that much about the changes in the value of the portfolio further away from σ0 . Hence, we look at the second derivative to see if the insensitivity holds in a neighborhood of σ0 . Thus, a low value of the second derivative is ”good” in this sense. If the second derivative is negative and αf 0 (σ0 ) is positive, this is also advantageous, as it implies that the sensitivity measure is getting closer to zero for higher σs. 23
The second derivative for different ESOs is illustrated in Table 1: ∂ 2 (αf (σ)) cf 00 (σ0 ) = ∂σ 2 f (σ0 ) σ=σ0 The constrained Asian ESO with high second strike price K2 exhibits the lowest sensitivity to increases in the volatility. A constrained Asian ESO with a low second strike price may have a low local sensitivity measure, but this increases (or decreases less) for higher volatility values compared to the former case. A similar result holds for the Parisian case. They have low local sensitivity values, which can eventually increase for higher volatilities. A plain vanilla call option with a high strike price is inferior, i.e. more risk-sensitive, than all the other options.
6. Conclusion We analyze two performance-vested executives’ stock option schemes: Parisian and Asian scheme. Both the Parisian scheme and the constrained version of the Asian option require a call option condition on the final payoff, and additionally impose restrictions on the path that the firm’s asset values can take. Parisian scheme requires the executives to ensure that the rate of return of the firm’s asset remains above a fixed level for at least d length of time, whereas the constrained Asian scheme requires them to achieve a certain average rate of return and simultaneously requires the total rate of return to outperform a reference level.
In order to compare different ESO schemes, we first choose the number of new ESOs which makes the price of the portfolio equal the value of a plain vanilla call option. Consequently, we fix an initial volatility level and examine how the market value of the ESOs changes when the manager decides to invest in a more risky (higher volatility) strategy. We 24
observe that under both new schemes, the market value of the ESOs falls below the vanilla call when the manager deviates to a more risky strategy. We show that both Parisian and Asian ESOs are less risk sensitive than the plain vanilla options. In other words, both schemes have the potential to prevent a firm’s executives from taking on too large risks and consequently make it also less likely for the executives to manipulate the executives’ stock options in their favor.
In this paper, the valuation of the new ESO schemes and the comparative statistics are both carried out under the risk neutral measure. The welfare implications in a utility-based framework are analyzed in a separate working paper of Chen and Pelger (2011).
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Appendix A. Valuation formulae of ESO under the Parisian scheme In this appendix, we apply the original “inverse Laplace transform” of Chesney et al. (1997) to compute the value of the manager’s Parisian ESO option. Note that S = sup {s ≤ t : S(s) = B(s)} gB,t
1 2 gs = sup s ≤ t : S(0) exp r − σ s + σWs = B(0)e 2 = sup {s ≤ t : S(0) exp {m σ s + σWs } = B(0)} = sup {s ≤ t : S(0) exp {σZs } = B(0)} = sup {s ≤ t : Zs = b} =gb,t . with m := σ1 (r − g − 12 σ 2 ), b :=
1 σ
ln
B(0) S(0)
and Zs := Ws + ms. {Zt }0≤t≤T is a martingale
under probability measure P , which is defined by the Radon-Nikodym density dP m2 T . = exp −mZT − dQ FT 2 Hence, the condition in (4) is equivalent to Tb+ := inf{t > 0|(t − gb,t )1{Zt >b} > d} < T.
(9)
Hereby we transform the event, the excursion of the stock price above the exponential barrier B(t) = B(0)egt , B(0) ≥ S(0), to the event, the excursion of the Brownian motion Zt above a constant barrier b. This simplifies the entire valuation procedure. Under the new probability measure P the asset price S(T ) can be expressed as n o 1 2 S(T ) = S(0) exp r − σ T + σWT = S(0) exp σZT exp{gT } 2 26
It is well known that the price of a T -contingent claim φ(S(T )) corresponds to the expected discounted payoff under the equivalent martingale measure Q, i.e., h i E e−rT φ(S(T ))1{T +
