extfieldsofstudy:
Computer Science
Psychology

text:
How context and design shape human-robot trust and attributions

In increasingly digitized working and living environments, human-robot collaboration is growing fast with human trust toward robotic collaboration as a key factor for the innovative teamwork to succeed. This article explores the impact of design factors of the robotic interface (anthropomorphic vs functional) and usage context (production vs care) on human–robot trust and attributions. The results of a scenario-based survey with = N 228 participants showed a higher willingness to collaborate with production robots compared to care. Context and design influenced the trust attributed to the robots: robots with a technical appearance in production were trusted more than anthropomorphic robots or robots in the care context. The evaluation of attributions by means of a semantic differential showed that differences in robot design were less pronounced for the production context in comparison to the care context. In the latter, anthropomorphic robots were associated with positive attributes. The results contribute to a better understanding of the complex nature of trust in automation and can be used to identify and shape use case-specific risk perceptions as well as perceived opportunities to interacting with collaborative robots. Findings of this study are pertinent to research (e.g., experts in human–robot interaction) and industry, with special regard given to the technical development and design.

Introduction
Responding to global trends, human-robot collaboration (HRC) is on the rise in many areas [1]. Especially areas which are affected by demographic change on the one hand and digitization on the other, as health care and production. In an aging society where there are fewer working citizens and more people in need of care [2] while, at the same time, innovative advances in automated technology become a tangible experience [3], the use of autonomous robots provides relieving assistance for future working and living environments [4].
Today, collaborative robots are already an integral and essential part of manufacturing industries where, for example, routine tasks in assembly lines are automated using robots under human supervision and control [5]. Next to gains in efficiency and effectiveness, robot roles and responsibilities include the execution of strenuous, risky, and dangerous tasks, such as handling heavy, toxic, or dangerous objects, to increase safety and enhance human labor [6]. Considering advanced HRC, humans and robots interact in close physical proximity by sharing their workspace [6]. This particularly applies to health care where robots can be used for daily nursing services (e.g., cleaning, feeding, and bathing), physical and cognitive coaching (e.g., physiotherapy or memory training), as well as psychological and social caregiving (e.g., communication and conversation to prevent loneliness in old age) [7].
Automation, however, may lead to risk perceptions among users due to several reasons, such as lacking experience [8], technology misuse [9], media priming [10], or the fear of innovation [11]. Also, especially in fragile usage contexts, as e.g., health care, the idea of being cared for by autonomous robots can, under some circumstances, be incompatible with the requirements of a human and human care which roots in a deep bond between caregiver and patient [12]. In this context, trust is a key factor for the development of users' acceptance toward automated technology and crucial for users' decisions to accept and use or reject a technology [12,13].
Previous studies on trust evolving and moderating factors are manifold (see, for example, [8,[14][15][16][17]), including the development of measurement scales to assess trust in automation. Yagoda and Gillan [18] provided a tool specially tailored to the many facets of humanrobot interaction (HRI) relevant to trust (e.g., context, task, and system), particularly suitable for empirical-experimental research to measure trust in human interaction with an automated robotic system. However, trust is often measured using Likert scales, leaving out other valuable instruments useful for a deep understanding and measurement of individual perceptions and mental models, such as semantic differentials which we have therefore included in this study.
Robot design and appearance has been revealed as a decisive factor for trust across different research settings, usage contexts, and user needs. Special research focus was given to anthropomorphism, meaning that the design of a robot, its physical form, but also its behavior, and the way it interacts mimics humans [19,20]. To examine the influence of anthropomorphism on peoples' attitudes and trust toward robots, embodiment was manipulated and evaluated in regard to acceptance, including humanlike (anthropomorphic), zoomorphic, caricatured, and functional designs [21]. In general, robot appearance (from anthropomorphic to functional) influenced users' likeability and empathy: less human looking robots were less liked and users' empathy was stronger toward anthropomorphic robots [22,23]. Further studies revealed that anthropomorphic design features (e.g., facial expression, gestures, communication) increased humans' trust and intention to use robots [24,25].
Particularly in social use cases, robot appearance seems to be important. Regarding care, Stuck and Rogers [26] found that older adults consider robot materials and clothing as trust indicators, presumably due to increased body contact during nursing. They further identified context-specific trust dimensions and features, such as easy handling and gentleness as a preferred robot characteristic [26]. Interestingly, using a robot in health care is not a clear-cut "yes" or "no" decision [12]. In a study, in which the participants were asked for which specific care situations they would prefer human vs robotic care, situations were identified in which humans were preferred (e.g., giving medication, spoon-feeding), but also care situations in which the care robot was selected as the care authority (e.g., putting someone to bed, helping someone to visit the toilet). Apparently, it is not only the context per se which shapes the acceptance of robotic support but also the specific function for which the robotic care is applied for [12]. To realize personalized robotic care, Portugal et al. [27] developed a mobile service robot with flexible function adaptive to (changing) user needs. The robot design considered size, shape, color, and acoustics to enable multimodal services for advanced HRI, including facial expressions for robotic emotional interaction [27]. The robot was perceived more machine-like than humanlike, which did not limit its acceptance but was subject to the demand for additional design elements (e.g., arms) [27], which altogether emphasizes the relevance of the functional and at the same time visual robot design.
In other contextual settings (organizational group work, for example), trust depended on environmental and situational factors (e.g., team collaboration and task type) [15]. These findings indicate a strong impact of the concrete application and usage context on users' perception and evaluation of HRC. However, most studies have primarily focused on single applications and usage contexts. Research that compares trust perceptions toward robots across different contexts and domains to identify key predictors is still rare and so far limited to other application fields, such as smart home and autonomous driving [28]. The present research connects to this research gap.

Questions addressed and logic of the empirical approach
The aim of this study was to derive scientifically based conclusions, with regard to robot design and behavior, for the development of use case-specific robots in order to increase peoples' trust and the willingness to use robots in different contexts. Our research addresses the user-centered evaluation of autonomous robots, depending on contextual and design factors. We study peoples' trust, use intention, and differences in attribution of autonomous robots in a care vs a production scenariotwo particularly relevant application fields for using robots due to global challengestaking into account design factors that have already been identified as trust decisive.
Due to a lack of knowledge of the influence of contextual factors on human-robot trust, we used an explanatory and structure discovering mix of methods, including a twostep empirical approach for the development and validation of theses. Based on qualitative data from preceding focus group discussions, we developed a quantitative online questionnaire to answer the following research questions: 1. How is the use of robots perceived and evaluated in the care vs production context? 2. What impact do contextual and design factors have on human-robot trust? 3. Which attributes do users associate with HRC?

Method
Data and questionnaire items ( Table 1) provided in the following sections have been translated from German. If not stated otherwise, the scales in the survey are based on self-developed items resulting from the qualitative preliminary study.

Empirical research design
The questionnaire consisted of three parts: first, we surveyed the participants' demographics (e.g., age, gender, and educational background) as well as explanatory factors, such as the affinity toward technology [29], attitude toward robots [30], interpersonal trust [31], and the disposition to trust robots (adapted from Beierlein et al.'s scale [31]).
Second, we introduced care and production as two usage contexts for robots and we measured the participants' intention to interact with the robots [32]. Also, we asked participants for their estimation of the role robots will play in these areas in the future (context as withinsubject factor). Here we wanted to know which developments in regard to HRC the participants expect and whether they personally are critical or optimistic toward this development.
Third, we presented two different robotic designs in two different usage scenarios to the participants and asked for an evaluation of the robots. Here we used a × 2 2 mixed design with Context (production vs care) as between-subject and Design (anthropomorphic vs functional) as within-subject factor. For each of the factors, we created a scenario with a textual description and an image illustrating the scenario. For both, the texts and the images, we ensured that they were similar in regard to factors not under investigation (for example, they had the same perspective, same image style, and same colors) and different only in regard to the experimental factors (that is, context and design). Figure 1 shows the robots presented in the survey for both designs and contexts. In this research design, we used two attributes for each of the two dimensions to keep the required sample size and duration of the survey reasonable.
We measured the effect of both factors by asking the participants whether they trusted the robot and for their perception of the robots. For measuring trust, we used the five items from Jian et al.'s scale [33] with the highest itemtotal correlation. For measuring the perception, we used a semantic differential that we developed through brainstorming sessions and focus groups. It consists of 22 items and captures various functional and nonfunctional aspects of the robots as social agents (e.g., reliable-unreliable, pretty-ugly, warmhearted-coldhearted; see Table 2).
We used 4-point Likert items (ranging from = min 0 I don't believe that/I don't appreciate that to = max 3 I believe that/I appreciate that) and 6-point (ranging from = min 0 totally disagree to = max 5 totally agree) as well as 6-point semantic differentials (with pairs of two opposing adjectives) to collect the responses. The order of the pairs was randomized in the survey but recoded to the canonical order for further analyses (lower numbers indicating more positive attributions). Responses were voluntary (no forced choice).
Sample acquisition took place in Germany. We distributed the online link to the questionnaire through our personal social networks by personal address, instant messaging, e-mail as well as Facebook and other computer-mediated social networks without incentives. As we did not seek a specific target group, we were interested in collecting answers of a broad random audience in order to receive an unbiased public perception on HRC scenarios. The median completion time of the survey was 16 min. Figure 2 illustrates the experimental design.

Description of the sample
In total, 388 participants started the web-based survey. Yet we had to exclude 160 participants of which the vast majority has not completed the survey (e.g., opened the survey without participating) or rushed through the survey. The rate of rejected responses is typical for this sampling technique [34].
The final sample ( = N 228) consisted of 125 women (54.8%) and 99 men (43.4%); 4 participants provided no gender information (1.8%). Age ranged between 15 and 80 years ( = M 34.6; = SD 14.4; = n 227). With 80 high school leavers (35.1%) and 71 university graduates (31.1%), the educational level was comparatively high [35]. The participants were slightly trustful toward other people ( = M 2.86; = SD 1.14; range 0-5) and had an affinity for technology ( = M 3.18; = SD 1.47; range 0-5). Asked for prior care experience, 12.7% ( = n 29) of the participants cared for other people regularly, followed by  [29]) I like to try out the functions of new technical systems I enjoy spending time getting to know a new technical system I try to understand exactly how a technical system works There will be more autonomous robots Perception (Believe/Judgment) Humans remain important in the future Robots will increasingly replace human labor Robots will increasingly assume responsible tasks Robots will relieve humans but not replace them Humans and robots will collaborate well I welcome the use of robots Use Intention (cf. [32]) I would like to collaborate with robots I consider robots to be useful I am suspicious of the system's actions Trust in Automation (shortened from [33]) The system's actions will have a harmful or injurious outcome The system is reliable The system is dependable I can trust the system 19.3% ( = n 44) who cared for another person in the past and 30.3% ( = n 69) care witnesses; 37.7% ( = n 86) had no care experience. Regarding production experience, 6.1% ( = n 14) of the participants worked in production contexts at the time of questioning, while 18.4% ( = n 42) worked there before, followed by 24.1% ( = n 55) with nonworking insights. The majority of the sample (51.3%, = n 117) reported to have no prior experience in production.
Regarding the use of toy, household, or industrial robots, 7.0% of the participants ( = n 16) collaborated with robots at work, followed by 11.8% ( = n 27) who used robots both in private and in business and 35.1% ( = n 80) who experienced robots in their free time. Nearly half of the participants (46.1%, = n 105) had no experience with robots.
The participants' disposition to trust robots was only slightly above the scale's midpoint ( = M 2.53; = SD 1.57; range 0-5). Their highest agreement was for the statement that one cannot rely on robots ( = M 3.08; = SD 1.47). Thus, the participants had only limited trust in robots (see Figure 3, upper part).
In regard to the general attitude toward robots (before any introduction of the concrete usage scenarios), the participants found that robots are necessary to relieve humans from heavy or dangerous work ( = M 3.97; = SD 1.35) but also that robots require careful handling ( = M 3.86; = SD 1.36; see Figure 3, lower part, range 0-5). The least agreed item from the scale was whether robots will lead to more employment opportunities ( = M 2.39; = SD 1.54): on the contrary, there is some fear that robots might destroy jobs ( = M 3.00; = SD 1.56). The question whether robots are good for the society was rated just slightly above the center of the scale ( = M 3.13; = SD 1.39). In summary, and as depicted in Figure 3, the participants see certain advantages of robots, on the one hand; on the other hand, disadvantages and risks, which justifies a deeper investigation of this phenomenon.

Results
Descriptive and inferential statics were used for data analysis. The level of significance (α) was set at 5%. For effect sizes, the partial eta-square (η 2 ) and Cohen's d were reported.
The results section is structured as follows: first, we analyze the expectations and evaluations regarding the future role of robots and use intentions in care vs production. Then we describe the influence of contextual and design factors on human-robot trust. Subsequently, to better understand the cross-domain trust dimensions, we compare how context and design shape properties attributed to the robots using the semantic differential.

Expectations and evaluations of future HRI
First, we analyzed the expectations of robots in care and production on two different scales (range 0-3): the question was whether the participants believed that the same scenarios for robot use (presented as items, such as there will be more autonomous robots) apply to the contexts of care and production (scale 1: Do you believe that?) and further, whether they considered them good or bad (scale 2: Do you appreciate that?).
According to the first scale, expectations about future HRC differed significantly between the usage contexts ( Figure 4a). Particularly as regards the rise of robots, item ratings showed strong differences, ( ) < t p 11.04 , 0.001, = d 0.731, provided that the use of robots was more likely assumed in the production scenario (    = d 0.684, and robots will increasingly assume responsible tasks, ( ) < t p 8.10 , 0.001, = d 0.536: in both cases, agreements were higher for the production scenario. In contrast, the assumption that humans remain important in the future was particularly expected for care, (− ) t 8.21 , < p 0.001, = − d 0.543. According to the second scale, evaluations of future HRC only partially differed significantly in the two contexts (Figure 4b). This included there will be more autonomous robots, ( ) < t p 3.93 , 0.001, = d 0.260, which was more likely accepted in production ( = M 1.87; = SD 1.06) than in care ( = M 1.55; = SD 1.15); although in both contexts, the level of agreement was rather low and only just above the scale center. Also, evaluations as regards robots will increasingly assume responsible tasks differed significantly, ( ) < t p 4.51 , 0.001, = d 0.298, with little appreciation in either scenario, particularly in care. This was different regarding humans remain important in the future: evaluations differed significantly, (− ) < t p 4.56 , 0.001, = d 0.302, but were favorable in both contexts. The participants appreciated a good teamwork of humans and robots, ( ) < t p 5.83 , 0.001, = d 0.386, particularly in production, though.
These findings point to an ambivalence between what the participants believed and what they thought was good: they expected an increased use of robots in the future, particularly in production, whereas the overall appreciation was rather restrained, especially in care which was considered to remain human. However, participants also indicated and supported an expected relief through the use of robots in working environments and were generally positive about future HRC.
In a next step, we analyzed the use intention of HRC (range 0-5) which, on an average, differed significantly  267. In addition, correlation analyses revealed significant relations between user factors and the intention to use robots in care and production. In particular, the participants' disposition to trust robots, attitude toward robots,  technology affinity as well as their highest educational attainment correlated positively with the use intention in both contexts (see Table 3).

Impact of context and design on human-robot trust
Next we analyzed whether the two presented usage contexts (production and care) and both robotic designs (anthropomorphic and functional) influenced the trust people attributed to the robots. Hereto, we calculated a repeated-measures (RM) analysis of variance (ANOVA) using context as the between-subject variable, the design as the within-subject variable, and perceived trust as the dependent variable (Section 3.1). Surprisingly, neither the usage context, ( )= , meaning that although we could not identify a direct effect of both factors, the combination of both context and shape affected trust in automation. As Figure 6 shows, the participants trusted robots with a functional appearance in production environments significantly more than anthropomorphic robots or robots in the care context. The effect by context and design (measured on the trust in robots scale) is small. We assume that this is due to the lack of an actual interaction with the robotic system and due to the missing reference frames in the scenario-based survey. Nonetheless, we consider this difference as a starting point to further investigate how design and context affect the robots' perception.

Perception of robots
As trust was influenced by context and design, we now analyzed whether design and context influenced the participants' attributions of the robots. Hereto, we calculated an RM-ANOVA with context as the between-subject variable, design as the within-subject variable, and average attribution from the 22-item semantic differential as the dependent variable. The difference in perception between the functional and anthropomorphic designs was less pronounced for the production scenario than for the care scenario. While the usage context as the between-subject factor had no effect on the perception of the robots, As Figure 7a shows, the functional design was perceived as more usable, more predictable, and stronger than the anthropomorphic design in the production scenario. In contrast, the anthropomorphic design was seen as much more warmhearted, curios, appealing, pretty, and social (see Figure 7a). Figure 7b illustrates that the differences between the functional and the anthropomorphic robot design were much bigger in the care context: here, the functional design was perceived as much stronger, but it was also much more associated with negative aspects than the anthropomorphic design. Specifically, the participants rated the functional design as rather evil, disgusting, and ugly. They believed that the robot is uninterested, unimaginative, and critical, while also thinking that it is inoperable. Finally, they rated the functional design as disagreeable, reserved, unsocial, and coldhearted.
In summary, despite some differences between anthropomorphic and functional design in the production context, the differences were greater in the care domain. In regard to care robots, the participants had rather negative views on robots with functional design, whereas the anthropomorphic design received more positive attributions. Table 2 lists the 22 items and respective arithmetic means of the semantic differential for evaluating the robotic designs and scenarios.

Discussion
This study revealed deeper insights into the topic of HRI by comparing two rising application fields: production and health care. Key findings indicate that collaborative robots are not only perceived differently across the investigated contexts but also depending on their physical shape and appearance. This demonstrates the importance of application-specific designs and evaluations, especially regarding research on and engineering of humanrobot trust. In the following, we discuss the research findings respecting their contribution to the state of the art and their implications for academia and industry.
The willingness to collaborate with robots was higher in the production compared to the care context. Also, expectations for the future use of robots differed considerably between the contexts: the participants expected a large increase in robotic coworkers in productiondifferent than in care. In line with previous research [36], it is not particularly surprising that expectations of robots in careas a highly intimate, socially interactive spaceare ambivalent: despite perceived benefits, roles and responsibilities were strongly considered to remain human. Regarding HRC in production, there seems to be a discrepancy between the participants' expectation of robots and their willingness to use them in practice: interestingly, participants indicated to like and be willing to collaborate with Table 2: Items and respective means of the semantic differential from 0 to 5 points for evaluating the robotic designs and scenarios. Items with (r) were reversed in the survey. Lower numbers represent more positive evaluations.

User factors UI care UI production
Attitude toward robots 0.472*** 0.513*** Trust disposition toward robots 0.350*** 0.419*** Technology affinity 0.226** 0.323*** Education 0.182** 0.169* robots in production, whereas they also showed only restrained support for actual robot use in this context (increasing robot use was commonly presumed but less appreciated). A possible explanation for this finding can be derived from our qualitative focus groups which were run prior to the questionnaire study: here, perceptions of production robots were closely related to a functional perspective by thinking of technical workflows or objects as in the manufacturing industry, for example, possibly giving the reduction in human workload a strong motive for use, while at the same time the fear of losing one's job became apparent. This is supported by Lotz et al. [37] who identified job loss as one key anxiety for HRC in industry. Therefore, it is advisable to look into this issue in more detail, including trade-offs between perceived barriers and benefits but also potential usage incentives. Concerning human-robot trust, we find that context and design are influencing factors. However, neither the area of use nor the robot appearance was decisive in themselves, but taken together they determined users' trust perceptions. Apparently, there is no design for all based on the context or based on the more or less anthropomorphic appearance. Rather, the very specific combination of application context, the robot's role, and its appearance seems to form human trust toward the robotic assistance. Robots with a functional design in production gained the highest level of trust. In care, anthropomorphic robots were perceived most trusting, reflecting the participants' attitude of care is human. This also becomes apparent with regard to perceptual differences in the scenario evaluations: in production, anthropomorphic design features were not an obstacle; but in the care context, they were essential. For the robot develop-ment, it is of great interest to further identify the design factors that are (and are not) accepted in detail and also to better understand users' (dis)trust in automation contexts. With a deeper look at the findings of the semantic differential, some dimensions can already be derived as initial input relating to the robot's visual appearance and aesthetics but also (supporting Stuck and Rogers [26]) to social attitudes, soft skills, and personality traits. Therefore, it is important for future works not only to look at robot design (What exactly is perceived as (un)attractive?) but also to consider which behavioral patterns (e.g., sociability) and characteristics (openness, creativity, maliciousness, etc.) might unfold a positive or negative influence on the perception and evaluation of HRC in specific use cases. These findings could then be integrated into research on the behavior design of robots for social interaction [38] but also on the cognitive capabilities of robots and their role as social agents as to recognize the needs and moods of the users and interact with them more intuitively [39,40] towardan improved and accepted HRC. Besides, such findings could also serve as an argument or caution against excessive humanization in robot design, since it has already been shown that robots that appear too human are perceived as deterrent (uncanny valley effect [41]).
With regard to the influence of individual differences on trust attribution patterns, particularly trust dispositions and attitudes toward robots were positively related to the use intention in two presented contexts. This confirms previous studies on the relation between trust and use intention [13], demonstrating the importance of trustexploring research in rising automation contexts. When it comes to the use of innovative robot technology, future research should investigate not only (dis)trust-related factors but also the extent to which these may be intrinsically motivated and related to people's interpersonal trust perceptions as basis for psychometric analyses (cf. [42,43]). As regards attitudes toward robots, the results obtained ( Figure 3) are similar to a study of the European Commission [30], demonstrating the overall positive and intentionally focused (e.g., in terms of (work) relief) assessments but also concerns, particularly as regards the handling of robots, which may be due to lacking experience. Follow-up studies are needed to gain deeper insights into attitude-related dimensions and identify potential predictors of the use intention. In this regard, implicit attitudes (as important implications for trust in automation [44]) should also be considered, possibly by taking into account the pairs of attributes set up in the semantic differential.
From a methodological perspective, the semantic differential as a preliminary scale for assessing robots' "personalities" represents a further contribution of this study. The scale with its adjectival pairs and dimensions has been developed on the findings of our focus group discussions and reacts sensitive in certain areas depending on robot design and usage context ( Table 2). Of course, the scale requires further refinements, but it is already a good starting point for other researchers in this field. Measuring trust and perceptions of robots is hard and offering more than Likert-type scales is mandatory to increase the participants' motivation and to capture broader aspects of HRI. Therefore, empirical approaches are necessary, which make it possible to measure users' trust and acceptance in close interaction with robots, such as in experimental trials in which robot materials, expressions, and gestures can be experienced in real-life environments.

Limitations
Of course, this study is not without its limitations. First, the sample of the current study is not representative for the general population. Nevertheless and despite its homogeneity, we were able to identify differences in the perception of robots in the two different contexts and between the two designs. Thus, we assume that the identified effects will get more pronounced in a larger and more representative sample. Of course, this is subject to a future study. However, as a starting point, it will be important that user factors are precisely addressed and balanced as regards the sample construction. This study's sample was rather young and we could not investigate age-related differences in robot perceptions. Consequently, further studies should investigate how age shapes the perception of robots or attributions. However, novel HRC in production and care is only slowly leaving academia and entering real-world environments. Consequently, the studied sample may not represent the current workforce but rather the future workforce of actual HRI scenarios. In this context, other user factors, as, e.g., innovation openness, risk behavior, technical self-confidence, or even anxiety, might be relevant personality constructs that might interact with the willingness to accept robotic assistance. In addition, care experience, gender, age, as well as aging concepts could significantly impact the HRC acceptance [45,46]. Furthermore, a comparison of perspectives could reveal exciting insights, for example, in care, by considering not only patients as future users but also caregivers as future coworkers [47]. This, of course, requires further studies that are focused on the specific task, context, and target audience, whereas this work studied the broader influence of context and design on perception and attribution.
Second, we used a scenario-based survey method instead of confronting participants with actual robots to measure robot trust and perception. Although studying real interactions would have been delightful, this approach enabled us to study a significantly larger sample. Furthermore, the visual illustrations and the textual descriptions of the robot usage scenarios enabled the participants to imagine their interaction with the robots and develop their judgments accordingly. In addition, the subjects evaluated only two of the four scenarios (between-subject design), which limits statistical procedures and underestimates potential effects. On the other hand, this shortened the length of the questionnaire and the participants were probably more motivated to respond conscientiously. In sum, this should have increased the quality of the quantitative data the study builds on. Yet further studies should validate the findings with tangible and interactive robots.
Third, in this study, we have only investigated two different design alternatives and two different usage scenarios. Next to design variations (e.g., zoomorphic, caricatured [21]), the context should continue to be considered in a broader sense. On the one hand, this refers to the integration of further automation contexts (e.g., smart living environments [48]) and, on the other, to the investigation of cultural and country-specific differences as the acceptance of robot technology and trust may be motivated both personally and socially [22,49]. Since the participants of this survey were acquired in Germany, it would be of great interest to examine the extent to which the obtained key findings are different or similar as regards the perception and evaluation of test persons with diverse cultural backgrounds, experience in dealing with, and attitudes toward robots, especially as there are already indications for cultural differences in users' likability and trust toward robots between Germany and Asian cultures, for example [22].


extfieldsofstudy:
Economics
Mathematics
Computer Science

text:
Semi-Static and Sparse Variance-Optimal Hedging

We consider hedging of a contingent claim by a 'semi-static' strategy composed of a dynamic position in one asset and static (buy-and-hold) positions in other assets. We give general representations of the optimal strategy and the hedging error under the criterion of variance-optimality and provide tractable formulas using Fourier-integration in case of the Heston model. We also consider the problem of optimally selecting a sparse semi-static hedging strategy, i.e. a strategy which only uses a small subset of available hedging assets. The developed methods are illustrated in an extended numerical example where we compute a sparse semi-static hedge for a variance swap using European options as static hedging assets.

INTRODUCTION
Semistatic hedging strategies are strategies that are composed of a dynamic (i.e., continuously rebalanced) position in one asset and of static (i.e., buy-and-hold) positions in other assets. Such hedging strategies have appeared in mathematical finance in several different contexts: the hedging of Barrier options (cf. Carr, 2011), model-free hedging approaches based on martingale optimal transport (cf. Beiglböck, Henry-Labordère, & Penkner, 2013), and-most relevant in our context-the semistatic replication of variance swaps by Neuberger's formula (cf. Neuberger, 1994). Compared with fully dynamic strategies, semistatic strategies have the advantage that no rebalancing costs or liquidity risks are associated with the static part of the strategy and hence even assets with limited liquidity can be used as static hedging assets.
Remarkably, for certain hedging problems, semistatic strategies allow for perfect replication even in incomplete markets-at least theoretically. Again, the most prominent example is the replication formula for a variance swap, given by Neuberger (1994) and Carr and Madan (2001) and discussed in more detail in Section 2.4: In any continuous martingale model, a variance swap can be replicated by dynamic hedging in the underlying and a static portfolio of European put-and call-options. This very replication formula is at the heart of the computation of the volatility index VIX, whose value is determined from a discretization of Neuberger's replicating option portfolio (cf. Chicago Board Options Exchange (2019)).
However, Neuberger's result relies on certain idealizations: Most importantly, the static part of the strategy consists of infinitesimally small positions in an infinite number of puts and calls with strikes ranging from zero to infinity. Any practical implementation of this strategy therefore has to decide on a certain quantization of the theoretical strategy, i.e., how to assign noninfinitesimal weights to the actually tradable put-and call-options. Rather than doing this in an ad hoc manner, our goal is to determine how to optimally implement a semistatic hedging strategy when a finite number of hedging assets is available. Our optimality criterion is the well-known variance-optimality criterion introduced by Schweizer (1984) and Föllmer and Sondermann (1986), i.e., we minimize the variance of the residual hedging error under the risk-neutral measure. As we show in Section 2, this criterion is perfectly compatible with semistatic hedging: The semistatic hedging problem separates into an inner problem, which is equivalent to the variance-optimal hedging problem with a single asset (as considered in Föllmer & Sondermann, 1986;Schweizer, 1984) and an outer problem, which is andimensional quadratic optimization problem, cf. Theorem 2.3.
After having analyzed the general structure of the variance-optimal semistatic hedging problem, we turn to another question in Section 3: How many assets < are enough to obtain a "reasonably small" hedging error? In case of Neuberger's formula for the variance swap-where infinitely many European options reduce the hedging error to zero-how good is using 12, 6, or even just 3 options? Beyond that, which 3 options should one select from, say, 30 that are available in the market? It turns out that this problem of finding a sparse semistatic hedging strategy is closely related to the well-known problem of variable selection in high-dimensional regression (cf. Hastie, Tibshirani, & Friedman, 2013, section 3.3), and more generally to sparse modeling approaches in statistics and machine learning. 1 Indeed, to solve the problem of optimal selection we will draw from methods developed in statistics, such as the LASSO, greedy forward selection and the method of Leaps-and-Bounds.
Finally, with the goal of a numerical implementation of sparse semistatic hedging in mind, we have to find tractable methods to compute variances and covariances of hedging errors, expressed mathematically as residuals in the Galtchouk-Kunita-Watanabe (GKW) martingale decomposition. Here, we build on results from Kallsen and Pauwels (2010) which enable us to calculate the GKW decomposition "semi-analytically," i.e., in terms of Fourier integrals, in several models of interest, such as the Heston model. The results of Kallsen and Pauwels (2010), which focus on calculation of the strategy and the hedging error in a classic variance-optimal hedging framwork, are however not sufficient for the semistatic hedging problem and we draw from some extensions that are developed in the technical companion paper, Di Tella, Haubold, and Keller-Ressel (2019).
We conclude in Section 5 with a detailed numerical example, implementing the sparse semistatic hedging problem for hedging a variance swap with put and call options in the Heston model. In particular, we compare the performance of the different solution methods for the subset selection problem, analyze the dependency of optimal hedging portfolio and hedging error on the number of static hedging assets, and study the influence of the leverage parameter on the optimal solution.

VARIANCE-OPTIMAL SEMISTATIC HEDGING
To set up our model for the financial market, we fix a complete probability space (Ω,  , ℙ) equipped with a filtration satisfying the usual conditions. We fix a time horizon > 0, assume that  0 is the trivial -algebra, and set  =  . We denote by = ( ) ≥0 the price process of a traded asset and set interest rates to zero to simplify the exposition of results. Consistent with market practice, we assume that a unique risk-neutral pricing measure ℚ, equivalent to ℙ, can be determined by calibration to vanilla option prices, regardless of market completeness. Expectations [.] denote expectations under this risk-neutral measure ℚ, unless otherwise indicated. Finally, we make the technical assumption that is a continuous square-integrable martingale under ℚ. More generally, we denote by  2 =  2 ( ) the set of real-valued -adapted square integrable ℚ-martingales, which becomes a Hilbert space when equipped with the norm ‖ ‖ 2  2 ∶= [ 2 ]. We also set  2 0 ∶= { ∈  2 ∶ 0 = 0}.

Variance-optimal hedging
Before discussing semistatic hedging, we quickly review variance-optimal hedging of a claim 0 in 2 (Ω,  , ℚ), up to the time horizon > 0, as discussed for instance in Föllmer and Sondermann (1986). We identify the claim 0 with the martingale which is an element of  2 . The set of all admissible dynamic strategies is denoted by where ⟨ , ⟩ denotes, as usual, the predictable quadratic variation of . The variance-optimal hedge with initial capital of the claim 0 is the solution of 2 = min ∈L 2 ( ), ∈ℝ [ ( The resulting quantity is the minimal hedging error. The minimization problem (1) can be interpreted as orthogonal projection (in  2 ) of the claim 0 onto the closed subspace spanned by deterministic constants (corresponding to the initial capital ) and by  2 ( ) ∶= {∫ 0 d , ∈ L 2 ( )} ⊂  2 0 , the set of claims attainable with strategies from L 2 ( ). The resulting orthogonal decomposition of 0 is known as the GKW decomposition of 0 with respect to , cf. Kunita and Watanabe (1967) and Ansel and Stricker (1993). From the financial mathematics perspective, (2) decomposes the claim 0 into initial capital, hedgable risk, and unhedgable residual risk. The orthogonality of to  2 ( ) in the Hilbert space sense implies orthogonality of to in the martingale sense, i.e., it holds that ⟨ , ⟩ = 0. Hence, the variance-optimal strategy can be computed from (2) as and can be expressed as the Radon-Nikodym derivative = d⟨ , ⟩∕d⟨ , ⟩.

The variance-optimal semistatic hedging problem
We are now prepared to discuss the variance-optimal semistatic hedging problem and its solution.
In addition to the contingent claim 0 , which is to be hedged, denote by = ( 1 , … , ) ⊤ the vector of supplementary contingent claims, all assumed to be square-integrable random variables in 2 (Ω,  , ℚ). Again, we associate to each the martingale The static part of the strategy can be represented by an element of ℝ , where represents the quantity of claim bought at time = 0 and held until time = . The dynamic part of the strategy is again represented by an element of L 2 ( ).
Definition 2.1 (Variance-Optimal Semistatic Hedging Problem). The variance-optimal semistatic hedge ( , ) ∈ L 2 ( ) × ℝ and the optimal initial capital ∈ ℝ are the solution of the minimization problem Note that ⊤ [ ] is the cost of setting up the static part of the hedge and its terminal value is ⊤ . The dynamic part is self-financing and results in the terminal value ∫ 0 d . Adding the initial capital and subtracting the target claim 0 yields the above expression for the hedging problem.
To solve the variance-optimal semistatic hedging problem, we decompose it into an inner and an outer minimization problem and rewrite (5) as The inner problem is of the same form as (1), while the outer problem turns out to be a finite dimensional quadratic optimization problem. To formulate the solution, we write the GWK decompositions of the claims ( 0 , … , ) with respect to as Similarly to (3) we get and introduce the vector notation ∶= ( 1 , … , ) ⊤ for the strategies and ∶= ( 1 , … , ) ⊤ for the residuals in the GKW decomposition. Finally we formulate the following condition.
Definition 2.2 (Nonredundancy condition). The supplementary claims = ( 1 , … , Intuitively, existence of a nonzero with ⊤ = 0 means that the number of supplementary assets can be reduced without changing the hedging error 2 in (5). We are now prepared to state our main result on the solution of the variance-optimal semistatic hedging problem.

∶= Var
Under the nonredundancy condition, is invertible and the unique solution of the semistatic hedging problem is given by The minimal squared hedging error is given by Moreover, the elements of , , and can be expressed as Corollary 2.4. If the nonredundancy condition does not hold true, then any solution ∈ ℝ of the linear system = , together with = [ 0 ] and = 0 − ⊤ is a solution of the semistatic hedging problem. The solution set is never empty, and the solution that minimizes the Euclidian norm of can be obtained by setting = † , where † denotes the Moore-Penrose pseudo-inverse of .
Notice that the minimal squared hedging error 2 in Theorem 2.3 is the Schur complement of the block in the 'extended covariance matrix' In particular, if ( 0 , ) has normal distribution, then 2 can be expressed as 2 = Var[ 0 | ].
Proof of Theorem 2.3 and Corollary 2.4. First, we consider the inner minimization problem in (6). This problem is equivalent to the variance-optimal hedging problem for the claim ∶= 0 − ⊤ . The solution is given by the GKW decomposition of the martingale ( 0 − ⊤ ) with respect to . By (3) we obtain using the bilinearity of the predictable quadratic covariation. Uniqueness of the GKW decomposition yields = 0 − ⊤ and the squared hedging error is given by Thus, the outer optimization problem in (6) becomes Because is positive semidefinite, the first-order condition = is necessary and sufficient for optimality of . Under the nonredundancy condition, Var( ⊤ ) > 0 for any ∈ ℝ ⧵{0}, hence is positive definite and in particular invertible. The unique solution of the outer problem is therefore given by = −1 , completing the proof of Theorem 2.3.
For the corollary, it remains to show that = has a solution, even when the nonredundancy condition does not hold. A solution exists, if is in the range of , or equivalently, if is in (ker ) ⊥ . By assumption ker is nonempty, and we can choose come ∈ ker , i.e., with ⊤ = 0. Because is the covariance matrix of is follows that ⊤ = 0, a.s. This implies that also ⊤ = Cov( ⊤ , 0 ) = 0, for all ∈ ker and hence that ∈ (ker ) ⊥ . □ Finally, we compute the hedge contribution of a single supplementary asset +1 . By hedge contribution, we mean the reduction in squared hedging error that is achieved by adding the asset +1 to a given pool of supplementary assets ( 1 , … , ). We denote by 2 and 2 +1 the minimal hedging error achieved with supplementary assets ( 1 , … , ) and ( 1 , … , +1 ), respectively.
Remark 2.6. The expression for the relative hedge contribution has an intuitive interpretation under the assumption that the residuals ( 0 , … , +1 ) have multivariate normal distribution. In this case the hedge contribution of +1 is equal to the partial correlation Cor( 0 , +1 | ) of 0 and +1 , given (cf. Muirhead, 2009, Ch. 5.3). Thus, roughly speaking, a supplementary asset has a high hedge contribution, if it is strongly correlated with 0 , even after conditioning on all claims that are attainable with semistatic strategies in and ( 1 , … , ).

Proof.
We ] , which is invertible due to the non-redundancy condition. Write = (Var[ +1 ] − ⊤ −1 ) for the Schur complement of in new . Using the Schur complement, the inverse of the block matrix new can be written as cf. Horn and Johnson (2012), and applying Theorem 2.3 yields and further algebraic manipulation yields (13). □

Hedging under ℚ vs. ℙ
The hedging criteria in (1) and (5) of minimizing the risk-neutral variance may seem "unnatural" at first sight, compared with the arguably more natural criterion of minimizing the variance under ℙ. After all, the ℙ-variance represents a quantification of the "true" risk associated to a hedging portfolio, which may be different from its risk-neutral counterpart. This discrepancy has been much discussed in the literature, with a good overview to be found in Cont and Tankov (2004, section 10.4.3). We also remark that the variance-optimal hedging principle has indeed been extended to the "physical setting" under ℙ (see the surveys of Pham, 2000 andSchweizer, 2001), albeit at the cost of an even more demanding mathematical theory. Instead of pursuing the path of variance-optimal hedging under ℙ, we present two arguments why the risk-neutral objectives (1) and (5) are a sensible choice after all, in particular with applications in mind: • Minimizing the hedging error under ℙ requires an accurate statistical estimation of the drift of the underlying asset(s), which is a challenging task from an econometric point of view. The risk-neutral pricing measure ℚ on the other hand can usually be inferred without difficulty from observed options prices ("model calibration"). Also the fact that an infinitude of equivalent local martingale measures exists in an incomplete market setting is of little concern in practice, as the choice of a parameterized model (e.g., the Heston stochastic volatility model) reduces the choice of ℚ to an optimization problem in a low-dimensional parameter space, which can easily be solved by numerically, cf. Cont (2010).
• Contrary to portfolio optimization, where long time horizons prevail and views on the growth rate of assets (i.e., on the "ℙ-world") are indispensable, the manager of a hedging portfolio remains focused on shorter time horizons and will avoid to make his/her choice of hedging portfolio dependent on subjective views on asset growth. Therefore, risk-neutral variance minimization may be in better correspondence to the actual objectives of the manager of a hedging portfolio.

The variance swap and long/short constraints
We review the semistatic hedging of a variance swap with an infinite pool of European put-and calloptions, as discussed in Neuberger (1994) and Carr and Madan (2001). We will apply the methods developed in this paper to this hedging problem in Section 5. It also serves as a motivation to add long/short constraints to the semistatic hedging problem.
Recall that a variance swap is a contingent claim on an underlying traded asset , which at maturity pays an amount swap ∶= [log , log ] − , where ∈ ℝ. Usually, is chosen such that the value of the contract is zero at inception, and the corresponding value * = [[log , log ] ] is called the swap rate. Recall that our only assumption on the discounted price process is that it is a squareintegrable strictly positive continuous martingale. Applying Itô's formula to log we get Hence, that is, to hedge the variance swap, it is enough to dynamically trade in the stock and enter a static position in the "log-contract" with payoff log 0 , cf. Neuberger (1994). Furthermore, from Carr and Madan (2001), we have which inserting into the above equation yields This equality can be interpreted as a semistatic replication strategy for the variance swap, which uses a dynamic position in and a static portfolio of infinitesimally small positions in an infinite number of out-of-the-money puts and out-of-the-money calls. We make several observations: (a) For any practical implementation the "infinitesimal portfolio" has to be discretized and portfolio weights have to be assigned to each put and call.
(b) As they are calls and puts on the same underlying asset, the static hedging assets are highly correlated.
(c) The static positions in puts and calls are long positions only.
To address point (a) different ad hoc discretizations of the integrals in (16) are possible (e.g., left or right Riemann sums, trapezoidal sums). However, it is not obvious which discretization is optimal in the sense of minimizing the hedging error. The choice of an optimal discretization in the varianceminimizing sense is precisely given by Theorem 2.3. Point (b) suggests that given a moderate number (say 30) of puts and calls as static hedging assets, many of them will be redundant in the sense that their hedge contribution (given the other supplementary assets) is small. This observation motivates the sparse approach of the next section and will be confirmed numerically in the application Section 5.
Point (c) finally motivates the addition of short/long constraints, or more generally, linear constraints of the type where ∈ R is fixed, to the outer problem in (6). With these constraints, the outer problem is a linearly constrained quadratic optimization problem, which can still be efficiently solved by standard numerical software.

SPARSE SEMISTATIC HEDGING
We now focus on the problem of optimal selection of static hedging assets, as outlined in the introduction and motivated in the previous section. Note that the subset selection only affects the static part of the strategy and hence only the outer problem in (6). Recall the 1 -norm ‖ ‖ 1 = ∑ =1 | | on ℝ and the (nonconvex) 0 -quasinorm ‖ ‖ 0 , which counts the number of nonzero elements of , cf. Foucart and Rauhut (2013).
Definition 3.1 (Sparse Variance-Optimal Semistatic Hedging Problem). The sparse variance-optimal semistatic hedge ( , ) ∈ L 2 ( ) × ℝ with effective portfolio size < and its optimal initial capital ∈ ℝ are the solution of the minimization problem (6), with the outer problem replaced by The 1 -relaxation of this problem is given by where > 0 is a tuning parameter that replaces . In both problems, we allow for long/short contains of the form (17).
The minimization problem (18) is of course equivalent to the extensively studied subset selection problem in linear regression and (19) to its convex relaxation in Lagrangian form, usually called LASSO. We refer to Hastie et al. (2013) for a general overview and to Tibshirani (1996) for the LASSO. We emphasize that • The 0 -constrained subset selection problem (18) is nonconvex and hard to solve exactly if the dimension is high.
• The 1 -penalized minimization problem (19) is convex and efficient numerical solvers exist even for large . Its solution is usually a good approximation to the exact subset selection problem, but no guarantee of being close to the solution of (18) can be given in general. 2 To illustrate the effect of the 1 -penalty, denote by * the solution of the unpenalized hedging problem (5) and assume for a moment that all GKW-r esiduals ( 0 , … , ) are uncorrelated. This assumption is highly unrealistic in the hedging context, but leads to a simple form of the solution of the penalized problem, cf. Tibshirani (1996): It is given by ′ = sign( * )(| * | − ) + , i.e., all static positions are shrunk toward zero by and truncated when zero is reached. This nicely illustrates the sparsifying effect of the penalty and the role of .
While (19) is frequently used as a surrogate for (18), the following alternatives exist for solving (18) directly, or for approximating its solution. Again, we refer to Tibshirani (1996) for further details on the described methods: Brute-Force Solve the quadratic optimization problem for each possible subset of cardinality . As there are ( ) of these subsets, this approach is usually not efficient and becomes completely infeasible for large .
Leaps-and-Bounds "Leaps-and-Bounds" is a branch-and-bound algorithm introduced by Furnival and Wilson (1974) for subset selection in linear regression, which gives an exact solution to (18) without testing all possible subsets. Essentially, a tree of nested subsets of hedging instruments (or regression variables) is formed, some branches of which can be discarded a priori, see Furnival and Wilson (1974) for details. While the worst case performance is no better than in the brute-force approach, it tends to be much faster in many practical settings.
Greedy Forward Selection A simple greedy approximation to (18) is to assume that the optimal subsets of different cardinality are nested. In the forward approach the problem (18) is first solved for = 1, which is easy. Then, iteratively, the supplementary claim with the largest relative hedge contribution (see (2.5)) is added to the set of active static positions in each step. The same procedure could be used backward ("greedy backward selection") i.e., starting with = and then removing iteratively the supplementary claim with the smallest hedge contribution. In general, no guarantee of being close to the exact solution of (18) can be given for these methods.
We will compare the practical performances of the different solution methods in Section 5.

STOCHASTIC VOLATILIT Y MODELS WITH FOURIER REPRESENTATION
The final ingredient still missing for a numerical solution of the (sparse) semistatic hedging problem is an efficient method to compute the quantities , , and from Theorem 2.3. One possible approach would be to compute transition densities of and 0 , … , by Monte Carlo simulation and to compute the GKW decomposition by sequential backward regression, cf. Föllmer and Schweizer (1988). Due to the fact that the joint distribution of and all price processes of supplementary claims is needed, we expect a heavy computational load in order to obtain reasonably high accuracy with this method. An interesting alternative method has been suggested by Kallsen and Pauwels (2010) (see also Hubalek, Kallsen, & Krawczyk, 2006;Pauwels, 2007) for the classic variance-optimal hedging problem (1) of European claims. This alternative is based on the well-known Fourier method for pricing of European claims, cf. Carr and Madan (2001); Raible (2000); Kallsen and Pauwels (2010).

Fourier representation of strategies and hedging errors
We stay close to the framework of Kallsen and Pauwels (2010) and assume that the payoff of some option is given by = ( ), where is the log-price process of the underlying stock, i.e., we also assume = exp( ). The payoff of a call for example can be written as ( ) = ( − ) + , but it is not necessary to restrict to this specific case. Furthermore, we assume that the (rescaled) two-sided Laplace transform̃( of the payoff exists at some = ∈ ℝ and is integrable over the strip in the complex plane. If the integrability condition [ ] < ∞ holds, then the risk-neutral price of the claim at time ∈ [0, ] can be recovered by the Fourier-type integral along ( ), where we denote the conditional moment generating function (analytically extended to the complex plane) of by Note that ( ) is well defined on ( ) due to the integrability condition imposed on . In the important cases of European puts and calls, the two-sided Laplace transform̃is given bỹ with > 1 for calls and < 0 for puts, (cf. Hubalek et al., 2006, section 4).
The key insight, pioneered by Hubalek et al. (2006) for variance-optimal hedging in models with independent increments and by Kallsen and Pauwels (2010) and Pauwels (2007) for affine stochastic volatility models, is that the Fourier representation (21) of European claims can be extended to their GKW decomposition (2). More precisely, both the strategy and the hedging error 2 = [ 2 ] of the variance-optimal hedging problem (1) can be expressed in terms of Fourier-type integrals, similar to (21). For our problem of interest, the semistatic hedging problem (5), the results of Hubalek et al. (2006), Kallsen and Pauwels (2010), and Pauwels (2007) are not sufficient: To obtain the quantities , , and of Theorem 2.3, we also need to compute the covariances [ ] between the GKW residuals of different claims. In the companion paper, Di Tella et al. (2019), we extend the results of Hubalek et al. (2006), Kallsen and Pauwels (2010), and Pauwels (2007) to the semistatic hedging problem. Moreover, we show that the method can be used in any stochastic volatility models where the Fourier transform of the log-price is known (e.g., the Heston, the 3/2 or the Stein-Stein model, cf. Lewis, 2000). Here, we only need a special case of the more general results in Di Tella et al. (2019), which is condensed into Theorem 4.1(i) below.

The Heston model
In the Heston model (cf. Heston, 1993) the risk neutral price process is given by where 1 and 2 are two Brownian motions such that ⟨ 1 , 2 ⟩ = , ∈ [−1, 1]; , , > 0. The joint moment generating function of the Heston model is known explicitly and of the form well defined for real arguments in the set and with analytic extension to the associated "complex strip" To represent ( , ) and ( , ), we introduce Δ( ) = ( − ) 2 − 2 ( 2 − ), Then, the explicit expression of is given, for ( , ) ∈  by (cf. Alfonsi, 2015, proposition 4.2.1), with the convention whenever the denominator of is equal to zero. Moreover, ( , ) is given by The following theorem specializes Theorem 4.1 to the Heston model and gives (up to integration) explicit expressions for the quantities , , and from Theorem 2.3. The proof of the theorem is given in Appendix A.
Theorem 4.2. Let ( , ) be given by the Heston model (28) and let the claim 0 be a variance swap, i.e., with payoff 0 = [ , ] at maturity . Let the supplementary claims ( 1 , … , ) be European puts and calls with payoffs and two-sided Laplace transforms̃, integrable along strips ( ), as in (21). If [ 2 ] < ∞ for all = 1, … , then the quantities , , and , defined in Theorem 2.3 are given by Remark 4.3. Note that the common leading factor 2 (1 − 2 ) of , , and also becomes the leading factor of the minimal squared hedging error 2 , cf. (12). This makes perfect sense, as it makes the hedging error roughly proportional to vol-of-vol and shows that the hedging error vanishes in the complete-market boundary cases = ±1 of the Heston model. However, and also appear inside , and therefore their influence on 2 is not limited to the leading factor 2 (1 − 2 ) alone.

NUMERICAL RESULTS
The following numerical implementation should be considered in terms of a 'stylized financial market' setting, i.e., while we do not calibrate the model to current market data, we use parameters that are realistic in a market setting. More specifically, we use the Heston model parameters from Gatheral (2006): In Section 5.4 we vary the leverage parameter , but keep all other parameters fixed. The current stock price is normalized to 0 = 100 and we use a time-to-maturity of = 1 (years) for the variance swap and the call options. The price of a variance swap (i.e., the swap rate * = [[log , log ] ]) can be readily calculated as * = ∫ 0 The supplementary assets are OTM-puts and OTM-calls with strikes ranging from min = 50 to max = 150 in steps of Δ = 5.
We focus on three aspects of the semistatic hedging problem: • Comparing the different methods that were proposed in Section 3 to solve the sparse semistatic hedging problem; • Analyzing the dependency of hedging error and optimal portfolio composition on effective portfolio size ; • Analyzing the dependency of hedging error and optimal portfolio composition on the leverage parameter .

Comparison of methods
As a first step, we computed , , and the matrix from Theorem 2.3 using the Fourier representation in Theorem 4.2 by adaptive integration in MATLAB. Next, we implemented the methods, described in Section 3, i.e., (1) Greedy forward selection (with and without short-sale constraints) (2) Leaps-and-Bounds (with and without short-sale constraints) (3) LASSO in the statistical computing environment R; using the function lars in the package lars with option type="lasso" for the computation of the LASSO solution. While computationally most demanding, the Leaps-and-Bounds solution can serve as a benchmark solution, as it is (up to numerical error) the exact solution of the sparse semistatic hedging problem (18). The other methods, in contrast, only return a "reasonably close" solution to (18). In all cases, we report the relative hedging error ∕ * , i.e., the hedging error normalized by the price of the variance swap. A challenge that is faced by all methods is the bad condition of the matrix . With parameters chosen as in (35) the reciprocal condition number of is 1.11 × 10 −6 . While small, this number is still several orders of magnitude larger than the machine precision of 2.22 × 10 −16 (double precision arithmetic) on the computer that was used. The bad condition of is not surprising, as put and call options with neighboring strikes are highly correlated. This effect is likely amplified by the fact that contains the correlations of the GKW residuals and not the correlations of the option prices themselves. While we have considered preconditioning of , along the lines of Neumaier (1998), we have found that greedy forward selection and Leaps-and-Bounds perform well even without additional conditioning. Also the addition of short-sale constraints seems to have a regularizing effect on the methods. Figure 1 shows the relative hedging error (as percentage of the variance swap price) attained with the optimal portfolio returned by Methods 1-3 for different effective portfolio sizes = 0 … 21. Notice that the implementation of LASSO adds and removes supplementary assets from the active set, such that the graph can show multiple solutions for the same effective portfolio size (e.g., for = 15). Focusing on the comparison of methods, we find that • The Leaps-and-Bounds method returns the solution with the smallest hedging error, consistent with the fact that it solves (18) exactly. It is remarkably fast, but further numerical experiments indicate that its runtime is sensitive to the choice of model parameters.
• The greedy method is the fastest method and the residual hedging error of its solution is only slightly higher than the hedging error of the Leaps-and-Bounds solution. Moreover, the performance of the greedy method is stable with respect to parameter choice.
• The LASSO methods seems to be severely affected by the bad condition of . This is not surprising, as it has been remarked, for instance, in Bühlmann & Van De Geer (2011, section 2.6) that the LASSO method has problems with highly correlated data.
Summing up, we can recommend the greedy method as fast, reliable, and easy to implement. The Leaps-and-Bounds method is useful as an efficient way to compute an exact benchmark solution. We cannot recommend LASSO, as it cannot deal well with the bad condition of . 3 Interestingly, this observations are contrary to the usual wisdom in variable selection for regression problems, where greedy forward selection often has unstable performance and LASSO yields superior results (cf. Bühlmann & Van De Geer, 2011, chapter 2). We attribute these findings to the highly correlated nature of the matrix , which is untypical in regression scenarios, but a natural feature of our hedging problem.

Analysis of the hedging error
We return to Figure 1 to analyze the hedging error resulting from the sparse variance-optimal semistatic hedging problem (18) for different effective portfolio sizes . We consider the benchmark solution returned by the Leaps-and-Bounds method with short-sale constraints. First, we note that dynamic hedging in the underlying , without using any static positions in puts and calls ( = 0), results in a relative hedging error of 59.7%. This error is already reduced to 5.7% by just adding three supplementary assets ( = 3) and can be further reduced to 3.4% by selecting six supplementary assets ( = 6). Finally, the error levels off to 1.6% when the full range ( = 21) of puts and calls between min = 50 and max = 150 is used. Further substantial reductions of the hedging error can only be achieved by extending the range of available strikes; adding more options within the current range has only negligible effects.
The sharp decrease of the hedging error between = 0 and = 3 affirms the basic premise of sparse semistatic hedging: That selecting only a small number of supplementary assets already leads to a significant reduction of the hedging error. On the other hand, the poor performance of the LASSO solution shows that a suboptimal choice of supplementary assets does not result in a satisfactory reduction of the hedging error. In other words, it is important that the sparse subportfolios are chosen optimally, and not arbitrarily.

Composition of the hedging portfolios
We now turn to the composition of the static hedging portfolio, i.e., the vector ∈ ℝ with the constraint ‖ ‖ 0 ≤ , that is returned by the solution methods for the sparse semistatic hedging problem (18). Recall that the element is the nominal size of the position in the supplementary asset , with negative sign indicating a short position. In our setting, the elements of can simply be indexed by the strike of the corresponding put/call. The optimal portfolios returned by the different solution methods, along with their dependency on effective portfolio size are shown in Figure 2. We make the following observations: • With the exception of the put = 55 only long positions are observed; • Positions in OTM puts ( < 100) are larger than in OTM calls ( > 100), in line with Neuberger's replicating portfolio (16); • The general pattern (going from effective portfolio size = 1 to 21) for all methods can be described as follows: Start with an (approximately) ATM option. Proceed by selecting both OTM puts and calls, going outwards as increases and putting more weight on OTM puts, until the limit min = 50 is reached. Continue by adding OTM calls and by filling up the gaps from earlier stages.
We suspect that the rare short positions are numerical artifacts, rather than belonging to the true optimal solution of (18). Indeed, their effect on the hedging error is minuscule, and we hence recommend to use a priori short-sale constraints, in the case of hedging a variance swap. Figure 2 gives a good overview of the portfolio composition, but it is difficult to assess the precise size of the individual positions . For this reason, we provide in Figure 3 an additional plot of the portfolio weights indexed by strike for the optimal portfolios of effective sizes = 3, 6, 12 in doubly logarithmic coordinates. Note that Neuberger's replicating portfolio (16)  i.e., the portfolio weights should form a line of downward slope −2. Figure 3 shows reasonable agreement with this asymptotic result, even for effective portfolio size as small as = 3. For = 12 numerical errors from the bad condition of the matrix seem to accumulate and could explain the unruly shape of the graph.

The role of correlation
Finally, we turn to the role of the correlation parameter , which is interesting for several reasons: First, the value of does not affect the theoretical price of the variance swap. Second, also does not affect the infinitesimally optimal strategy (16). Finally, allows to tune the degree of market incompleteness, as the Heston model becomes a complete market model in the boundary cases = ±1. Despite of the first two points, it turns out that has a significant effect on the attainable hedging error and the composition of the optimal portfolio in the sparse semistatic hedging problem. This influence can already be suspected from the leading factor 1 − 2 appearing in Theorem 4.2, which propagates to the (squared) hedging error itself, see also Remark 4.3. Indeed, as Figure 4 shows, the dependency of the relative heading error on is very close to a "semi-circle law" ( ) = √ 1 − 2 , with different constants for different effective portfolio sizes .

CONCLUSIONS AND OUTLOOK
In this paper, we have introduced a general framework for semistatic and sparse variance-optimal hedging. We have also demonstrated the numerical feasibility of our approach in the concrete example of hedging a variance-swap in the Heston model. In future work, we think it would be interesting to examine the degree of model-dependency of these results, by implementing the same approach for other models with known Fourier transform, e.g., the 3∕2 model of Carr and Sun (2007) or the recent rough Heston model of El Euch and Rosenbaum (2019). Even in stochastic volatility models without an explicit Fourier transform, a combination of PDE and Monte Carlo techniques should in principle allow to compute the quantities , , and in Theorem 2.3 and hence to determine optimal semistatic hedging strategies.
Finally, we identify two other directions, in which we see potential for relevant extensions of our results: In the setup of Section 2 we have assumed that all supplementary assets ( ) as well as the hedging target 0 mature at the same time. In practice, also assets with maturity mismatch can be useful for hedging. For example, a variance swap with fixed maturity could be hedged-not perfectly, but variance optimally-by a portfolio of European options with shorter, larger, or even mixed maturities. Moreover, because the supplementary assets ( ) are typically less liquid than the underlying, it seems reasonable to include a bid-ask spread for these assets. Such bid-ask spreads should only lead to additional linear inequality constraints to the outer problem in (6). As a quadratic optimization problem with linear constraints, the hedging problem should remain numerically feasible, even under bid-ask spreads.

ACKNOWLEDGMENTS
MKR thanks Johannes Muhle-Karbe for discussions on variance-optimal semistatic hedging. We acknowledge funding from the German Research Foundation (DFG) under grant ZUK 64 (all authors) and KE 1736/1-1 (MKR, MH) ENDNOTES 1 These connections to linear regression should not come as a surprise: It has been noted already in Föllmer and Schweizer (1988) that variance-optimal hedging in discrete time is equivalent to a sequential linear regression problem. 2 We remark that conditions for the perfect recovery of solutions of (18) by solving (19) can be given within the theoretical framework of compressive sensing, see, for instance, Foucart and Rauhut (2013). 3 It should be said, in all fairness, that the R-function lars also provides the option "stepwise" instead of "lasso" which effectively corresponds to the greedy method.


s2fieldsofstudy: Business Economics

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The Effect of Financial Variables on Systematic Risk in Six Industries in Thailand

Risk is classified into two types which are systematic risk and unsystematic risk. Unsystematic risk is known as a diversifiable risk that can be avoided or managed. On the other hand, systematic risk is the market-related risk which cannot be controlled or diversified away. Between these two types of risks, the systematic risk becomes the major concern of firms and investors as this type of risk cannot be avoided or diversified away, but need to be strategized and managed accordingly. The purpose of this study is to examine the main factors influence on the behavior of systematic risk in six industries of Thailand, the period of study is 15 years from 2002 to 2016 and consist of 372 non-financial listed firms. This study employs the panel data analysis, comprising of the random effect model (REM), fixed effect model (FEM), and pool ordinary least square (POLS). The overall findings show some common financial variables such as financial leverage, liquidity, firm size, firm growth, and profitability are considered as the main factors affecting systematic risk in Thai consumer goods, technology, telecommunication, utilities, and health care. However, Thai consumer service is reported as an insignificant relationship between financial variables and systematic risk. Apart from financial variables, there is an impact of the financial crisis (2009) on systematic risk in all industries. Findings in this study extent in the finance literature on systematic risk, different internal industries may have different factors influencing the behavior of systematic risk.

INTRODUCTION
Risk is known as uncertainty which involves any activity of investment. Normally, the risk is classified into two types which are unsystematic risk and systematic risk. Unsystematic risk is known as a specific risk to firms which can control and reduce through diversification, this type of risk caused by internal factors such as mismanagement and so on. Systematic risk is the market relates risk which cannot avoid and will affect the market as a whole, it is known as un-diversifiable risk (Markowitz, 1952).
According to the nature of systematic risk and unsystematic risk, the most concern for firms' managers and investors is a systematic risk since it cannot diversify away. Firms' managers and investors have played more attention in finding a way to understand and managed systematic risk in orders to prevent their investment from a lost (Gitman & Zutter, 2012;Alaghi, 2012). In the finance literature stated that, the higher profit can be achieved by reducing the portion of the risk, that is, it is very important for firms' managers and investors to understand the nature of risk especially systematic risk since unsystematic risk can reduce through diversification. By understanding the nature of systematic risk may help firms' managers and investors can develop new strategies dealing with this type of risk (Iqbal & Ali Shah, 2012) .
This study examines the specific factor affecting systematic risk in six industries of Thailand by focusing on all non-financial listed firms within 15 years from 2002 to 2016. Additionally, this study also examines the effect of the global financial crisis (2009) on systematic risk. The findings of this study may provide new evidence and useful information to extend in the literature which may help to increase the current understanding of systematic risk. Understanding the source of systematic risk of different industries may assist firms to know how to handle systematic risk issue during the financial crisis and normal economic situation. This may result in an increase in firms' efficiency and profitability in the future.

LITERATURE REVIEW
According to the capital asset pricing model, systematic risk is very important as compare to unsystematic risk. That is because systematic risk plays an important role in determining the expected return of stocks, this type of risk is known as the external impact which cannot be avoided or reduced through diversification. Systematic risk indicates the risk of firms related to market risk, so if the systematic risk of firms is higher than market risk. This will affect to return of shareholders when there is a loss by the movement of the stock price, that is, the stock price can be changed depending on the change of systematic risk (Alaghi, 2013).
Previously, there are several studies attempted to find ways to understand the nature of the sources of systematic risk by using firm-specific variables such as firm size, liquidity, profitability, growth, operating efficiency and financial leverage to determine systematic risk in financial and non-financial firms of a particular market. The results of these studies have shown a relationship between systematic risk and firm-specific variables (see for example ;Mandelker & Rhee, 1984;Borde, 1998;Gu & Kim, 2002;Hamada, 1972;Lee & Jang, 2007;Aruna & Warokka, 2013). Iqbal and Ali Shah (2012) studied the effect of financial variables on systematic risk by focusing on non-financial firms listed in Karachi Stock Exchange within 4 years from 2005 to 2009. Their results show that firm size, profitability and growth are positively associated with systematic risk, but leverage liquidity, operating efficiency is negatively related to systematic risk. Similarly, Aruna and Warokka (2013) examined the specific factors affecting systematic risk in the manufacturing industry of Indonesia within 2 years from 2005 to 2007. They found that only growth is positively associated with systematic risk while leverage, firm size and liquidity is negatively associated with systematic risk. Alaghi (2013) investigated a relationship between firm-specific variables and systematic risk of non-financial firms in the Tehran Stock Exchange. Their finding showed that there is no any relationship between firm size and systematic risk, but liquidity, profitability, operating efficiency, and leverage have a significant impact on systematic risk. Furthermore, a study conducted by Lee and Jang (2007) focused on firm-specific factors determining systematic risk in the US airline industry. Their study has shown that there is no relationship between liquidity and systematic risk while firm size, safe, leverage, profitability and growth are related to systematic risk. However, even though previous studies found there is a relationship between systematic risk and firm-specific variables but previous studies only provided a mixed result or inconclusive result which cannot specify which are the main factor affecting systematic risk. Therefore, this may not provide sufficient information to make a sound decision for firms' managers and investors. Systematic risk is the market relates risk which cannot diversify away, it is the main factor affecting stocks' return. Thus, understanding the sources of systematic risk is very crucial for firms' manager and investors in order to develop new approaches dealing with this type of risk in the future.

METHODOLOGY
This study includes 372 non-financial listed firms which derived from six industries such as consumer goods, consumer service, telecommunication, utilities, health care and technology. Financial firms are excluded from this study due to a different capital source and rule. This study uses panel data approach such a fixed-effect model, random effect model, and pooled ordinary least squares to estimate the parameter of interests. There are three tests to select which is the best model such as Chow test, Hausman test, and Breusch and Pagan test. The regression for this study is shown below: , = 0 + 1 , + 2 , + 3 ℎ , + 4 , + 5 , + 6 , 7 + 7 2008 , + The development of hypothesis in this study base on previous studied which shown as below: H1: Operating efficiency is negatively related to systematic risk. H2: Financial leverage is positively related to systematic risk. H3: Firm Growth is positively related to systematic risk. H4: Profitability is negatively related to systematic risk. H5: Firm size is positively related to systematic risk H6: Liquidity is positively related to systematic risk. H7: Financial crisis in 2008 has a significant influence on systematic risk.

FINDINGS AND DISCUSSION
According to a summary of full sample for six industries of Thailand in Table 2, it evidences that financial variables are not significantly related systematic risk in consumer service while other industries found at least one variable is associated such as Thai consumer goods shown financial leverage as the main factor, technology is profitability, telecommunication is liquidity, utilities are firm growth and firm size, and health care is firm growth. The less financial variables which did not mention as the main factor are insignificant related to systematic risk. Aruna and Warokka (2013) use some of the accounting variables such as leverage, liquidity, firm size and firm growth to determine systematic risk in Indonesian manufacturing, they found there is not any significant relationship between accounting variables and systematic risk. Therefore, it can be said that systematic risk can be explained by financial variables but not all of the financial variables can be used to influence the behaviour of systematic risk, it depends on the characteristic of industries as well. Besides financial variables, the global financial crisis (2009) is significantly correlated to systematic risk in all industries. This result is supported the seventh hypothesis (H7) of this study.

CONCLUSION
Previously, numerous studies have been done on examining the impact of firm-specific variables on systematic risk in local and international firms. However, these studies focused on particular industries such as the airline industry, restaurant industry, manufacturing, banking, tourism and so on. By focusing only on a particular industry, these studies have provided an inconclusive finding which cannot specify clearly which is the main factor affecting systematic risk. Therefore, previous studies may not provide sufficient information for firms to understand the nature of systematic risk issue or sources of systematic risk and have a sound decision on their business. This study examines the relationship between financial variables and systematic risk in non-financial listed firms across various industries in Thailand within 15 years from 2002 to 2016. This study may help firms' managers and investors to enhance their understanding of the nature of the systematic risk or an impact of financial leverage, firm size, growth, operating efficiency and profitability on systematic risk. By understanding the sources of systematic risk in non-financial firms across various industries in Thailand may assist firms' managers and investors to have a good strategy dealing with systematic risk issue in the future which may lead to increase in the profitability and efficiency of the firms. Moreover, this study also examines the impact of the global financial crisis in 2009 on systematic risk. Therefore, understanding the sources of systematic risk issue which relates to the financial crisis in 2008 may help firms' managers and investors to be able to generate a healthy plan dealing with risk.