Hurdle rate: executive stock options.

Abstract:

Executive stock options with a rising strike price are a recent innovation in executive compensation in Australia and New Zealand. These options combine a dividend protection feature and a strike price that increases at a hurdle rate set with reference to a cost of capital estimate. With a constant dividend yield, the strike price becomes a path-dependent function of the stock price and exact analytic valuation becomes intractable. However, path-dependent American options can be valued using a Monte Carlo approach proposed in Longstaff and Schwartz (2001). We examine procedures for valuing these options and compare them with Black and Scholes (1973) and Merton (1973) formula valuations.

Keywords:

EXECUTIVE STOCK OPTIONS; MONTE CARLO METHODS.

1. Introduction

Performance and reward measures such as economic value added (EVA) that explicitly take into account the cost of capital have gained worldwide acceptance. (1) Several prominent companies in Australia and New Zealand have extended this concept to the design of executive stock options with two distinguishing features: (1) a strike price that drifts upward at a hurdle rate usually set with reference to a cost of capital estimate; and (2) dividend protection that adjusts the strike price downward by the amount of any dividends paid. We refer to these executive stock options as hurdle rate options. Companies trading on the Australian Stock Exchange (ASX) or New Zealand Stock Exchange (NZX) that have issued hurdle rate options include Fisher and Paykel Appliances, Sky City Entertainment Group, Pumpkin Patch, Santos Limited, and others.

A traditional executive stock option plan grants call options on a firm's shares, with a strike price typically fixed at the share price on the grant date. (2) Since the grantee benefits by any increase in the share price, such options have often been criticised as being too generous. Critics complain that stock options with a constant strike price can unfairly reward executives for broad market advances irrespective of individual firm performance. A hurdle rate option represents an attempt to link executive compensation to individual company performance, as exercise is only profitable when the cum-dividend return on the firm's shares surpasses a predetermined hurdle rate.

Standard executive stock options normally do not include dividend protection. Since dividend payments reduce the share price, executives holding stock options may have an incentive to keep dividend payments low. Hurdle rate options remove this adverse incentive via their dividend protection feature.

As an example of a typical hurdle rate option, in 2004 Fisher and Paykel Appliances (FPA) proffered an executive stock option grant to its Managing Director and Chief Executive Officer that specified annual hurdle rate and dividend adjustments to the strike price. Salient key terms for the option grant were:

* The exercise price will be calculated using a base share cash price adjusted to account for the company's cost of capital and deducting dividends paid.

* On the grant date a 'base price' will be determined by the volume-weighted average share price for the 10 business days prior.

* At each anniversary of the grant date a new base price will be calculated by multiplying the last base price by a percentage amount representing the company's cost of capital (anticipated to be between 10% and 12% per annum); and reducing the resulting figure by the amount of any cash dividends paid (not including any imputation credits) in the immediately preceding 12-month period.

* However, to ensure that the base price does not decrease, if there were circumstances where the base price is less than the last calculated base price, the new base price shall be the last calculated base price.

* One third of the options granted become exercisable on each of the second, third and fourth anniversaries of the grant date and all unexercised options expire on the fifth anniversary of the grant date.

* Options are not transferable and do not participate in dividends or other distributions of the company.

The FPA grant limits the dividend adjustment to no more than the hurdle rate adjustment. Thus, the strike price can never be reset below a previous level. As a practical matter, this provision is unlikely to have much effect, since hurdle rates set according to the cost of capital are normally higher than observed dividend yields, and so strike prices for hurdle rate options are usually monotonically increasing.

While the key features of the FPA grant attempt to better align the interests of executives with shareholders, they also present a daunting valuation task for financial analysts or anyone interested in valuing these options. Valuation difficulties arise from two factors: 1)regular increases in the strike price may trigger early exercise even though the options are dividend-protected; and 2) dividend payments linked to the share price yield a strike price that is a path-dependent function of the share price. A hurdle rate option valuation formula based on Black and Scholes (1973) and Merton (1973) is presented in section 2. However, this formula is an approximation to a considerably more difficult valuation problem. In section 2 we also discuss the difficulties involved in applying standard option pricing methodology to value hurdle rate options.

In section 3, we present a practical method to value hurdle rate options based on numerical methods originally developed in Bossaerts (1989) and refined in Longstaff and Schwartz (2001). In actual practice (e.g. financial statement reporting), these options are typically valued using a variation of the formula of Black and Scholes (1973) that includes an adjustment for dividends similar to that specified by Merton (1973). However, a simple Black-Scholes-Merton formula may incorrectly value hurdle rate options because no account is made for the path dependency of the dividend-adjusted strike price or for the optimal early exercise that may occur anytime after vesting. Our conclusions and a summary follow in section 4.

2. Black-Scholes-Merton Formula Valuation

The development of an arbitrage-free formula for the value of a European call option by Black and Scholes (1973) revolutionized financial derivative securities valuation. One measure of the importance of this development is the immense number of subsequently published research papers developing option pricing theory and methods that augment and extend the original Black-Scholes framework to a broad spectrum of theoretical and practical applications. Other classic theoretical papers that bear directly on the content of this paper are Merton (1973), Magrabe (1978), Fischer (1978), Geske, Roll and Shastri (1983), and Longstaff and Schwartz (2001).

2.1 Early Exercise for a Dividend-Protected Call Option

Merton (1973) shows that a dividend payment may trigger optimal early exercise of an unprotected American call option. However, hurdle rate options are dividend protected. Geske, Roll and Shastri (1983) show that if the stock price falls by the amount of a dividend payment, then a dividend-protected American call option will not be exercised prematurely. Their argument is straightforward. If the stock price and strike price fall by the same amount then the value of exercise immediately before or after the dividend payment is the same. Thus, if early exercise is not optimal immediately after a dividend payment, it is also not optimal immediately before the payment.

2.2 Early Exercise for a Call Option with an Increasing Strike Price

In a continuous time framework, Margrabe (1978) and Fischer (1978) show that it is not optimal to exercise an American-style call option on a non-dividend paying stock when the exercise price grows at a rate no greater than the riskless interest rate. However, this result does not extend to a discrete time framework. To see this, suppose a current stock price is $105 and an option on the stock has a strike price of $100 that is about to be increased by $5 to reflect a 5% riskless interest rate. If exercised immediately before the adjustment the option yields a payoff of $5, which may be more or less than the value of the option after the strike price is increased to $105. If the ex-adjustment option value is less than $5, the option would be optimally exercised before the adjustment.

2.3 Early Exercise for Hurdle Rate Executive Stock Options

Hurdle rate executive stock options have strike prices adjusted in discrete time, usually annually, to reflect a hurdle rate and any dividends paid since the last adjustment. For example, the Fisher and Paykel options described above have their strike prices adjusted annually; first upwards to reflect a cost of capital estimate, then downward by the amount any dividends paid. Thus, letting K denote the strike price immediately before the adjustment, a hurdle rate adjustment of H and a dividend adjustment of D yield a strike price immediately after the adjustment of K + H - D. Exercise is triggered when the value of immediate exercise exceeds the value of the option after the adjustment; that is, when S - K > C(S - D, K + H - D), where C(S - D, K + H - D) denotes the value of the call option at the new share price of S - D and strike price of K + H - D immediately after the strike price adjustment.

With fixed hurdle rate adjustments and a known sequence of dividend payments, option valuation using a Black-Scholes-Merton formula is straightforward using the formula stated immediately below, in which C denotes the call option value, [S.sup.*] and [K.sup.*] denote the dividend adjusted stock price and the dividend and hurdle rate adjusted strike price, h is the hurdle rate, r is the riskless interest rate, T is the time to option maturity, [sigma] is the annual return volatility of the stock, and N(d) indicates the standard normal distribution probability for the value d.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

In equation (1) above, [S.sup.*] is computed as the initial stock price less the present value of a known sequence of dividends, i.e., [D.sub.1], [D.sub.2], ..., [D.sub.T-[epsilon]], where [D.sub.T-[epsilon]] is the last dividend payment made before option expiration. [K.sup.*] is the initial strike price reset by a known sequence of hurdle rate and dividend adjustments.

The Black-Scholes-Merton formula in equation (1) represents the value of a European call option with a future sequence of dividends known on the grant date. (3) The assumption of known dividends is plausible over a short horizon of, say, a year or two. However, executive stock options are granted with maturities typically ranging from five to ten years. Over such long horizons, the assumption of known dividends is unrealistic. A more plausible assumption consistent with observed practice is that of a constant dividend yield, where the cash dividend is a constant percentage of the observed share price. However, the assumption of a constant dividend yield has the effect of making the strike price a path-dependent function of the stock price. Path dependency generally complicates option valuation considerably. Indeed, path dependence and the possibiliaty of early exercise make exact analytic valuation of hurdle rate options intractable. (4)

Since an exact solution for the value of hurdle rate options presents intractable analytic difficulties, we turn instead to numerical Monte Carlo methods. Monte Carlo simulation was formally introduced as an option pricing methodology by Boyle (1977). For some time, the Monte Carlo approach has not always been regarded as suitable for valuing American-style options. However, recent advances in numerical methods permit accurate Monte Carlo valuation of American options. In this paper, we adopt the least squares Monte Carlo (LSM) approach proposed by Longstaff and Schwartz (2001). This approach allows path-dependent European and American options to be valued via Monte Carlo simulation experiments. The Monte Carlo approach is often considered inefficient because it is based on brute force. However, the speed and power of modern desktop computers is now such that accurate price calculations can be accomplished within seconds.

3. Least Squares Monte Carlo Valuation

The Black-Scholes-Merton formula stated in equation (1) does not properly value hurdle rate options when the future sequence of dividend payments is uncertain. To value hurdle rate options in the case of uncertain dividend payments, we use a Monte Carlo approach. Monte Carlo simulation has long been used to estimate the conditional expectation of the payoff of a European-style option (see Boyle 1977). However, for some time it was not clear how American options could be valued using Monte Carlo simulations. This was so because to value an American option it is necessary to compare the payoff from immediate exercise at each decision date with the expected payoff from continuation without exercise. Longstaff and Schwartz (2001) credit Bossaerts (1989) for providing the earliest solution to the problem of valuing an American option with Monte Carlo simulations. Further development is provided in Tilley (1993), Barraquand and Martineau (1995), Carriere (1996), Broadie and Glasserman (1997), Broadie, Glasserman, and Jain (1997), Raymar and Zwecher (1997), and Carr (1998). Longstaff and Schwartz (2001) make the important contribution of developing a parsimonious algorithm that significantly increases the efficiency of the algorithm. Moreno and Navas (2003) and Stentoff (2004) find that the least squares Monte Carlo approach proposed by Longstaff and Schwartz (2001) is quite robust and provides an efficient method to value American options when alternative methods are infeasible.

Indeed, the least squares Monte Carlo approach is sufficiently flexible to account for a host of factors that might affect the value of executive stock options. For example, Carr and Linetsky (2000) and Szimayer (2004) examine changes in the payoff structure of executive stock options caused by the early departure of an executive or a company takeover. Szimayer (2005) examines the impact of event risk on executive stock option exercise strategies and option values in a general setting. In his model, the occurrence of an event terminates the option's promised payoff schedule and is replaced with a rebate determined by the event. Default risk and career change are examples for this type of event. (5)

The key insight of the least squares Monte Carlo (LSM) approach to estimating conditional expectations is that the information required for estimating expected payoffs from continuation without exercise is already contained in the set of simulated price paths. To process this information, the LSM approach uses a cross-sectional least squares regression of in-the-money options on each decision date. Fitted values from the cross-sectional regression provide estimates of the expected payoffs from continuation without exercise. Comparing these expected payoffs with the value of immediate exercise results in an exercise strategy that maximizes the simulation value of the option.

Let [X.sub.k,t] denote the non-negative intrinsic value of the option on date t along the [k.sup.th] simulated price path. Also let [V.sub.k,t] be the optimal value of the option on date t along the [k.sup.th] path. The LSM algorithm begins by setting option values equal to intrinsic values at maturity, i.e., [V.sub.k,T] = [X.sub.k,T]. Then the set of option values at date T along the price paths with strictly positive intrinsic values at date t-1, that is, the set with [X.sub.k,T-1] > 0, are regressed on a polynomial function of the strictly positive intrinsic values [X.sub.k,T-1]. Fitted values from the polynomial regression represent conditional expected values used to make an optimal exercise decision. Early exercise is optimal if a fitted value is lower than the corresponding value of immediate exercise, in which case option value is updated by the immediate exercise value. Specifically, if option intrinsic value at date T-1 along the [k.sup.th] path is greater than the discounted expected value from continuing without exercise, then the option value [V.sub.k,T-1] is set equal to the intrinsic value [X.sub.k,T-1]; otherwise, option value is set equal to the discounted value of continuing without exercise, i.e., [e.sup.-r][V.sub.k,T], where r is the riskless single-period discount rate. Options with zero intrinsic value on date T-1 along the k path that were not included in the regression also have their values set equal to the discounted value from continuing without exercise.

[V.sub.k.T-1] = {[K.sub.k,T-1] if [X.sub.x,T-1] > [e.sup.-r] E([V.sub.k,T]) [e.sup.-r][V.sub.k,T] otherwise

The procedure is repeated at each decision date recursively from option maturity back to the original grant date. In our simulations, optimal early exercise opportunities are checked with each dividend payment made after the end of the vesting period.

The dependent variable in our polynomial regressions comprises the discounted values of continuing without exercise, while the independent variables are polynomials of the strictly positive values of immediate exercise. The use of polynomials as independent variables poses the question as to which type of polynomial and what degree is appropriate. Moreno and Navas (2003) examine the robustness of the least squares Monte Carlo (LSM) algorithm and report that different polynomials of the same degree all yield very similar results. They also report that polynomial degrees between 3 and 20 also yield similar option prices. Stentoff (2004) found that while Legendre and Laguerre polynomials yield similar accuracy, Legendre polynomials have a small advantage in computational speed over Laguerre polynomials. In our LSM calculations we use the IMSL routine RCURV, which uses orthogonal Legendre polynomials as independent variables. Like Moreno and Navas (2003), we found that computed American option values tended to increase slightly as the polynomial degree was increased. For reasons of simplicity and conservatism, we report results obtained with 3rd degree polynomials. In general, using a 10th degree polynomial yielded American option values about one percent higher than those obtained with 3rd degree polynomials. Of course European option values are unaffected, since LSM is not used to calculate their prices.

Stentoff (2004) reports that increasing the number of simulated stock price paths not only reduces the standard error of an option price estimate, but also reduces any potential bias in the LSM algorithm. Each option value reported below is based on one million simulated stock price paths performed using a compiled Fortran program on a desktop computer with an Intel Pentium CPU running at 2.8 GhZ. No more than a few seconds were required to calculate each price.

3.1 Applying the Least-Squares Monte Carlo Approach to Value Hurdle Rate Options

Table 1 reports option contract values obtained from Monte Carlo simulation experiments along with values calculated using the Black-Scholes-Merton formula in equation(l). To implement equation(1) with a constant dividend yield assumption, dividends are computed by their expected value under a risk-neutral probability measure. This yields a dividend at time t of [D.sub.t] = y[Se.sup.(r-y)t]. Values reported in Table 1 are based on 1,000-share contracts with stock and strike prices of S = K = $1, a contract maturity of T = 6 years with 2-year vesting, an interest rate of r = 6%, and a volatility of [sigma] = 25%. In Table 1, we set the hurdle rate at 10% and let dividend yields vary from 0% to 8% in increments of 2%.

The accuracy of the Monte Carlo simulation methodology can be assessed by comparing prices in the first row of Table 1 obtained with a zero dividend yield. In this case, equation (1) yields an exact option price. The European-style option price of $157.68 obtained by simulation is 19 cents, or 0.012% higher than the exact price of $157.49. This difference is economically insignificant as it is well within the standard tick size of one dollar for option contracts traded on the Australian stock exchange. (6)

When the dividend yield is zero, the Black-Scholes-Merton formula in equation (1) yields an exact European option value. However, with a non-zero dividend the formula is not exact and becomes increasingly inaccurate as the dividend yield increases. For instance, with a 10% hurdle rate and a 2% dividend yield the Black-Scholes-Merton formula in equation (1) yields a value of $135.56, which is 9.91% lower than the European option value of $150.48 obtained by simulation.

The least squares Monte Carlo (LSM) methodology also reveals that early exercise premiums can be substantial, even with a zero dividend yield. For example, in table 1 with a zero dividend yield and 2-year vesting the American option price of $171.04 is $13.36 larger than the European option price of $157.68. With a 2% dividend yield the early exercise premium is $13.11 = $163.59 - $150.48, which is 8.71% of the European option value. Thus when accurate valuation that accounts for the possibility of early exercise is critical, the LSM methodology is preferred. The LSM methodology can account for the combined effects of path dependency and early exercise, without which severe valuation biases may ensue.

Notice that the 'dividend-protection' feature of hurdle rate options does not provide complete protection. Table 1 shows that while American option value is $171.04 with a zero dividend yield, option value falls monotonically as the dividend yield increases. This is true for both European and American hurdle rate options. Consistent with this phenomenon, the early exercise premium rises with the dividend yield. In table 1, the early exercise premium rises from $13.36 = $171.04-$157.68 with a zero dividend yield to $31.67 = $136.57 - $104.90 with 8% dividends.

Table 2 reports option contract values with varying hurdle rates. These values are based on 1,000-share contracts with stock and strike prices of S = K = $1, a contract maturity of T = 6 years with 2-year vesting, an interest rate of r = 6%, and a volatility of [sigma] = 25%. The dividend yield is y = 3% and hurdle rates vary from 0% to 12% in increments of 3%.

In table 2 we see that the Black-Scholes-Merton formula in equation (1) is most accurate when the hurdle rate is set equal to the riskless interest rate. However, in practice hurdle rates are normally set with reference to a cost of capital estimate, which are generally higher than the riskless interest rate. In this case, as the hurdle rate rises above the riskless interest rate, values calculated from equation (1) become increasingly inaccurate. For example, with a hurdle rate of 9% equation (1) yields an option value of $142.91, while the European option value obtained by simulation is $164.04--almost 15% higher. The American option value of $174.11 is almost 22% higher.

Table 3 reports executive stock option values for options with maturities ranging from 3 to 10 years. These values are based on 1,000-share contracts with stock and strike prices of S = K = $1, an interest rate of r = 6%, a volatility of [sigma] = 25%, a dividend yield of y = 3%, a hurdle rate of h = 10%, a vesting period of two years and maturities ranging from 3 to 10 years.

Table 3 reveals how European and American option values rise with increasing maturities. For example, with a maturity of just 3 years the European option value obtained by simulation is $123.38. This European option value then rises to $143.45 with a 5-year maturity and then to $167.94 with a 10-year maturity. The corresponding American option values rise from $127.53 with a 3-year maturity, to $153.61 with a 5-year maturity, and then to $190.14 with a 10-year maturity.

4. Conclusion and Summary

Hurdle rate options granted by Australian and New Zealand companies provide an interesting example of an innovative design for executive stock options that explicitly takes into account the cost of capital in setting rewards for managers. In this paper, we have examined problems associated with valuing hurdle rate executive stock options. Valuation difficulties arise because the strike price of a hurdle rate option is a path-dependent function of the stock price whenever dividend payments are linked to the stock price. To overcome these valuation difficulties we adopt a least squares Monte Carlo (LSM) approach recently proposed by Longstaff and Schwartz (2001).

Our analysis reveals that using a Black-Scholes-Merton formula to value hurdle rate options often leads to potentially significant valuation biases. This result is important, as Black-Scholes-Merton valuation formulas feature prominently in recognised accounting standards and are routinely used to value executive stock options. Results from numerical experiments suggest that the Black-Scholes-Merton formula typically undervalues hurdle rate options by a substantial amount.

The exact magnitude of the bias depends on particular parameter values. Security analysts or anyone interested in knowing an accurate value for executive stock options should be aware of these potential biases and exercise caution when using a Black-Scholes-Merton formula to estimate the value of hurdle rate executive stock options.

(Date of receipt of final transcript: May 10, 2005. Accepted by Doug Foster & Garry Twite, Area Editors.)

References

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Broadie, M., Glasserman, P. & Jain, G. 1997, 'Enhanced Monte Carlo estimates for American option prices', Journal of Derivatives, vol. 5, pp. 25-44.

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Hall, B.J. & Murphy, K.J. 2000, 'Optimal exercise prices for executive stock options', American Economic Review, vol. 90, pp. 209-14.

Longstaff, F.A. & Schwartz, E.S. 2001, 'Valuing American options by simulation: A simple least-squares approach', Review of Financial Studies, vol. 4, pp. 1131-47.

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(1.) For example, see www.eva.com for a partial list of companies worldwide that have adopted the EVA measure.

(2.) Hall and Murphy (2000) report that 95% of option grants to S&P 500 executives in 1998 were at-the-money grants of plain vanilla options.

(3.) With known dividends, the valuation of an American option allowing optimal early exercise is readily accomplished using standard binomial tree methodology. The use of a binomial tree to value an option was originally proposed by Bachelier (1900); a modern reference is Cox, Ross, and Rubinstein (1979).

(4.) Strike price resets are not new to Australia and New Zealand. Gray and Whaley (1999) examined the reset feature of a put option attached to some geared equity products sold by Macquarie Bank. Another example of an executive stock option with a strike price that is a path-dependent function of the stock price occurs with the common practice of option repricing, whereby the strike price is reset to be equal to the stock price after a fall in the stock price. Corrado, Jordan, Miller and Stansfield (2001) provide an option pricing formula for options subject to multiple repricings.

(5.) See, for example, Elliot, Jeanblanc, and Y or (2000), Bielecki and Rutkowski (2001), and Blanchet-Scalliet and Jeanblanc (2004).

(6.) Option prices are commonly reported on a per-share basis. On this basis the difference between Monte Carlo and formula prices would only be a few cents.

Joe Cheung, University of Auckland, New Zealand.

Charles Corrado, Department of Commerce, Massey University, Albany, Private Bag 102 904 NSMC, Auckland, New Zealand. Email: [e-mail]

J.B. Chay, Sung Kyun Kwan University, Korea.

Do-Sub Jung, Sun Moon University, Korea.

Table 1

Hurdle Rate Option Contract Values with Varying Dividend Yields

Option contract values are based on 1,000-share contracts
with stock/strike prices S = K = $1, contract maturity T = 6
years with 2-year vesting, interest rate r = 6%, volatility
[sigma] = 25%, hurdle rate h = 10%, and dividend yields as
indicated. Monte Carlo prices are averages from one million
simulation experiments. The assumed dividend at time t used in
equation (1) is [D.sub.t] = y[Se.sup.(r-y)t].

Dividend     Monte Carlo Prices    Black-Scholes-Merton
Yield (y)                            Prices--Eq. (1)

            European    American

0%           $157.68     $171.04        $157.49
2%           $150.48     $163.59        $135.56
4%           $142.13     $156.52        $116.79
6%           $127.45     $148.81        $100.78
8%           $104.90     $136.57         $87.17

Table 2

Hurdle Rate Option Contract Values with Varying Hurdle Rates

Option contract values are based on 1,000-share contracts
with stock/strike prices S = K = $1, contract maturity T = 6 years
with 2-year vesting, interest rate r = 6%, volatility [sigma] = 25%,
dividend yield y = 3%, and hurdle rates h as indicated. Monte Carlo
prices are averages from one million simulation experiments. The
assumed dividend at time t used in equation (1) is
[D.sub.t] = y[Se.sup.(r-y)t].

Hurdle     Monte Carlo Prices    Black- Scholes-Merton
Rate (h)                            Prices--Eq. (1)

           European   American

0%         $271.54    $276.46        $341.69
3%         $259.44    $263.99        $269.00
6%         $218.80    $223.41        $201.50
9%         $164.04    $174.11        $142.91
12%        $115.11    $135.09         $95.54

Table 3

Hurdle Rate Option Contract Values with Varying Maturities

Option contract values are based on 1,000-share contracts
with stock/strike prices S = K = $1, interest rate r = 6%,
volatility [sigma] = 25%, dividend yield y = 2%, hurdle rate
h = 10%, a vesting period of two years and maturities as indicated.
Monte Carlo prices are averages from one million simulation
experiments. The assumed dividend at time t used in equation (1)
is [D.sub.t] = y[Se.sup.(r-y)t].

Maturities (7)   Monte Carlo Prices    Black-Scholes-Merton
                                          Prices--Eq. (1)

                 European   American

3                 $123.38    $127.53         $116.67
4                 $134.67    $142.18         $125.54
5                 $143.45    $153.61         $131.53
7                 $155.87    $171.48         $138.16
10                $167.94    $190.14         $140.35