Pricing executive stock options with averaging features under the Heston–Nandi GARCH model

• Author: Xingchun Wang,Zhiwei Su
• Published date: 01 September 2019
• Date: 01 September 2019
• DOI: http://doi.org/10.1002/fut.22036

J Futures Markets. 2019;39:1056–1084.wileyonlinelibrary.com/journal/fut1056

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© 2019 Wiley Periodicals, Inc.

Received: 19 April 2018

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Revised: 29 May 2019

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Accepted: 29 May 2019

DOI: 10.1002/fut.22036

RESEARCH ARTICLE

Pricing executive stock options with averaging features

under the Heston–Nandi GARCH model

Zhiwei Su

1

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Xingchun Wang

2

1

PBC School of Finance, Tsinghua

University, Beijing, China

2

School of International Trade and

Economics, University of International

Business and Economics, Beijing, China

Correspondence

Xingchun Wang, School of International

Trade and Economics, University of

International Business and Economics,

Qiuzhen Building, 100029 Beijing, China.

Email: xchwangnk@aliyun.com

Funding information

University of International Business and

Economics, Grant/Award Number:

CXTD9‐01; China Scholarship Council,

Grant/Award Number: 201806645036;

National Natural Science Foundation of

China, Grant/Award Numbers: 11671084,

11701084

Abstract

In this paper, we focus on the pricing issue of four types of executive stock options

(ESOs) in the Heston–Nandi generalized autoregressive conditional heteroskedas-

ticity option pricing model. Based on the derived benchmark strike prices in the

proposed framework, we obtain the closed‐form pricing formulae for four types of

ESOs. In the numerical part, we investigate the sensitivity and cost efficiency of

ESOs and suggest that systematic risk (stock β) and the fraction of wealth invested

in restricted stock could impede the cost efficiency of ESOs.

KEYWORDS

Asian options, executive stock options, GARCH models, indexed options

JEL CLASSIFICATION

G13, G30, J33, M52

1

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INTRODUCTION

Executive stock options (ESOs) are typically granted by corporations worldwide to link their executives’payments to

the stock performance in hope of better aligning the interest of executives with that of shareholders and alleviating the

principal‐agent problem (see, e.g., Haugen & Senbet, 1981; Jensen & Meckling, 1976). However, many studies provide

empirical evidence that the abnormal stock returns are negative before unscheduled ESO awards and positive afterward

(see, e.g., Chauvin & Shenoy, 2001; Lie, 2005). Heron and Lie (2007) conclude that most of the abnormal return pattern

around option grants is attributable to backdating of option grant dates, by exploiting the changes in the reporting

requirements for ESO grants. In addition, in some cases, executives are rewarded abundantly due to market movements

instead of their own efforts (see, e.g., Garvey & Milbourn, 2003). Indexed Asian ESOs (AESOs) may be more effective

than the traditional ones (see, e.g., Tian, 2013). In this paper, we consider ESOs with averaging features or indexing

features or both in the generalized autoregressive conditional heteroskedasticity (GARCH) framework and provide

closed‐form pricing formulae of four types of ESOs, based on the derived explicit benchmark strike prices.

ESOs have been studied heavily in the existing literature. Meulbroek (2001) brings forward the portfolio concerns

when assessing the cost efficiency of ESOs. Exploiting a GARCH framework, Duan and Wei (2005) take a different

perspective by breaking down the total risk into idiosyncratic and systematic components and demonstrate that option

values depend on the amount of systematic risk. Chong, Ding, and Li (2015) investigate ESOs in the stochastic volatility

framework using the adjusted binary tree model. Under a principal‐agent framework, Hall and Murphy (2000) and Hall

and Murphy (2002) acknowledge the gap between option costs and subjective values of a nontradable option. Dittmann,

Maug, and Spalt (2010) and Dittmann, Maug, and Spalt (2013) further investigate the optimal compensating structure

for the executives and conclude that existing option compensating schemes are not of the best arrangement.

ESOs are commonly used by corporations and are proclaimed to align the interest of executives with that of

shareholders. However, Tian (2004) shows that the incentive to increase stock price does not always increase as more

options are granted, based on a utility‐maximization framework. In addition, the author finds that granting options to

executives creates incentives to reduce idiosyncratic risk but creates incentives to increase systematic risk, and the

combined risk incentive can be either positive or negative, which depends on the characteristics of the firm and the

executives. Ross (2010) also shows that granting options to executives does not necessarily make them more willing to

take risks. Further, the author finds necessary and sufficient conditions under which incentive schedules make

executives more or less risk averse. Chaigneau (2013) argues that the regulator should set compensation to provide first‐

best incentives. In static models, the literature states that the structure of managerial compensation matters for risk

taking (see, e.g., Dittmann & Maug, 2007). By taking into account the fact that risk taking can be adjusted dynamically

over time, Leisen (2015a) studies dynamic risk taking by a risk‐averse manager for a given bonus scheme, and motivates

a power bonus scheme that incentivizes the manager to implement the socially optimal risk level. Leisen (2015b)

studies continuous‐time risk taking of an expected utility maximizer under bonus schemes without/with deferral, and

extends the single‐period analysis in Ross (2010) to a continuous‐time analysis.

To reduce excessive compensation, the indexing scheme has been proposed in the literature. Namely, the indexed

option pays off only if the firm’s stock price exceeds a specified benchmark strike price. Johnson and Tian (2000) design

a European‐style ESO with a benchmark strike price and derive the pricing formulae in the Black–Scholes model.

Jørgensen (2001) extends the framework in Johnson and Tian (2000) and allows for the early exercise feature to price

American‐style indexed ESOs (IESOs). However, Garvey and Milbourn (2003) provide evidence that IESOs are less cost

effective than vanilla ESOs due to asymmetric benchmarking in compensation. Concretely, the executive’s pay is

benchmarked more often when the market is down than when the market is up, indicating that such an indexing

scheme does not achieve its intended purpose. In a recent paper, employing a principal‐agent analysis, Tian (2013)

incorporates the averaging feature in the payoff structure of ESOs in the Black–Scholes model, and considers both the

cost and incentive efficiency of AESOs.

1

In this paper, we consider ESOs with averaging features or indexing features or both in the GARCH framework. The

discrete‐time GARCH models (see, e.g., Bollerslev, 1986; Byun, Jeon, Min, & Yoon, 2015; Chiang & Huang, 2011; Duan

& Wei, 2005; Heston & Nandi, 2000; Ritchken & Trevor, 1999) can capture the volatility clustering and leverage effects.

Moreover, as noted by Duan (1997) and Nelson (1990), the continuous‐time stochastic volatility models of Heston and

Nadi (2000), Hull and White (1987), and Scott (1987) can in fact be approximated by GARCH processes.

In this paper, following Heston and Nandi (2000) Duan and Wei (2005), we adopt the GARCH processes to describe

the market index and stock price dynamics. Following , we break down the underlying price risk into systematic and

idiosyncratic parts, allowing us to better investigate the effects of systematic risk.

2

However, different from Duan and

Wei (2005), we do not assume the one‐factor model for the stochastic discount factor, and do not use the benchmark

strike price derived in the Black–Scholes model. Instead, we derive the corresponding benchmark strike price in the

proposed GARCH framework. Additionally, motivated by the capital asset pricing model (CAPM), we adopt the βto

represent the underlying asset’s sensitivity to systematic risk. Tzang et al. (2016) work under a similar CAPM–GARCH

model to study the impact of systematic risk on volatility skew, under the assumption that unsystematic risks can never

be completely eliminated and there is a nonzero idiosyncratic risk premium. Different form Tzang et al. (2016), we

assume that idiosyncratic risk can be diversified and not being paid. Additionally, following Heston and Nandi (2000)

and Duan and Wei (2005), we directly write the returns of the market index and the underlying asset in the form of risk‐

free interest rate plus a return premium compensating the systematic risk. In the proposed framework, we derive the

explicit pricing formulae for ESOs with averaging features or indexing features or both by ignoring the possibility of

early exercise of the options and assuming away the departure risk of the executives. Actually, we follow Duan and Wei

(2005) and Tian (2013) to treat ESOs as a European option. The derived pricing formulae extend the existing pricing

formulae for Asian options in Kemna and Vorst (1990) and indexed options in Johnson and Tian (2000) by taking time‐

varying variances into consideration. Finally, we numerically compare the differences between the prices in the

Black–Scholes model and the ones obtained in the proposed GARCH framework, and investigate the sensitivities and

the cost efficiency of ESOs.

1

Note that we use AESOs and ESOs with averaging features/schemes interchangablely.

2

Throughout this paper, we refer to stock βas systematic risk following the literature (see, e.g., Duan & Wei, 2005; Tian, 2013).

SU AND WANG

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