Elena Kornyshova
email: elena.kornyshova@cnam.fr

Rebecca Deneckère
email: rebecca.deneckere@univ-paris1.fr

Kaoutar Sadouki
email: kaoutar.sadouki@lecnam.net

Eric Gressier-Soudan
email: eric.gressier_soudan@cnam.fr

Sjaak Brinkkemper
email: s.brinkkemper@uu.nl

Smart Life: Review of the Contemporary Smart Applications

Keywords: Smart Life, Smart Application, Systematic Mapping Study

Research efforts in various fields related to Smart life increase constantly. There are very well-established fields like Smart home, Smart city, and Smart grid, and more emergent ones, like Smart farming, Smart university, and Smart tourism. Smart life intends to enhance human life and serves as an umbrella term for all Smart topics. However, the research domain of Smart life is very diverse and multifaceted. Our main goal is to systematize the existing research work around Smart life to provide direction for the development and maintenance of Smart artefacts and applications, thus, to move towards Smart life engineering. To achieve this goal, a mandatory step is to understand and organize all the different Smart topics studied in the scientific literature by means of a systematic mapping study. We analyzed 2341 existing Smart state-of-theart works and research agendas. We propose a taxonomy of Smart applications, study their evolution over time, analyze their venues, specific terminology, and driving factors. The resulting overview is useful to researchers and practitioners to improve the positioning of their work and to identify the research opportunities in all types of smartness for humans, organizations, and society.

Introduction

The number of research works in "Smart" fields grows continuously. Many related fields have appeared over the last decades. For instance, Smart home is defined in [START_REF] Lutolf | Smart home concept and the integration of energy meters into a home based system[END_REF] as "the integration of different services within a home by using a common communication system. It assures an economic, secure, and comfortable operation of the home and includes a high degree of intelligent functionality and flexibility." In [START_REF] Chamee | Historicizing the smart cities: Genealogy as a method of critique for smart urbanism[END_REF], Smart city is "a city that collects and utilizes data gathered from distributed sensors and video cameras that connect everything from trash bins to streetlights." Since then, appeared new concepts like Smart farming or Smart agriculture [START_REF] Hidayat | Method of Systematic Literature Review for Internet of Things in ZigBee Smart Agriculture[END_REF], Smart university [START_REF] Al-Shoqran | A Review on Smart Universities and Artificial Intelligence[END_REF], Smart tourism [START_REF] Mehraliyev | A state-of-the-art review of smart tourism research[END_REF], and so on. Numerous state-of-the-art works exist in all these fields, as for the Smart home domain for which we identified almost 180 state-of-theart and research agenda papers, for instance [START_REF] Tiwari | Changing World: Smart Homes Review and Future[END_REF] [7] [START_REF] De Silva | State of the art of smart homes[END_REF], or [START_REF] Alaa | A review of smart home applications based on Internet of Things[END_REF]. Despite a huge number of works in Smart fields, works aiming at classification, organization, and systematization of the different Smart fields are neglected. We unify all these Smart fields within the concept of Smart life that we define as "a system of integrated Smart application systems for the purpose of an enriched life experience". In our vision, Smart life is a set of different Smart applications (e.g., Smart home, Smart city, Smart transportation, and so on) which, in their turn, are composed of different Smart artefacts (e.g. Smart car, Smart ship, Smart railway, etc. for the Smart transportation application). We believe that optimization could and should be done to design, develop, and maintain all these various applications in a smarter way. In our vision, different methodologies, technologies, and tools could be shared and reused between the Smart fields. This leads us to promote a new field we called Smart life engineering.

To start to investigate this research direction, we should first analyze the existing works. In the already published literature on Smart life, we did not identify any detailed taxonomy on Smart applications. Most of the state-of-the-art works on Smart life deal with a particular sub-topic within this context: IoT application for energy consumption [START_REF] Wang | Energy management solutions in the Internet of Things applications: Technical analysis and new research directions[END_REF], Security in Internet-of-Things (IoT) [START_REF] Harbi | A Review of Security in Internet of Things[END_REF], IoT usage in Smart cities with a classification of Smart technologies [START_REF] Tai-Hoon | Smart City and IoT[END_REF], computer technologies from the viewpoint of artificial intelligence [START_REF] Yamane | Deductively verifying embedded software in the era of artificial intelligence = machine learning + software science[END_REF], machine to machine usage [START_REF] Severi | M2M technologies: Enablers for a pervasive internet of things[END_REF], and an integrative study about IoT [START_REF] Baruah | Impact of IOT in Current Era[END_REF]. Only two papers are generic: one source gives a classification of Smart technologies [START_REF] Mizintseva | Smart Technologies for Smart Life[END_REF]. The second one suggests a systematic literature review on Smart systems [START_REF] Romero | Towards a Characterisation of Smart Systems: A Systematic Literature Review[END_REF]. In addition, [START_REF] Bouchrika | A Survey of Using Biometrics for Smart Visual Surveillance: Gait Recognition[END_REF] reviews Information and Communication Technologies (ICT) within a "Smart Earth" concept.

In the current work, we focus on a study of Smart applications. We analyzed the titles and abstracts of 2341 State-of-the-Art-and Research Agenda-oriented papers in order to establish a taxonomy of Smart applications, analyze their venues, understand the underlying terminology, and, finally, understand what are driving factors of Smart life. We used the Systematic Mapping Study method [START_REF] Petersen | Systematic mapping studies in software engineering[END_REF] to develop our research approach.

We explain our research approach in section 2. We highlight the obtained results in Section 3. In Section 4, we conclude the paper and outline our future research.

Research Approach

We used a systematic mapping design [START_REF] Petersen | Systematic mapping studies in software engineering[END_REF] to study the Smart life field of research. Systematic Mapping Studies (SMS) are similar to other systematic reviews (like SLR systematic literature review synthesizing the existing research in established fields), except that they employ broader inclusion criteria to select a wider range of research papers and are intended to map out topics with a field classification rather than synthesize study results. The study presented here covers the existing work in the field of Smart life. We followed the process presented in [START_REF] Petersen | Systematic mapping studies in software engineering[END_REF] which includes five steps: defi-nition of research questions, conducting search for primary studies, screening of papers, keywording of abstract and data extraction and mapping of studies.

Based on [START_REF] Petersen | Systematic mapping studies in software engineering[END_REF], we followed the procedure on SMS summed up in Table 1. We named differently Steps 2 and 4 as more compliant with our research process. Step 2: Finding Papers. This step aims at identifying a set of papers based on a relevant search string. We searched and selected papers in the SCOPUS scientific database using the SCOPUS Search API in November 2021. (This database includes all "articles being published in virtually all scholarly journals of any significance in the world"1 ). Firstly, we searched for papers containing the term "Smart" only in the title, and the number of obtained papers was more than 126 000. Thus, we limited our search to secondary research papers like state-of-the-arts and research agendas. This reduced search allowed to obtain 2410 sources with DOIs. The inclusion criterion related to the search string is given in Table 2.

Step 3: Screening of Papers. We analyzed the titles and, if needed, abstracts and papers content. We excluded 69 sources, not representing research papers or not relevant to Smart life topics papers. We obtained 2341 papers 2 . In Table 2 we summarize the exclusion criteria used to obtain the list of relevant papers. The title includes the term «smart» and at least one of the terms "research agenda", "state-of-the-art", "review", or "survey": Search string: TITLE(smart) AND (TITLE("research agenda") OR TITLE("State-ofthe-art") OR TITLE(review) OR TITLE(survey)). Exclusion criteria The source is not a research paper (erratum, retracted, etc.).

(2341 sources selected)

The source is related to an abbreviation SMART, like, for instance, SMART (stroke-like migraine attacks after radiation therapy) syndrome. The source mentions the term «smart», which is used in its ordinary sense, like "working smart and hard".

Step 4: Classification Scheme Definition. The goal of this step of SMS is to identify the classification scheme to be applied to the obtained results. To answer the defined research questions, we classified all relevant papers accordingly to the following criteria:

• Criterion related to Smart topics. Smart life is a set of Smart applications. This qualification allowed us to establish a detailed taxonomy of Smart life applications: ─ Smart Application • Criteria related to publication venues: ─ Year ─ Venue ─ Type of Venues

In addition, we used the titles and abstracts to provide additional representations.

Step 5: Data Extraction and Mapping Process. For RQ1, the main extraction was done with relation to the Smart life topics as it required additional work to classify Smart life applications into a taxonomy. Then, we classified the identified applications into four Smart domains: Persons, Society, Environment, and Enterprises, and we analyzed the distribution of publications in these domains over time. For RQ2, the data on years and issues were already available, we completed them by qualifying the most used issues by their type (Survey-oriented, Generic, Domain-specific). For RQ3, we created keyword clouds based on papers titles and papers abstracts to have a visual representation of the most used terms in the scientific literature related to Smart life and we extracted the main used terminology. For the two clouds, we deleted the stop words, the numbers, and the special characters. Moreover, on the second one (set of abstracts), we made a pre-processing of the data to change plural nouns into singular ones if the two forms were present in the dataset.

Validity Threats. Qualitative research is based on subjective, interpretive, and contextual data. Thus, we analyzed the potential biases, which could threaten the validity of our research. [START_REF] Thomson | Qualitative Research: Validity[END_REF] proposes five categories of validity. To minimize the impact of the validity threats that could affect our study, we present them with the corresponding mitigation actions in Table 3.

Table 3. Validity Threats.

Validity

Actions Descriptive validity refers to the accuracy of the data.

-We unified the concepts and criteria used in the study and structured the information to be collected with a data extraction form to support a uniform recording of data. Theoretical validity depends on the ability to get the information that it is intended to capture.

-We used a search string and applied it to a library including the most popular digital libraries on computer sciences and software engineering. -A set of inclusion and exclusion criteria have been defined. -We combined two different search methods: an automatic search and a manual search (backward and forward snowballing), to diminish the risk of not finding all the available evidence. -The choice of English sources should be of minimal impact concerning the discard of other languages. Generalization validity is concerned with the ability to generalize the results.

-Our set of research questions is general enough to identify and classify the findings on Smart applications. Evaluative validity is achieved when the conclusions are reasonable given the data.

-Two researchers studied the papers, working independently but with an overlap of studies to identify potential analysis differences. -At least two researchers validated every conclusion. Transparency validity refers to the repeatability of the research protocol.

-The research process protocol is detailed enough to ensure it can be exhaustively repeated.

Analysis of Smart Application Research

We present the obtained results in the following sub-sections organized accordingly to the defined research questions.

Taxonomy and Evolution of Smart Applications

Many applications are interconnected to each other. However, a structuration should be done. We studied the topics of the selected 2341 papers. Firstly, we followed the bottom-up approach: we studied the titles of all papers, however, when titles were not sufficient to identify the application field, we read the abstracts, and, if needed, looked inside the papers. We attributed from one to three levels of applications classification to each paper (for instance, Smart energy -> Smart grid -> Smart grid communication), except papers dealing with Smart artefacts, technologies, or systems without attachment to a concrete application field. The attribution of each paper to a Smart application was done by an expert. When the mapping of a topic to a Smart domain was not obvious, the paper was analyzed in more detail by two or more experts to find a consensus. Once papers qualified, we started to organize them within a taxonomy using a top-down approach: we took the major fields of Smart life like Smart home, Smart city, Smart industry, Smart healthcare, etc. and we organized the identified application fields within major fields. Considering the number of the major application fields, we organized them within four categories that we called Smart life domains: Smart applications for persons, Smart applications for environment, Smart applications for society, and Smart applications for enterprises (See Fig. 1.). To give some more detailed illustrations, the Smart city concept is used to solve the challenges in urban cities by providing a good livable environment for the citizens [START_REF] Al-Ani | Smart city applications: A survey, Perva-siveHealth[END_REF]. Therefore, in the taxonomy, the Smart city application domain contains sub-applications addressing security, growth, lighting, parking, and so on. With regards to Smart agriculture, new technologies are more and more used in modern agriculture and IoT farmers can remotely monitor their crop and equipment by phones and computers [START_REF] Mekala | A Survey: Smart agriculture IoT with cloud computing[END_REF]. The Smart agriculture application domain contains usual agricultural types like Smart farming, Smart fishery, Smart crop management, and Smart livestock farming, but also some other domains influencing agriculture, like climate-smart agriculture, Smart pest management, and Smart irrigation. In addition to this main taxonomy, we identified the presence of "generic clusters" that can be used in different Smart applications. For instance, Smart energy is an independent application domain, within Smart industry. However, we can find it in other domains: Smart city, Smart home, etc. Another identified cluster is the Smart transportation one. This specific cluster identifies the Smart applications related to the transportation domain. These applications can be relative to several other applications, like Smart vehicle charging which can be applied in Smart cities (to identify the flow of electric vehicles in a specific area and the distribution of energy in the grid), or in a Smart home (to analyze the consumption of a specific house).

Fig. 4. Smart Life Applications for Enterprises.

We observed that new notions are introduced in the literature, even if they are not completely implemented yet. Smart earth is defined as an innovative approach to environmental monitoring in [START_REF] Bouchrika | A Survey of Using Biometrics for Smart Visual Surveillance: Gait Recognition[END_REF]. Smart island is an attempt to classify the existing smart-related methodologies in the urban context to analyze their possible application to the islands [START_REF] Desogus | Smart Islands: A Systematic Review on Urban Policies and Smart Governance[END_REF]. Smart state is described in [START_REF] Hortz | The Smart State test: A critical review of the Smart State Strategy 2005-2015's Knowledge-Based Urban Development[END_REF] as a direction of "Knowledge-Based Urban Development". In [START_REF] Kaur | A Systematic Literature Review of Blockchain Technology for Smart Villages[END_REF], a literature review is done on the usage of the blockchain technology in Smart villages. Finally, the term Smart world appeared in the literature [START_REF] Kumari | A Review on Hybrid WSN-NGPON2 Network for Smart World[END_REF] [27] as a unifying term of various Smart applications. For instance, [START_REF] Liu | A review of the smart world[END_REF] defines Smart world as "an attractive prospect with comprehensive development of ubiquitous computing involving penetrative intelligence into ubiquitous things". In our vision, this term corresponds to our vision of Smart life.

When constructing the Smart life applications taxonomy, we excluded 328 sources dealing only with Smart artefacts, Smart technologies, and Smart systems not related to specific application domains. The taxonomy does not either include Smart infrastructure-related applications as we foresee them as transversal issues, we can find infrastructure almost in all major Smart applications fields.

The evolution of Smart domains is shown in Table 4 for the period from 1986 to 2022 (publications already available in 2021). We grouped the first 21 years in the same column as the number of publications was very law for this period. We complete this chronological vision by a graphical representation illustrating the distribution of Smart domains (See Fig. 5).

Sources of Papers Publishing on Smart Applications

One of our goals is to analyze the venues of papers published on Smart life. This question leads us to study the distribution of publications by type of venue: conferences, journals, or workshops, resulting in the distribution presented in Table 5. The most used type is journals (66%) with conferences at second place (27%), and a very small number of venues are presented by books and workshops. We identified the principal venues publishing on Smart life (Table 6). We selected venues that published 10 and more papers on Smart life state-of-the-arts and research agendas. Three first places are taken by Renewable and Sustainable Energy Reviews (74 venues), IEEE Access (65 venues), and Energies (43 venues), thus 117 venues in

Smart Applications Underlying Terminology

We created two keyword clouds: one based on the title of the selected publications (Fig. 6) and the second one based on their abstract (Fig. 7). The size of the word in the cloud indicates its frequency in the dataset. The titles contained 2023 occurrences of the word "Smart", which put it in the first place before the others. Table 7 shows the frequency of the most popular words in this title dataset (over 200 occurrences). Some words concern the main objective of this study which is "Smart" of course. We can identify also that we are in the domain of "internet", "data", "system", "things", "IoT", and "systems" (1475 occurrences). What is more interesting however is the frequency of the set of terms "grid" and "energy" (901 occurrences) which means that this specific topic is at the heart of the selected research works in this study. The term "city" appears 342 times, "security" 210 times, and "management" 249 times which are still important frequencies. The dataset based on the selected publication abstracts contained 8098 occurrences of the word "Smart", four times more than the next frequent word. Table 8 shows the frequency of the most popular words in this dataset (over 1000 occurrences). Apart from the "Smart" word, we can also identify words concerning our research method, like "papers", "research", "study", "review" and "challenges" (7941 occurrences), as we restricted our SMS to papers which were literature studies. As in the title dataset, we identify words about our domain of "system", "technology", "data", "network", and "application" (9655 occurrences). Again, the frequency of the set of terms containing "grid", "energy", and "power" (5384 occurrences), reiterates our preceding finding that this domain is massively present in the selected publications. The term "city" appears 1890 times which is still an important frequency.

Based on the study of the keywords, we established that, except for the research study terms and the development-related ones, the most used application areas are home, agriculture, health, energy, and parking. The main purposes are metering, monitoring, manufacturing, and intelligence. The most present characteristics are mobile, electric, sustainable, and renewable. And, finally, the more frequent technologies are sensors, grids, state of control, and network.

Smart Life Driving Factors

Addressing Smart life applications raises some specific requirements. The most important ones, in our humble point of view, are related to driving perspectives. What are the main issues we should take care of to be able to successfully build this new generation of applications? There is many, but let's focus on what we expect to be the most significant ones through application domains.

The first one corresponds to the ability to scale. From Smart cities to Smart homes, applications and technologies have to face numerous units: sensors, computers, communication devices, people or users, buildings, data produced and consumed… Their considerable quantity implies the ability to scale, eventually to scale to the world. The panoptic model from Jeremy Bentham [START_REF] Bentham | An Introduction to the Principles of Morals and Legislation[END_REF], already contradicted by Michel Foucault, is no longer a reference. This has already been considered by Cloud providers as they adopted the CAP Theorem stating that, among the 3 properties of Consistency/Adaptability/Partitioning, you can only insure two of them. This is due to the deep nature of Cloud Computing and its inherent dimensions of distribution and scaling.

Application designers will address safety and security as key issues. They are mandatory, we fully agree and there is no discussion about these requirements. It is very general for any application that involves human beings in someplace. From our point of view, ubiquity -sometimes also addressed as pervasivity -belongs to the deep heart of Smart life applications. Nowadays, through the pandemic crisis, most people have experienced collaborative tools such as Zoom, Teams, Google meet, or whatsoever. These tools bring more than connectivity. They took a part of ourselves and put it in some shared void enforcing belongingness. We are at the beginning of a new era where we are in the physical world and one or more virtual worlds at the same time. From Building Information Modelling (BIM) to Metaverses, the feeling of being "more than one" and being "only one" at the same time is enforced. Maybe we are moving forward to the empathy box of the mercerism described in "Do androids dream of electric sheep" from Philip K. Dick [START_REF] Dick | Do Androids Dream of Electric Sheep[END_REF]?

The relationship between human beings and technology is an emerging issue. From a human perspective, we are driving the system, not the other way round. But this is not just black or white; there is a blurred frontier between what human needs and what Smart life brings. Sentience is the key issue dealing with requirements. Sentience has been defined a long time ago, especially in the context of animal rights. Such a concept can be derived and adapted to technology. It is easier to understand in the context of Robots, more precisely when you apply it to robots' consciousness. Recently, this concept has started to be refined in [START_REF] Kornyshova | Introducing Sentient Requirements for Information Systems and Digital Technologies[END_REF]. ICT should be able to sense human expectations without any explicit specification and adapt to these expectations consistently. Modern application designers will take care of sustainability and ecological footprint. We deeply believe that a sentient-ready application or technology will be able to fully take care of this requirement if properly designed and self-aware.

Conclusion and Future Research

Our main goal is to promote the field of Smart life and Smart life engineering to provide fundamentals for more efficient and sustainable research efforts. In this paper, we summarize the research work on "Smart" state-of-the-arts and research agendas to identify and classify Smart applications. We carried out a systematic mapping study based on 2341 publications to define a taxonomy of Smart applications, understand their distribution, analyze sources of the research works, and identify the underlying terminology. Finally, we discuss driving properties allowing us to move forward the Smart life.

Based on the obtained results, we plan to investigate how the identified driving properties could be considered to identify possible relationships between different Smart applications. We will carry out a bibliometric study for this purpose. We also intend to go further with the identification of research challenges in the field of Smart life and Smart life engineering. Our main goal is to discover if common methodological basics could be identified in the field of Smart life.

Fig. 1 .

 1 Fig. 1. Smart Life Domains.

Fig. 2 .

 2 Fig. 2. Smart Life Applications for Persons.

Fig. 3 .

 3 Fig. 3. Smart Life Applications for Environment and Society.

Table 4 .

 4 Smart Domains Evolution. We can see that the first reviews appearing in the literature concern Smart applications for enterprises (in 1986). Then literature shows interest in two other domains, namely Smart applications for persons and for society, in the 2000s. More recently, Smart applications for the environment came out more prominently. In the Smart applications for enterprises domain (1142 papers), Smart industry dominates (891 papers), essentially with Smart energy-related topics, followed by Smart agriculture (104 papers), Smart transportation (78 papers), and Smart business management (60 papers). Smart home and Smart healthcare are equally represented in Smart applications for persons (191 and 190 papers respectively over 408), with a little part for Smart welfare. Smart applications for Environment are only represented by 16 papers, half of them focused on Smart natural resource management (9 papers). Smart city is taking the big part of the Smart applications for society (384 papers over 440), followed by Smart education (43 papers).

Fig. 5 .

 5 Fig. 5. Smart Domains Distribution.

Fig. 6 .

 6 Fig. 6. Word Cloud Based on the Papers Titles.

Fig. 7 .

 7 Fig. 7. Word Cloud Based on the Papers Abstracts.

Table 1 .

 1 Five-Steps Procedure of SMS on Smart Applications.

	Number SMS Process Step	Outcome
	1	Definition of Research Questions	Review Scope
	2	Finding Papers	All Papers
	3	Screening of Papers	Relevant Papers
	4	Keywording using Abstracts (Classifi-	Classification Scheme
		cation Scheme Definition)
	5	Data Extraction and Mapping Process Systematic Map

Step 1: Definition of research questions. The

	goal of this SMS is to identify and
	structure different Smart applications. To detail this goal, we established the follow-
	ing research questions:
	RQ1. What are the different applications studied in the Smart life field and their
	evolution?
	RQ2. In what venues are published research papers on Smart life applications?
	RQ3. What is the specific terminology used in Smart life applications?
	RQ4. What are the main driving factors of Smart life based on data about Smart
	applications?

Table 2 .

 2 Inclusion/Exclusion Criteria for the Study on Smart Topics.

	Selection Criteria Criteria Description
	Inclusion criteria
	(2410 sources
	identified)

Table 5 .

 5 Papers Distribution by Publication Type.

	Publication Type	Number	Percentage
	Journal	1553	66%
	Conference	629	27%
	Book or Book chapter	149	6%
	Workshop	10	1%

Table 7 .

 7 Popular Words Frequency in the Titles.

	Word	Frequency	Word	Frequency Word	Frequency
	Smart	2023	Internet	266	Things	245
	Grid	476	Data	255	IoT	241
	Energy	425	System	252	Systems	216
	City	342	Management 249	Security	210

Table 8 .

 8 Popular Words Frequency in the Abstracts.

	Word	Frequency	Word	Frequency Word	Frequency
	Smart	8098	Energy	1806	Review	1456
	System	2803	Research	1795	Study	1287
	Technology 2403	City	1890	Power	1252
	Papers	2330	Data	1736	Challenges 1073
	Grid	2326	Application 1669	Network	1044

https://www.elsevier.com/__data/assets/pdf_file/0007/917179/Scopus-User-Community-Germany-API-final.pdf

[START_REF] Chamee | Historicizing the smart cities: Genealogy as a method of critique for smart urbanism[END_REF] The complete list of these papers is available at http://cri-dist.univ-paris1.fr/rcis22/RCIS2022_Appendix%20A%20-%20references%20list.pdf.

the field of energy, and 65 generic venues. 262 venues have between 2 and 9 papers, and 931 venues have only one paper.


Philippe Bertrand
email: philippe.bertrand@univ-amu.fr

Black-Scholes Approximation of Warrant Prices: Slight Return in a Low Interest Rate Environment

Keywords: JEL classication: G11, G13, G21, D03 Warrant, Option, Black-Scholes

The objective of this paper is to emphasize the di¤erences between a call and a warrant as well as the di¤erent valuation methods of warrants which have been introduced in the nancial literature. For the sake of simplicity and applicability, we only consider a debt-free equitynanced rm. More recently a formal distinction between structural and reduced form pricing models has been introduced. This distinction is important whether one wishes to price a new warrant issue or outstanding warrants. If we are interested in pricing a new issue of warrants, e.g. in the context of a management incentive package, one has to rely on a structural model. However most of practitioners use the simple Black-Scholes formula. In this context, we analyze the accuracy of the approximation of the "true" price of a warrant by the Black-Scholes formula. We show that in the current low interest rate environment, the quality of the approximation deteriorates and the sensitivity of this approximation to the volatility estimate increases.

Introduction

Warrants are long term call options to buy the rm's own shares and are issued by the rm itself.

Because of this latter characteristic, warrants are more complex to value than exchange-traded options.

As in credit risk studies [START_REF] Abínzano | Pricing levered warrants with dilution using observable variables[END_REF] , warrant pricing models fall into either the "structural" or the "reduced form" category. Strictly speaking, this distinction has been rst put forward quite recently in the literature on warrant pricing by Jarrow and Trautmann (2011). But the works of Bensoussan, Crouhy and Galai (1994, 1995a and 1995b) already made this distinction without using the same terminology. The starting point for the analysis of a structural model is the evolution of structural variables of rms, such as the value of assets and debt, to determine among other things the value of warrants. These models deal directly with the dilution e¤ect, that is the increase in shares if warrants are exercised. To the contrary, a reduced-form model starts with the assumptions about the stock price process and the outstanding warrants, meaning that they have already been issued (see Jarrow and Trautmann, 2011).

The distinction between these two types of models shows up particularly useful when we want to distinguish the valuation of an outstanding warrant from that of a warrant to be issued. A structural model, based on the value and on the volatility of the rm's assets, is well designed to price a new issue of warrant because in this case the stock price does not already reect the fact that the warrant exists. As rst shown by Crouhy and Galai (1991b) and Handley (2002) and then developed in more depth by Jarrow and Trautmann (2011), a reduced form model is more appropriate for outstanding warrant because in a e¢cient market, "the stock price of the underlying rm conditionally reects dilution at all times following announcement of a warrant issue". Therefore, "pricing a warrant as a call option on the stock of the underlying rm does not actually ignore dilution because dilution already should be reected in the underlying stock price before expiration of the warrant".

In the present paper, we don't address the issue of possible sequential optimal exercise of American warrants before maturity. The treatment of this issue is available in Chapter 19 of Ingersoll (1987). In addition, Koziol (2006) addresses extensively the issue of optimal exercise strategies for warrants issued by levered rms. Instead, we make here the assumption that American warrants are exercised as a block at maturity. Thus, we suppose that no dividends are paid (see A2 Schulz and Trautmann 1994).

Another point that needs to be addressed has to do with the cash received by the rm from the sale of warrants. The objective here is to ensure that these cash ows do not modify the rate of return and the risk of the rm's assets. Several possibilities are available : the issue proceeds are immediately distributed as cash dividends to shareholders, alternatively the rm can repurchase its stocks or use the proceeds to invest in a scale expansion of the rm. Following Galai and Schneller (1978), we retain the cash dividends assumption [START_REF] Barthélémy | Mixed-asset portfolio allocation under mean-reverting asset returns[END_REF] .

When a new issue of warrants has to be priced, the issuer need to know which model he can use and, if so, which approximation is relevant. It is particularly useful in the context, for example, of warrants as support for incentive management programme. Such warrants are usually long term out-of-the-money warrants. On the contrary when dealing with outstanding warrants, the price of the underlying security already reects the dilution. In this paper we are primarily interested in the pricing of new warrant issues in the simple case of a no-debt rm and the type of approximate formula that can be used by an issuer.

The remaining parts of the paper are organized as follows. In section 2, the warrant mechanisms are recalled. In section 3, the di¤erent warrant pricing models derived in the literature are presented and discussed. The quality of the approximation of the price of a warrant by a simple application of the Black-Scholes model is assessed in Section 4, with an emphasis on these results in a low interest rate environment. Section 5 summarizes the conclusion of this article.

Warrants mechanisms

In this section, we follow the standard presentation of Galai and Schneller (1978). A warrant is a call option written by a rm on its own common stock. As such, it is the right to buy a share [START_REF] Black | The pricing of options and corporate liabilities[END_REF] of the rm at a certain price (exercise price) during a given time period. If a warrant is exercised, new shares are issued by the rm, and the cash payment that is made by warrant buyers may or may not increase the assets of the rm depending of the scenario we are considering [START_REF] Crouhy | Warrant valuation and equity volatility[END_REF] .

While the call option is issued by an outside business organization such as the CBOE, the warrant is issued by the rm and its proceeds are a part of the rms equity. If a warrant is exercised, it increases the number of outstanding shares of the rm and thus dilutes the equity of its stockholders.

For comparison purpose, we analyze the same rm, i.e. a rm with same value of assets, but with two di¤erent capital structures: a pure equity nanced rm and a rm nanced both by equity and warrants. Moreover, we make the unrealistic assumptions that the rm has no debt. For the introduction of debt into the analysis of warrant pricing, we refer to Crouhy and Galai (l994), Abínzano and Navas (2013) and Simonato (2015). The rm operates over the time period

[�� � ],
at time � warrants are issued with expiration at time � . The time � price of a warrant is denoted by � � . The pure equity rm has � shares of stock outstanding at time � and no debt. The rms assets value at time � is denoted by �� . The value of this rm at time � is denoted by

� ¤ � with � ¤ � = � � ¤ � .
The features of this rm are displayed in Table 1 rst three rows. The price of a share at time � is �¤ � and it is equal to �� �� . For the rm which plans to issue � European warrants with an exercise price equal to �, some additional assumptions are in order. As already said, it is assumed that the proceeds received from the warrant buyers are distributed as cash dividends to the existing shareholders [START_REF] Crouhy | Common Errors in the Valuation of Warrants and Options on Firms with Warrants[END_REF] .

If at time � the warrants are exercised, the rm will be worth �� + �� and there are (� + �)

shares of stock outstanding. The price of share is now, �� = �� +� � � +� . Thus, each warrant is exercised if :

�� = �� + � � � + � � � , �� ¡ �� � + � � 0 , � � + � ³ �¤ � ¡ � ´� 0 (1)
Finally, the warrants is exercised if �¤ � � �. Table 1 shows that on the liability side of the balance sheet, the two rms have the same equity value at time � , �� and thus it must also be true at time

�� � � = � � � + �� � = �� ¤ � = � ¤ � .
Thus, initial shareholders wealth is not a¤ected by the issuance of warrants. On the assets side [START_REF] Crouhy | The interaction between the nancial and investment decisions of the rm: the case of issuing warrants in a levered rm[END_REF] ,

� � = � ¤ � + � ¡�� E Q � [1 �¤ � �� � �],
that is the value of the rm with warrants is higher than the value of the pure equity rm. At rst glance, this seems to contradict Modigliani-Millers Proposition I (1958) which states that in perfect capital markets, the value of a rm is determined by the value of its assets and is independent of its capital structure, i.e. how the rm nances its investments.

But what matters is shareholders wealth, which is not a¤ected. Moreover, because we take into account the present value of the additional expected cash ow arising at time � due to the potential exercise of the warrant, we must also take into account that the number of shareholders at time � is also expected to be higher than � , so that equality on both sides of the balance sheet is restored.

The price of a share of the rm with warrants is lower than the price of a pure equity rm:

� � = � ¤ � ¡ (��� ) � � .
This is due to the cash payment of �� � and again reects expectations of the dilution that could occur at time � .

Valuation of Warrants

Three general approaches to valuing warrants have been introduced in the academic literature (Black and Scholes 1973, Galai, and Schneller 1978 and Handley 2002) :

1. First, warrants are analyzed as call options on the value of the rm (i.e., value of its shares of common stocks and its warrants), 2. Second, warrants are analyzed as call options on the stock of an otherwise identical all-equity rm 7 , [START_REF] Crouhy | Common Errors in the Valuation of Warrants and Options on Firms with Warrants[END_REF] In this case, the price of the share must adjust downward because some money went out of the rm while at the same time the equity increases at time �, conditionally on the possibility of warrants exercises at time � , see Handley (2002). [START_REF] Crouhy | The interaction between the nancial and investment decisions of the rm: the case of issuing warrants in a levered rm[END_REF] Q denotes the risk neutral probability measure such that the price of each risky cash ow whose value depends on the only source of risk, i.e. the rms assets value which is supposed to follow a geometric Brownian motion, is equal to the discounted expectation of its future value under this measure. The symbol 1 �¤ � �� denotes the indicator function which is equal to one if �¤ � � � and to zero if �¤ � • �. The risk-free interest rate is � and the time to expiration of the warrant is � = � ¡ �. [START_REF] Bensoussan | BlackScholes approximations of warrant prices[END_REF] An identical rm is a rm with the same assets value, � .

�¤ � = �� �� �¤ � = �� �� (c) Value of rm � ¤ � = �� ¤ � � �¤ � = �� � �¤ � = �� 2) Firm with warrants (a) Number of shares � � � + � (b) Value of a warrant � � 0 �� +� � � +� ¡ � = � � +� ( �¤ � ¡ �) (c) Value of a share � � �� = �¤ � �� = �� +�� � +� (d) Value of rms equity � � = � � � + �� � = � ¤ � ´�� � �� = �� � �� + �( �� ¡ �) = �� (e) Value of rm � � = � ¤ � + � ¡�� E Q � [1 �¤ � �� ��] �� �� + �� 3.
Third, warrants are analyzed as call options on the stock of the underlying rm.

These three approaches are all valid but in di¤erent contexts. They are general in the sense that they dont depend on a specic dynamics for the underlying security, either the value of the rm or the value of the stock.

As already mentioned, a more recent and distinct typology classies the warrant pricing models according to their "structural" or "reduced-form" nature. The latter is not particularly useful for the issuance of new warrants. Therefore, the structural approach is the more appropriate when a new warrant issue has to be priced, for example in the context of a management package.

Warrant as a call option on the value of the rm

The Black-Scholes model applied to the value of the rm

This model belongs to the structural approach because the exogenous variables are the value of the rm and its volatility and, consequently, the value of the stocks and warrants of the rm are endogenous to the model.

In table 1, row (2.b) displays the value of the warrant at maturity, � :

� � = 1 � + � ( �� ¡ ��) + (2)
= � � + � ( �¤ � ¡ �) + (3)
In the Black and Scholes (1973) framework, the following assumptions are made: the nancial markets are perfect and complete and transactions takes place in continuous time. There is no arbitrage opportunity. The risk-free interest rate is constant, the stock does not pay a dividend and its price follows a Geometric Brownian Motion (GBM hereafter) with constant drift and volatility.

Here, this model is applied not to the stock price but to the rms asset value, �� , whose dynamics is described by a GBM. It follows from all the previous assumptions that a unique equivalent martingale measure (commonly referred-to the "risk neutral probability") exists under which the stochastic process for � becomes:

�� � � � = ��� + � � � ��
where �� is a standard Brownian motion with respect to the equivalent martingale measure, Q.

� � is the volatility of the instantaneous return of

� � �� = � � + (���) � � ,
Thus, the price of any assets in this complete market is simply the discounted expected value of its future payo¤ under Q.

Then, the following formula for the warrant price is obtained:

� � (� � � � � ; � � �� �) = 1 � + � (� � ©(� 1 ) ¡ ��� ¡�� ©(� 1 )) = � � + � •µ � � + � � � � ¶ ©(� 1 ) ¡ �� ¡�� ©(� 2 ) = � � + � � �� � (� � ��� � � ; � � �� �) (4) with � 1 = ln µ � � + � � � � � ¶ + (� + � 2 � �2)� � � p � � 2 = � 1 ¡ � � p � where ©(�)
is the cumulative distribution function of the standard normal distribution.

Expression (4) for the value of the warrant corresponds to the standard Black-Scholes formula in which:

-the value of the equity per share � � �� = � � + (��� ) � � replaces the stock price � � � -the volatility � � is the volatility of the equity of the rm (i.e., shares of stock plus warrants) instead of just the stocks volatility, � � , -the formula is multiplied by �� (� + � ).

This valuation formula is known in the warrant literature as the classical warrant formula or the "correct warrant valuation model" (Veld 2003). The main obstacle to the application of this formula is that it depends on the value of the rm, � � , and on its instantaneous volatility, � � , which are not observable. Moreover, the value of a warrant depends on its own value through the value of the rm (i.e., � � = �� � + � � � ) on the right hand side of expression (4). This simply requires the use of a numerical root nding routine.

As shown by Crouhy and Galai (1991a) and Bensoussan, Crouhy and Galai (1994, 1995a and 1995b), the fact that the model starts from the exogenously given stochastic process of the value of the rm, �� which is not observable and not from that of the stock price, �, implies that the stock price itself must be analyzed as a contingent claim on the rms asset, � . This in turn implies that the stochastic process for the stock price is no longer stationary which means in particular that its volatility is not constant over time. This comes from the impact of warrants issuance on the capital structure of the rm. As mentioned by Bensoussan et al. (1995): "By issuing warrants, part of the total asset risk is shifted from the stockholders to the warrant-holders, and the degree of this risk-sharing changes continuously as a function of the asset value and the time-to-maturity of the warrants." The formula (4) is not really useful in practical applications because it relies on non observable variables, � � and � � . Its why, based on the previous model, the authors cited above in the title of the section introduce a method to express the warrant value as a function of only observable variables, namely the current stock price and the volatility of the stock return. Therefore, we must nd a way to express � � and � � as function of � � and � � :

� � ´� (� � � � � ) and � � ´�� (� (� � � � � )) � 1.
We already have the relationship between � � and (� � � � � ) (see Table 1, line 2.d.):

� � = � � �� ¡ (���) � � (� � � � � ) (5)
2. To nd the relationship between � � and (� � � � � ), we start with the observation that, for a $1 change in the rms value (¢ � = 1), we have from ( 5):

¢ � = 1 � ¡ � � ¢ � = 1 � ¡ � � (� + �) © (� 1 )
If we analyze the stock as a contingent claim on the value of the rm, the expression of the elasticity of the stock price with respect to the value of the rm is given by: ���

= (����� ) ¢ (� � �� � ) = ¢ � (� � �� � ).
A standard result from stock option theory states that the the standard deviation of the rate of return of a call equals its elasticity times its underlying stock volatility (see Cox and Rubinstein, 1985). This translates here into the following [START_REF] Bensoussan | Stochastic equity volatility related to the leverage e¤ect I: Equity volatility behaviour[END_REF] :

� � (� (� � � � � )) = ��� ¢ � � (6)
Expression (6) shows that the volatility of the stock is not constant, it is a function of the value of the rm and of time. Furthermore, because stock and warrant returns are perfectly positively correlated, the volatility of the assets of the rm is simply the weighted average of the stocks volatility and of the warrants volatility:

� � = � � ��� � � � + � � ��� � � � (7)
If the rms asset risk is assumed to be constant over time, � � = ����, and because the warrant is riskier than the stock, relationship (7) implies that the volatility of the stock is less than the volatility of the rms assets. Note that when a rm is also nanced by debt, the volatility of the stock may be higher than the volatility of the rms assets. As recalled by Bensoussan, Crouhy and Galai (1994 and 1995b): "While debt causes the risk of equity to increase relative to the volatility of the rm, the issuance of diluting securities like warrants has the opposite e¤ect". Moreover, expression (7) also implies that the stocks volatility varies over time. In fact, as a function of � � and therefore � � , the stocks volatility is stochastic.

Given � � and � � , the price of the warrant is obtained in two steps: 1. First, the following system of nonlinear equations for (� � � � � ) must be solved numerically:

( � � = � � �� ¡ (���)� � (� � � � � ) � � = ¢ � (� � �� � )� � (8)
2. Second, the warrant price is given by: 2004) proves that this system always admits a solution and Pechtl and Trautmann (2003) prove its uniqueness.

� � (� � � � � ) = � � ¡ � � � � Ukhov (

Warrant as a call option on the stock of an identical all-equity rm

The value of a call option on the stock of an identical ctitious all-equity rm at maturity, � � , is :

� � = ( �¤ � ¡ �) +
Relationship (3) allows to write:

� � = � � + � � �
Thus by virtue of the principle of no arbitrage opportunity:

� � = � � + � � �� � (� ¤ � � � � ¤ ; � � �� �)� 8� � �
The main limitation in applying this approach is that it is unlikely to nd an identical all-equity rm to the one that issues warrants.

Warrant as a call option on the stock of the rm

This model belongs to the reduced-form approach. In this case, the warrants can be simply valued as a call options on the stock of the rm:

� � = � �� � (� �� � � ; � � �� �)� 8� 2 [0� � ]
The use of a Black-Scholes model with a GBM for the stock price dynamics seems to contradict what was said in section 3.1.1 on the variation over time of stock volatility in a structural model. But, Jarrow and Trautmann (2011) showed that a reduced-form model could be compatible with a GBM, in particular with a constant volatility of the stock. Their argument rely on the di¤erence of the information set [START_REF] Bensoussan | Stochastic equity volatility related to the leverage e¤ect II: valuation of European equity options and warrants[END_REF] in a structural and a reduced form model.

For this formula to be valid, the share price must already reect the issue of warrants and therefore the dilution that it entails. As pointed out by Handley (2002), when a warrant is issued for cash and the proceeds reinvested in a scale expansion of the rms assets, the stock price should not adjust. On the contrary, if the proceeds are paid to shareholders in the form of dividends, the share price should adjust downward by an amount close to that of the warrant price.

Thus either the warrants are already outstanding or the issue is about to occur and, in an e¢cient market, this information is already incorporated into the stock price. Note that in the case considered here, it is not necessary to explicitly adjust the call formula by a dilution factor since the dilution is already implicitly reected in the stock price.

If the warrant is valued as a function of the stock price before the market has this information, then an explicit adjustment for dilution must be made (Handley, 2002).

Approximation quality of a simple Black-Scholes model

Because only � � and � � are observable, most practitioners base their Black-Scholes type warrant pricing model on � � and a constant � � . But instead of solving the system of equations ( 8), they simply value a warrant as a call option on the stock of the rm as in section 3.3. Then, it is interesting to evaluate the quality of such an approximation compared to the correct valuation model of section 3.1.1. Before that, we perform numerical simulations to highlight the non-stationarity of the stocks volatility.

Stocks Volatility

As rst shown by Crouhy and Galai (1991a), in a structural model like the one in section 3.1.1 the stocks volatility is a function of � and � given by � � ´�� (� � (� � � � � ) � �). Then, it would be interesting to analyze how stocks volatility changes with � and �. To this end, we start from given � and � � and derives the corresponding � and � � . Using the approach of Schulz and Trautmann (1994), we use the system of equations ( 8) with the following parameter values: � � = 0�3� � = 0�01� � = � = 1� � = � �� = 100 (we consider a rms value of � = 100 as our base case). We perform simulations for a range of values of � from 0 to 500 and � from 0 to 10. For each pair (�� � ), we compute the corresponding (�� � � ). The results are shown in Figure For � 6 = 0, stocks volatility rst decreases with �, reaches a minimum then becomes increasing and converges toward � � as � becomes very large. As the expiration date approaches (����� � ! 0), stocks volatility behavior becomes unstable around ��� = � = �. This is due to the behavior of the options delta which converges either to 1 for � �� � �� 1�2 for � �� = � and 0 for � �� � �. The volatility of the stock 10 is equal to � � as long as the option is out-of-the-money, then it jumps to � � = [START_REF] Black | The pricing of options and corporate liabilities[END_REF] 4 � � at � = � and reaches a minimum value equal to � � = 1 2 � � for � ! � + after which it increases and converges to � � .

We also display on Figure (2) the e¤ect of a lower level of dilution by considering � = 1� � = 10 with � = � �� = 10.

10 By using the expression for the stocks volatility,

�� = � � � 1 � ¡ � � (� +� ) © (�1)

�

�� � and the fact that

lim �!0 + ©(�1) = � � � � � 1 if � �� � � 1 2 if ��� = � 0 if � �� � �
we obtain the desired result The general shape of stocks volatility surface is the same as the one with � = 1. As expected, the only di¤erence is that the minimum stocks volatility is now much higher and is equal to 27�27%. This comes from the much lower dilution. Likewise, the range of variation of the stocks volatility is narrower. The fact that the dilution is 10 times lower than in the previous case means that the share of the value of the warrant in the total value of the rm is much lower. Thus, there is less of the risk of the rm that is transferred from stockholders to warrant holders, which explains the higher level of stocks volatility.

lim �!0 + �� = � � � � � 1 2 �� if � �� � � 3 4 � � if � �� = � �� if � �� � � .
The previous discussion remains exactly the same for other values of the rms volatility.

As a function of time to maturity, the stockss volatility is decreasing as long as � � � � = ���.

For � � ¸� = � ��, it becomes an increasing almost "linear" function of time to maturity. Lets go into more detail.

The behavior of the volatility of the stock with respect to the time to maturity for a given (�� � � ) is driven both by the stocks price � and by its delta ¢ � . Ceteris paribus, the stock price increases when the expiration date approaches [START_REF] Galai | Pricing warrants and the value of the rm[END_REF] and the delta behavior depends on the moneyness of the warrant. Taking into account both e¤ects, we can conclude that the volatility of the stock decreases almost linearly as we get closer to maturity and for deep in-the-money warrant.

As the warrant get closer to the money, the relationship is still decreasing but "less linear".

For out-of-the-money warrant, the stocks volatility increases when the warrant approaches the expiration date. Note that for the slightly in-the-money warrant, the range of variation in the volatility of stock is the greatest and is about 1 2 � � for very long term warrants. An estimate of the standard deviations of the time series of the stocks rate of return instead of the volatility of the rms asset will lead to an underestimation of the latter and therefore of the [START_REF] Galai | Pricing warrants and the value of the rm[END_REF] Because of the warrant theta.

price of the warrant. This underestimation is all the more important as the warrant is close to the money and near maturity [START_REF] Escobar | Portfolio optimization under Solvency II[END_REF] .

Black-Scholes approximation

It is tempting for practitioners to value a warrant by simply applying the Black-Scholes formula without any adjustments. Formally, this is correct if the stocks price already reects the information about the warrants issuance. But if we try to evaluate the issue price of a warrant before the market has this information, then that is no longer correct.

Another practice would be to make the wrong adjustment as highlighted by Crouhy and Galai (1991b). For example, the model of section (3.2) can be applied, but on the rm with warrants whereas it is only a correct model for an identical pure-equity rm. By doing this, the additional dilution factor in the expression � = ³ � �+� ´� will biased the valuation of the warrant. In any case, we will analyze the accuracy of the Black-Scholes valuation of the price of a warrant compared to the correct valuation of a warrant for which we use information available to the market, (�� � � ).

In this section unlike the previous one, given (�� � � ), we numerically solve the system of equations ( 8) to obtain (�� � � ) and then the warrant price, � . Proceeding in this way for each value of the parameters, we are led to compare rms with di¤erent asset values, which does not seem very consistent. But remember that our objective here is to compare the accuracy of the Black-Scholes approximation for a given set of parameter values. Moreover, we still rely on the same parameter values as those in the previous sections [START_REF] Handley | On the Valuation of Warrants[END_REF] . Specically, we assess the accuracy of the approximation by calculating the following error term:

�(� �� � � ; � � �� �) = � �� � (� �� � � ; � � �� �) (��(� + �))� �� � (� � ��� � � ; � � �� �) ¡ 1 (9)
First of all, it is worth noting that most of the time the error term is positive, which means that the Black-Scholes approximation overestimates the true value of the warrant. By inspecting expression (9), we see that the sign of �(� �� � � ; � � �� �) depends on two e¤ects working in opposite directions:

-rst, given that

� � • � � and � • ��� : � �� � (� � ��� � � ; � � �� �) � � �� � (� �� � � ; � � �� �)� -second, ��(� + � ) � 1.
We simulate the error term for the following set of parameter values:

� = 1%, � 2 [50; 150]� � � 2 [20%; 100%]� ��� 2 [0�1; 1]
and � 2 f0�5� 5� 10g. The Black-Scholes approximation works very well for at-the-money and in-the-money warrant. The percentage price di¤erence is in the range [¡1�41%� 3�61%] for � 2 [100; 150]. [START_REF] Ingersoll | Theory of Financial Decision Making[END_REF] As expected, the approximation is all the more precise [START_REF] Escobar | Portfolio optimization under Solvency II[END_REF] This remark does not apply to the case where an estimate of �� is used to nd �� by solving the system of equations (8). [START_REF] Handley | On the Valuation of Warrants[END_REF] Because the current level of interest rates in the markets is very low, we use � = 1%. To assess the e¤ect of a higher interest rate, we consider in the next section the e¤ect of a � = 10% on the approximation error committed. [START_REF] Ingersoll | Theory of Financial Decision Making[END_REF] The lower bound of ¡1�41% is obtained for ��� = 100%, � � = 20%� � = 0�5 and � = 107. The upper bound of 3�61% is obtained for ��� = 100%, �� = 88%� � = 5 and � = 100.

the lower the dilution. For out-of-the-money warrant, the range of the approximation error is wider and is equal to [0�022%� 100�11%] for � 2 [50; 99]. [START_REF] Jarrow | A reduced form model for warrant valuation[END_REF] Note that in this case, the approximation error is always positive. This is only true because the interest rate is 1%. For example, if the interest rate is 10%, negative approximation errors occur as soon as � ¸96 and � = 0�5.

We display on gures 3, 4 and 5 the approximation error as a function of both the dilution (��� 2 [0�1; 1]) and the stock volatility � � 2 [20%; 100%]. On each gure, we draw three surfaces for the time to maturity � 2 f0�5� 5� 10g.

The analysis of the e¤ect of the stocks volatility on this price di¤erence is complex as can be seen on Figures 3, 4 and5. The quality of the approximation depends also on the dilution, on the time to maturity and on the moneyness of the warrant. We will distinguish three cases according to the moneyness of the warrant:

1. � = 50� � = 10 and � = 100: for a high dilution of 100% and a low volatility level of 20%, the price di¤erence is very high at 18�82%. This happen for a relatively high level of warrant price 3�25 and a corresponding call value of 3�86. For this dilution of 100%, the price di¤erence is rst decreasing then slightly increasing and nally decreasing again with � � as can be seen on Figure 3. It is still equal to 4�3% for � � = 60%. When � = 5, the behavior is the same but the highest di¤erence is now equal to 47�89%� but for a much smaller warrant value of 0�69 and a corresponding call value of 1�02. Although, the price di¤erence looks very important for � = 0�5 (about 100% !), it has no economic meaning because the prices of both the Call and the warrant are negligible. Notes: This gure shows the approximation error for a deep out-of-the-money Warrant (� = 50 and � = 100) as a function of the volatility, �, and of the dilution, ��� , for three maturities, � = 0�5, 5 and 10 years. [START_REF] Jarrow | A reduced form model for warrant valuation[END_REF] The lower bound of 0�022% is obtained for ��� = 10%, �� = 20%� � = 5 and � = 99. The upper bound of 100�11% is obtained for ��� = 100%, �� = 22%� � = 0�5 and � = 56.

Thus, for out-of-the money warrants, the use of a simple Black-Scholes formula instead of the correct valuation formula for warrants leads to an overestimation of the warrant price which can be very important even for time to maturity equal to 5 and 10 years. In this case, only overestimates occur.

2. � = 100� � = 10 and � = 100 (see Figure 4) [START_REF] Jarrow | Structural versus reduced form models: a new information based perspective[END_REF] : for a dilution of 100% and a low volatility level � � = 20%, the price di¤erence is much lower than in the previous case at 1�25%, then it increases and reaches a maximum at 3�55% for a stocks volatility of 65% and then decreases. When � = 5, we have the same behavior but the price di¤erence is lower at 0�71% (for ��� = 1 ) with a maximum of 3�613% for � � = 88%. For � = 0�5. the price di¤erence can be negative for low dilution and low volatility leading to an underestimation of the true value by use of the simple Black-Scholes formula. Notes: This gure shows the approximation error for at-the-money Warrant (� = 100 and � = 100) as a function of the volatility, �, and of the dilution, ���, for three maturities, � = 0�5, 5 and 10 years.

3.

The behavior for in-the-money warrants is the same except that the range of variation is narrower as can be seen on Figure 5. But here, we must distinguish between in-the-money warrants and deep in-the-money warrants. Indeed for in-the-money and near maturity warrants, the price di¤erence can be negative for a large range of dilution and volatility values.

Approximation Error at issuance in a low interest rates environment

The current low level of interest rates can have a substantial impact on many economic variables. Typically, when a warrant is about to be issued as, for example, part of a management package [START_REF] Koziol | Optimal exercise strategies for corporate warrants[END_REF] , it is very out-of-the-money with a fairly long maturity (typically, � = 5 years) and it is very likely that the volatility of the stock is moderate to high and is likely to be estimated with high level of uncertainty. The approximation error can therefore be substantial as shown in Figure 3, even with a moderate level of dilution. Moreover in this particular case, the pricing error is very sensitive to the stocks volatility. This means that an error in estimating volatility can have a signicant impact on the pricing error of the warrant price approximated by the simple Black-Scholes value. This features is reinforced by the low level of interest rates as can be seen in the Figures 6.

It is worth noting that in the current low interest rates environment, the pricing error problem is more pronounced. When the rst simulation studies of the pricing error were undertaken some thirty years ago, interest rates were high and most authors ran their simulations with a typical interest rate of 10%. Today, typical interest rates, even for a long maturity, are around 1%. And as we can see in the Figures 6, this matters. First, the level of the pricing error is much higher, ceteris paribus, with an interest rate of 1% than with a rate of 10%. Second, the range of pricing error for volatilities between 20% and 60% is much higher with low interest rates. And this remains true whether the dilution is 10% or 50%. S=50, T=5, K=100, M/N=10%, r=1% S=50, T=5, K=100, M/N=10%, r=10% 2 (obtained with ��� = 50%) gives a more precise idea of the magnitudes. By looking at it for � = 1% and � � = 30%, the Black-Scholes formula overestimates the warrant price by 10�2%. For the same volatility but with � = 10%, the overestimation is now only 1�83%. For a stock volatility of 50%, the Black-Scholes formula overestimates the warrant price by 3�11% for � = 1% and by only 1�63% for � = 10%. Its only for very high stocks volatility, such as � � = 80% and above, that the impact of an interest rate di¤erence is very small. We can verify that for a dilution between 10% and 100% the impact of an interest rate di¤erence is always signicant up to a volatility level of around 60%.

To better understand the e¤ect of the interest rate on the approximation error, we display in the Figures 7 the pricing error as a function of the level of the interest rate for two values of the volatility (� = 20% and 40%) and for two values of the dilution (M/N=10% and 50%). It is inversely related to the level of the interest rate and is very important for a dilution of 50% and a volatility of 20%.

Note also the strong sensitivity of the pricing error to the interest rate, as shown by the steepness of the blue curve. S=50, T=5, K=100, M/N=10%, � = 20% S=50, T=5, K=100, M/N=10%, � = 40% In te r e s t R a te , r ( % )

Concluding Remarks

The objective of this paper was rst to recall the rich nature of warrants compared to traded options. This is so because these securities modify the capital structure of rms. Even in the simple case where we assume that rms do not nance by debt, the valuation of warrants is complex, and several approaches are possible as recalled in the text. We then focused on the quality of the approximation of the issue price of a warrant by a simple Black-Scholes call formula, a common practice of practitioners. In particular, we have shown that the approximation error is higher and more sensitive to the stockss volatility parameter in the current low interest rate environment.

3. 1 . 2

 12 TheSchulz and Trautmann (1994), Bensoussan, Crouhy and Galai (1995a) andUkhov (2004) model

Figure 1 :

 1 Figure 1: Stock Volatility (� = 1)

Figure 2 :

 2 Figure 2: Stock Volatility (� = 10)

Figure 3 :

 3 Figure 3: Price Di¤erence (%) for � = 50 and � = 100�

Figure 4 :

 4 Figure 4: Price Di¤erence (%) for � = 100 and � = 100�

Figure 5 :

 5 Figure 5: Price Di¤erence (%) for � = 150 and � = 100� Notes: This gure shows the approximation error for deep in-the-money Warrant (� = 150 and � = 100) as a function of the volatility, �, and of the dilution, ���, for three maturities, � = 0�5, 5 and 10 years.

0. 1 0Figure 6 :

 16 Figure 6: Pricing Error, Call and Warrant Prices as a function of volatility

Figure 7 :

 7 Figure 7: Pricing Error, Call and Warrant Prices as a function of the interest rate

Figure 8 :

 8 Figure 8: Pricing Error as a function of the interest rate and of the volatility

Table 1 :

 1 Shares, Warrants and Firm Values

	�	�
	�¤ � • �	�¤ � � �

Table 2 :

 2 Pricing Error, Call and Warrant Prices This Table displays the approximation error, the Call and Warrant prices for ��� = 50%, � = 1% and 10% and for stocks volatilities between 20% and 80%.

	� � (%)	20	30	40	50	60	70	80
					r=1%
	Call	1.02	4.02	8.06 12.47 16.90 21.18 25.20
	Warrant	0.82	3.65	7.68 12.09 16.49 20.70 24.62
	C/W-1 (%) 24.43 10.20 4.99	3.11	2.47	2.33	2.34
					r=10%
	Call	5.34	9.78 14.19 18.44 22.45 26.19 29.62
	Warrant	5.17	9.60 13.97 18.14 22.05 25.68 29.01
	C/W-1 (%) 3.35	1.83	1.56	1.63	1.80	1.97	2.09
	Notes:

See e.g.[START_REF] Jarrow | Structural versus reduced form models: a new information based perspective[END_REF] for an analysis of this distinction in credit risk models.

If one of the other two cases is retained, some of the relationships established below must be adjusted but not the warrant valuation formula.

In order to simplify the exposition and without loss of generality, we assume a conversion ratio of 1:1. That is, a warrant gives the right to buy a share.

Note that the scenario considered has no e¤ect on the wealth of the shareholders nor on the value of the warrant. See below for more details.

The detailed derivation can be found in the three papers already cited. The most rigorous proof can be found in Bensoussan,[START_REF] Bensoussan | BlackScholes approximations of warrant prices[END_REF].Crouhy and Galai (1991) were the rst to prove that stock volatility is no longer stationary if rms equity is analyzed as a contingent claim on the rms assets.

By information set, they refer to the technical denition of the ltration of the probability space.

Note that the order of the three surfaces according to � is not the same as in Figure[START_REF] Black | The pricing of options and corporate liabilities[END_REF].

The term management package refers to the remuneration systems for executives, particularly in LBO operations. The management directly invests part of its assets in the company through warrants, convertible bonds or shares. The idea is to incentivize managers to succeed and therefore to set up asymmetric remuneration schemes. Stock warrants are therefore the perfect tool for this purpose.

¤ This work was supported by French National Research Agency Grant ANR-17-EURE-0020.