Production, Manufacturing and Logistics
Mean–variance analysis of option contracts in a two-echelon supply chain

https://doi.org/10.1016/j.ejor.2018.05.033Get rights and content

Highlights

  • Studying option contracts with risk considerations in a two-echelon supply chain.

  • Constructing the mean–variance models of option contracts.

  • Investigating the channel coordination of option contracts with risk constraints.

  • Designing a new minimum option quantity commitment for the supplier.

Abstract

This paper studies the implications of risk considerations for option contracts in a two-echelon supply chain. Under the mean–variance framework, we first investigate the conditions for coordinating the supply chain by using option contracts. We find that supply chain coordination is not always achieved, contrasting with the result that properly designed option contracts can always coordinate a supply chain in the absence of risk considerations. Second, we analyze the Stackelberg game for a decentralized supply chain in two cases, depending on whether the retailer's risk aversion threshold is known to the supplier. We show that when the threshold is public information, there exists a unique equilibrium in which the supplier with a higher risk tolerance prefers to reduce the exercise price, and thus, the retailer's order quantity increases. When the retailer's risk aversion threshold is private information, the retailer has an incentive to pretend to be less risk averse. To curb this incentive distortion, we design a new minimum option quantity commitment for the supplier. We complement our theoretical results with numerical simulations.

Introduction

Risks are pervasive in firms’ operational decisions, and thus, risk management plays a vital role in the success of firm operations. Indeed, inappropriate risk management may lead to significant financial losses. For example, rapidly weakening demand coupled with locked-in supply agreements incurred a $2.5 billion inventory write-off for Cisco Systems, Inc. in the second quarter of 2001 (Norrman & Jansson, 2004). In the third quarter of 2001, Nike lost $100 million in sales revenue due to an inventory shortage (Norrman & Jansson, 2004). Therefore, incorporating risk factors into supply chain decisions has drawn heightened attention from practitioners. For instance, Hewlett-Packard established a procurement risk management system to evaluate and control supply chain risks. Through this system, Hewlett-Packard saved at least $100 million in sourcing costs in 2008 (Nagali et al., 2008). Although firms have begun to realize the importance of supply chain risk management, determining how to make a tradeoff between profit and risk remains a significant challenge.

Previous studies have mainly focused on two types of supply chain risk: disruption risks, such as those associated with wars, earthquakes, diseases and terrorist attacks (Qi, Bard and Yu, 2004, Ray and Jenamani, 2016, Sodhi, Son and Tang, 2012), and operational risks, such as those emanating from supply reliability and demand uncertainty (Fan, Li, Sun and Cheng, 2017, Liu and Nagurney, 2011, Wei and Choi, 2010, Xue, Choi and Ma, 2016, Zeng and Yen, 2017). Researchers have proposed various methods for managing a supply chain under risk constraints. The most widely used method is the mean–variance framework, originating from Markowitz's portfolio theory. Markowitz originally proposed the mean–variance framework to analyze risk diversification of financial assets and to help investors design an optimal portfolio (Markowitz, 1959).

The mean–variance framework is widely explored within the realm of operational decisions to address various supply chain risks, particularly those arising from uncertain market demand (Chiu and Choi, 2016, Choi and Chiu, 2012, Choi, Li and Yan, 2008a, Liu, Cao and Salifou, 2016, Tomlin, 2006, Wu, Li, Wang and Cheng, 2009). Specifically, Choi et al. (2008a) carry out a mean–variance analysis for the newsvendor problem in which decision makers are risk-averse, risk-neutral, or risk-taking. They analytically investigate the effective frontiers for each case. With the same objective function, Choi and Chiu (2012) study the mean-downside-risk and mean–variance newsvendor models for both the exogenous and endogenous retail price cases. They find that the retailer orders the same stocking amounts in the mean-downside-risk and mean–variance models. Some studies have also analyzed how to achieve supply chain coordination within the mean–variance framework (Chiu, Choi and Li, 2011, Choi, Li and Yan, 2008c, Choi, Li, Yan and Chiu, 2008b, Gan, Sethi and Yan, 2004, Gan, Sethi and Yan, 2005, Wei and Choi, 2010). For instance, Choi et al. (2008b) study the coordination of a buyback contract under the mean–variance framework. They find that channel coordination is not always achievable under risk constraints, in contrast to results indicating that the conventional buyback contract can always coordinate a supply chain (Lee and Rhee, 2007, Pasternack, 1985, Tsay, 2001). Chiu et al. (2011) investigate the channel coordination problem for a target sales rebate contract in which supply chain parties make decisions based on a mean–variance analysis. Wei and Choi (2010) explore the coordination of both a wholesale price and a profit-sharing contract under the mean–variance model. They analytically characterize the necessary and sufficient conditions under which supply chain coordination is achieved. The present paper contributes to this line of research by providing a mean–variance analysis of option contracts, with a focus on supply chain coordination and equilibrium analysis for a two-echelon supply chain.

Contract arrangements provide a useful risk management mechanism in supply chains under demand uncertainty. For example, option contracts allow a buyer to determine how much to purchase according to the realized market demand, while providing a supplier with upfront payments. Option contracts have seen widespread application in a number of industries, including IT, telecommunications, semiconductors, and electricity (Anderson, Chen and Shao, 2017, Wu and Kleindorfer, 2005). For example, many giant companies, such as IBM, Sun Microsystems and Hewlett-Packard, have taken a portfolio procurement strategy with options (Tsay & Lovejoy, 1999). In addition, option contracts have been used in the agriculture industry (Wang and Chen, 2017, Zhao, Ma, Xie and Cheng, 2013). For example, to facilitate vegetable sales in Ishikawa, Japan, some local farmers established an agriculture agency as their representative to negotiate with vegetable buyers such as stores and restaurants. The agency has offered the option contracts of vegetables to its buyers since 2008. The buyers first place an option quantity before the growing season and pay 4% of the total value as deposits. The farmers then grow the vegetable based on the buyers’ orders. During the selling season, the buyers purchase an amount of the vegetable up to the option quantity from the farmers at a pre-determined exercise price to satisfy the realized market demand. Fig. 1 illustrates this transaction process.

An increasing number of studies have focused on the application of option contracts to procurement risk management (Anderson, Chen and Shao, 2017, Liu, Chen and Li, 2014, Nosoohi and Nookabadi, 2016, Paul, Sarker and Essam, 2016, Sawik, 2016, Wang, Chu, Wang and Kumakiri, 2012, Wang, Tang and Tsao, 2006, Wu and Kleindorfer, 2005, Zhao, Ma, Xie and Cheng, 2013, Zhao, Choi, Cheng and Wang, 2018). Zhao et al. (2013) examine the feedback effects of the bidirectional option on the retailer's initial order strategy. Nosoohi and Nookabadi (2016) investigate the manufacturer's decisions in two stages under call, put and bidirectional options. Wang et al. (2006) analyze a call option contract and show that this contract improves the buyer's performance. Wang et al. (2012) show that in a two-stage model, the buyer has a higher expected profit in the first stage, whereas the supplier may have worse performance in the second stage compared with the case without option contracts. Those studies compare the optimal decisions relating to option contracts with the conventional newsvendor model within a risky environment. However, little research has examined how risk (as an exogenous factor) affects the supply chain decisions involved in option contracts. Introducing risk constraints into a mean–variance model, we examine supply chain coordination and optimal decisions for option contracts. We find that channel coordination depends on the retailer's risk attitude and may not always be achieved, in contrast to results indicating that properly designed option contracts can always coordinate a supply chain (Gomez_Padilla and Mishina, 2009, Wang and Liu, 2007, Wang et al., 2015, Zhao et al., 2010).

In this paper, we study the mean–variance model under option contracts in a two-echelon supply chain. We investigate the tradeoff between profit and risk faced by supply chain parties. Our proposed mean–variance model of option contracts is in sharp contrast to that of a buyback contract presented by Choi et al. (2008b) and that of a profit sharing contract presented by Wei and Choi (2010). Table 1 summarizes the difference of the three contracts with risk constraints.

In terms of model setup, the supplier in Choi et al. (2008b) only decides the returns price b while the wholesale price is fixed. The manufacturer in Wei and Choi (2010) decides the wholesale price w and the profit sharing ratio λ. In our option contract, the supplier decides the option price o and exercise price e. These three contracts therefore provide the supplier with different degrees of control in achieving supply chain coordination. As a result, the coordination outcomes are different as well. In Choi et al. (2008b), a higher buyback price offers the retailer higher expected profit and lower risk but imposes on the supplier lower expected profit and higher risk. The results seem inconsistent with classical investment theory, in which a high risk is often accompanied by a great expected profit. Wei and Choi (2010) show the necessary and sufficient conditions that coordinate the supply chain, and they characterize the equilibrium for the Stackelberg game in a decentralized supply chain when the proportion of the profit and risk sharing is predetermined. Our proposed model, however, shows that when the supplier (retailer) bears a lower risk, she (he) obtains a lower expected profit. In addition, in our mean–variance models, the proportion of risk sharing between the supplier and retailer is not predetermined; instead, it is determined by the option price and exercise price. By changing the prices, the allocation of expected profit and risk sharing will be altered. Our results are in line with the observation that with a higher risk tolerance, the supplier always prefers to reduce the exercise price, and the retailer increases the option quantity to enjoy more profit.

We summarize the key results of our paper as follows. We first study the channel coordination of option contracts with risk constraints. We find that supply chain coordination depends on the retailer's risk tolerance and, more importantly, may not always be achieved. We next study a decentralized supply chain in which the supplier and retailer are maximizing their own profits subject to risk constraints. We consider two cases: (1) the retailer's risk aversion threshold is public information; and (2) the threshold is private information of the retailer. In the former case, we find that changing the option price and exercise price reallocates expected profit and risk sharing between the retailer and the supplier. We also show that a unique equilibrium exists in the Stackelberg game, and the equilibrium outcomes depend largely on the supplier's and retailer's risk tolerance levels. In the latter case, we first show that the retailer benefits from pretending to be less risk averse. To avoid untruthful information reporting on the part of the retailer, we propose a minimum option quantity commitment for the supplier.

The contribution of our paper is twofold. First, our paper contributes to the option contract literature by investigating the implication of risk considerations for option contracts. In doing so, we develop insights into how risk considerations shape the coordination results and equilibrium outcomes. Second, this paper contributes to the literature on contract design with risks by studying supply chain coordination and equilibrium analysis in an option contract setting. In particular, we study both symmetric and asymmetric information settings in which the retailer's risk attitude may be known or unknown to the supplier.

The rest of this paper is organized as follows. Section 2 describes the notation and assumptions. Section 3 studies supply chain coordination with option contracts in the mean–variance model. Section 4 conducts the equilibrium analysis in the decentralized case under both symmetric information and asymmetric information cases. Section 5 concludes and provides some management insights.

Section snippets

Model description

We consider a two-echelon supply chain consisting of a supplier and a retailer in which the newsvendor-like retailer orders products from the supplier to satisfy an uncertain demand X. The probability density function (PDF) of X is f(x), the cumulative distribution function (CDF) is F(x), and the complementary CDF is F¯(x). The mean and standard variance of X are u and σ, respectively. For convenience, we refer to the supplier as “she” and the retailer as “he” in the following analysis.

We are

Supply chain coordination

In this section, we examine whether or not supply chain coordination can be achieved and, if achievable, under what conditions. Following the approach for supply chain coordination under the mean–variance model (Choi et al., 2008b), the supplier is supposed to play the role of supply chain coordinator. The supplier's objective is to set an optimal option price and an optimal exercise price such that the retailer's order quantity equals the supply chain's optimal quantity.

Decentralized supply chain

In this section, we study the strategic interaction between the supplier and the retailer in a decentralized supply chain. For this setting, each player's objective is to maximize their own expected profit subject to their respective risk constraints. In the centralized case, the information about the retailer's risk aversion threshold is common knowledge for the players. However, in the decentralized case, the supplier may not know this information. Therefore, we consider two information cases

Conclusion and management insights

In this paper, we analyze supply chain coordination and option contract design under the mean–variance model. In this model, each party aims to maximize their expected profits subject to constraints on the risk. Our main results are fourfold.

First, the supply chain is not always coordinated under option contracts with risk constraints. By leveraging the exercise price, the supplier achieves the channel coordination only when the retailer's risk aversion threshold falls within certain intervals.

Acknowledgments

This research is supported by the National Natural Science Foundation of China under Grant Nos. 71571065, 71790593 and 71521061; the Program for New Century Excellent Talents in University under Grant No. NCET-13-0193; and the Ministry of Education in China of Humanities and Social Science Project under Grant No. 14YJA630077.

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