Sabah Al-Fedaghi
email: salfedaghi@yahoo.com

State-Based Behavior Modeling in Software and Systems Engineering

Keywords: Conceptual modeling, state, thinging machine model, state machine, event

The design of complex man-made systems mostly involves a conceptual modeling phase; therefore, it is important to ensure an appropriate analysis method for these models. A key concept for such analysis is the development of a diagramming technique (e.g., UML) because diagrams can describe entities and processes and emphasize important aspects of the systems being described. The analysis also includes an examination of ontological concepts such as states and events, which are used as a basis for the modeling process. Studying fundamental concepts allows us to understand more deeply the relationship between these concepts and modeling frameworks. In this paper, we critically analyze the classic definition of a state utilizing the Thinging machine (TM) model. States in state machine diagrams are considered the appropriate basis for modeling system behavioral aspects. Despite its wide application in hardware design, the integration of a state machine model into a software system's modeling requirements increased the difficulty of graphical representation (e.g., integration between structural and behavioral diagrams). To understand such a problem, in this paper, we project (create an equivalent representation of) states in TM machines. As a case study, we re-modeled a state machine of an assembly line system in a TM. Additionally, we added possible triggers (transitions) of the given states to the TM representation. The outcome is a complicated picture of assembly line behavior. Therefore, as an alternative solution, we re-modeled the assembly line based solely on the TM. This new model presents a clear contrast between state-based modeling of assembly line behavior and the TM approach. The TM modeling seems more systematic than its counterpart, the state machine, and its notions are well defined. In a TM, states are just compound events. A model of a more complex system than the one in the assembly line has strengthened such a conclusion.

Introduction

The design of complex man-made systems mostly involves a modeling phase; therefore, it is important to ensure an appropriate analysis (understand, design, and evaluate) method for these models and their fundamental concepts. The analysis can be viewed as including software engineering and philosophy simultaneously [START_REF] Bringsjord | Philosophical engineering: Some reflections[END_REF].

A key concept for such analysis is the development of a diagramming technique because diagrams can describe entities and processes, provide documentation, communicate ideas, and emphasize important aspects of the systems being described [START_REF] Buede | The Engineering Design of Systems: Models and Methods[END_REF]. For example, The Unified Modeling Language (UML) and its profile are considered a suitable specification language for the design of systems.

In this paper, we focus on state machines, which are considered the appropriate basis for modeling system behavioral aspects. Besides the state-machine concept, other models have been invented, often inspired by the idea of states (e.g., the models used in PLC languages). The central idea of these models is a state. The state machine stands as the preferred model for describing systems' behavior [START_REF] Wagner | Misunderstandings About State Machines[END_REF].

State Machines

Finite-state machines (FSMs) are well-established computational abstract devices and are used at the heart of most digital design [START_REF] Wilson | Chapter 6 -Block Diagram Modeling and System Analysis, Model-Based Engineering for Complex Electronic Systems[END_REF]. FSM models are widely utilized to specify systems in such fields as sequential circuits, distributed systems, communication networks, and communication protocols. They can also be used to model software systems' behavior. A "state machine can solve a large number of problems, among which is electronic design automation, communication protocol design, parsing and other engineering applications. In biology and artificial intelligence research, state machines are sometimes used to describe neurological systems and in linguistics, to describe the grammars of natural languages" [START_REF] Obota | Modeling & Control of Event Based Behaviour Using State Machines[END_REF].

In software engineering, an FSM models the behavior of a single "object," specifying the sequence of "events" that an object goes through during its lifetime. It takes inputs and produces outputs by following a set of rules determined by the internal state of the system. Typically, "behavior" refers to how the software will respond to external events (sometimes called triggers). According to Wagner and Wolstenholme [START_REF] Wagner | Misunderstandings About State Machines[END_REF], "Probably, the state machine is the only known model (of the many used in software development) that really gives a designer a chance to verify a control system and thus, it is the only way to produce reliable control software."

Events trigger transitions between states. A state is established by its relations to other states and to inputs and outputs. A machine is in one indivisible state at a time. The current input plus the current state determine the following output and the machine's next state. FSMs have been extended by developing the so-called statecharts, which provide the possibility to model states at multiple hierarchical levels.

States

A state is understood as a static situation, such as waiting for some external event to occur. When a state is entered, it becomes active, and it becomes inactive if it is exited. In this paper, we focus on simple states (i.e., ignoring composite and submachine states). A state is also described as an "abstraction of the values and links of an object" [START_REF] Blaha | Object-Oriented Modeling and Design with UML[END_REF]. According to Blaha and Rumbaugh [START_REF] Blaha | Object-Oriented Modeling and Design with UML[END_REF], "sets of values and links are grouped together into a state according to the gross behavior of objects." States often correspond to verbs with an "ing" suffix (Waiting, Dialing) or the duration of some condition (Powered, Below Freezing). Also, events represent points in time, and states represent intervals of time; however, according to Blaha and Rumbaugh [START_REF] Blaha | Object-Oriented Modeling and Design with UML[END_REF], "Of course, nothing is really instantaneous; an event is simply an occurrence that an application considers atomic and fleeting." On the other hand, some thinkers consider events subtypes of states [START_REF] Chisholm | Person and Object: A Metaphysical Study[END_REF].

Difficulties

Despite its wide application in hardware design, the integration of the state machine model into a software system is accomplished with some "new ideas or reinventions" [START_REF] Wagner | Misunderstandings About State Machines[END_REF]. Some extensions and changes in the state machine terminology have increased the difficulty of graphical representation of state machines. According to Wagner and Wolstenholme [START_REF] Wagner | Misunderstandings About State Machines[END_REF], the definition of a finite state machine seems to "require discussion." The concept is still not well understood or interpreted in the software domain despite its broad application in hardware design [START_REF] Wagner | Misunderstandings About State Machines[END_REF]. Misunderstandings about state machines have produced several stories and half-truths. The concept of the state machine has (unintentionally?) been reinvented for software several times [START_REF] Wagner | Misunderstandings About State Machines[END_REF]. According to Steward [START_REF] Steward | The Ontology of Mind Events, Processes, and States[END_REF], a misunderstanding of the nature of states and of their role in causal explanation has led to a seriously distorted understanding of states. According to Baldawa [START_REF] Baldawa | Writing Maintainable Code Using State Machines in Python[END_REF], "We should bear in mind that even though state machines are powerful tools to solve certain kinds of problems, it is not a panacea for all your database modeling problems and not all problems can be modeled using state machines."

In this paper, we try to analyze critically the classic definition of a state in state machines, utilizing the Thinging machine (TM) model [START_REF] Al-Fedaghi | Change in Conceptual Modeling and Systems Reconfiguration[END_REF][START_REF] Al-Fedaghi | Conceptual Modeling of Events Based on One-Category Ontology[END_REF]. Conceptual modeling includes ontological concepts such as states and events that are used as a basis for the modeling process. Analyzing fundamental concepts in conceptual modeling allows us to understand more deeply the relationship between ontological concepts and modeling frameworks.

About this paper

The next section contains a brief description of TM modeling. In Sections 3 and 4, we project the states of an assembly line example in a corresponding TM machine, presenting a clear contrast between state-based behavior modeling and the TM approach. In section 5, we analyze a more complex system of a telephone line given in Blaha and Rumbaugh [START_REF] Blaha | Object-Oriented Modeling and Design with UML[END_REF].

Thinging Machine (TM)

TM views the world as thimacs (things/machines) constructed from nets of subthimacs. Modeling consists of a lower (static) structure of things that are simultaneously machines, and both merge into a thimac. At the upper level (dynamics), a time thimac combines with the static thimac to generate events.

The thimac is an encapsulation of a thing that reflects the unity and hides the thimac's internal structure, and a machine (see Fig. 1) shows the structural components (static: outside of time called region), including potential actions of behavior. The static "thing" does not actually exist, change, or move, but it has potentialities for these actions when combined with time. A TM event is an encapsulation of a region and a time.

A thimac is a thing. The thing is what can be created (appear, observed), processed (changed), released, transferred, and/or received. A thing is manifested (can be recognized as a unity) and related to the "sum total" of a thimac. The whole TM occupies a conceptual "space" that forms a network of interrelated thimacs that together form an organic whole.

The thimac forms a compositional structure, in which elementary thimacs combine in systematic ways to create compound new thimacs, allowing us to make infinite thimacs structure. The result is compositional "world" models built to represent things and understand their interactions and relations. The whole is a grand thing/machine. Thimacs can be "connected" only via flow connections among thimacs. Therefore, things are part of the TM static description (static model) and are part of the dynamic model when merged with time to form events.

The thimac is also a machine that creates, processes, releases, transfers, and/or receives. Fig. 1 shows a general picture of a machine. The figure indicates five "seeds" of potentialities of dynamism: creation, processing, releasing, transferring, and receiving.

All things are created, processed, released, transferred, and received, and all machines create, process, release, transfer, and receive other things. Things "flow through" (denoted by a solid arrow in Fig. 1) other machines. The thing in a TM diagram is a presentation of any "existing" (appearing) entity that can be "counted as one" and is coherent as a unity.

Fig. 1 can be described in terms of the following generic (having no more primitive action) actions: Arrive: A thing moves to a machine. Accept: A thing enters the machine. For simplification, we assume that all arriving things are accepted; therefore, we can combine the arrive and accept stages into one stage: the receive stage. Release: A thing is ready for transfer outside the machine. Process: A thing is changed, handled, and examined, but no new thing results. Create: A new thing is "coming into being" (found/manifested) in the machine and is realized from the moment it arises (emergence) in a thimac. Things come into being in the model by "being found." The "ceasing to be" of a thing can occur anywhere in the model and can be represented as a freezing storage (graveyard) in the model. Transfer: A thing is input into or output from a machine.

Additionally, the TM model includes the triggering mechanism (denoted by a dashed arrow in this article's figures), which initiates a flow from one machine to another. Multiple machines can interact with each other through the movement of things or through triggering. Triggering is a transformation from one series of movements to another.

State-based Modeling vs. TM Modeling

In this section, we project the states of an assembly line given by Huang [START_REF] Huang | Discrete Event System Modeling Using Sysml and Model Transformation[END_REF] (Fig. 2) in a corresponding TM machine. The resultant TM model shows that that the model does not cover many details besides states. We resolve this lack of details by adding possible triggers (transitions) of the given states. The outcome is a complicated picture of behavior. Therefore, we contrast this state-based method with a re-modeling of the assembly line in a TM.

According to Huang [START_REF] Huang | Discrete Event System Modeling Using Sysml and Model Transformation[END_REF], a state is a unique status of a system at a particular time. A discrete event system is a system in which the state does not change between consecutive events. An approach to reduce the number of system states is to describe components' states instead of all of a system's states. Huang [START_REF] Huang | Discrete Event System Modeling Using Sysml and Model Transformation[END_REF] gives an example (Fig. 2) of an assembly line that includes two machines, M1 and M2. Each machine has unit capacity and has a preceding buffer with a capacity of three items. The states in this system can be captured in four components (B1, M1, B2, and M2). The set of all possible state values of B1 is (0, 1, 2, or 3). Because the succeeding buffer could block M1, there are three possible state values, (idle, busy, or blocked). M2 has two possible state values, (idle or busy). The number of all possible system states is 96 [START_REF] Huang | Discrete Event System Modeling Using Sysml and Model Transformation[END_REF].

TM perspective of the assembly line states

Fig. 3 shows the TM model of this state-based modeling of an assembly line. Items are received in M1 (numbers 1 and 2) to trigger incrementing (3) the number of items in B1. We assume that initially, the number of items in B1 is zero [START_REF] Wilson | Chapter 6 -Block Diagram Modeling and System Analysis, Model-Based Engineering for Complex Electronic Systems[END_REF]. If the number of items in B1 is greater than zero [START_REF] Obota | Modeling & Control of Event Based Behaviour Using State Machines[END_REF], then an item is released (6) from B1 to be processed [START_REF] Chisholm | Person and Object: A Metaphysical Study[END_REF] in M1. Also the number of items in B1 is decremented [START_REF] Steward | The Ontology of Mind Events, Processes, and States[END_REF]. These processes of incrementing and decrementing trigger the creation (9) of new values for the number of items. When M1 receives an item from B1, it blocks any further release (10) because it will be busy processing the received item.

When M1 releases [START_REF] Al-Fedaghi | Conceptual Modeling of Events Based on One-Category Ontology[END_REF] an item, after processing it, to M2, assuming that M2 is not blocked [START_REF] Huang | Discrete Event System Modeling Using Sysml and Model Transformation[END_REF], the blocking of further release from B1 is lifted [START_REF] Whitehead | Adventures of Ideas[END_REF], so it can check whether B1 contains any items [START_REF]UML 2 Tutorial -State Machine Diagram[END_REF]. When M2 receives the item (15), the number of items in B2 is incremented (16). If the number of items in M2 is greater than zero (17), an item is released from B2 to M2 (18). Also, that number is decremented (19). These operations of incrementing and decrementing trigger the creation of the number of items in B2 (20). When M2 receives an item from B2, it blocks further release (21). When M2 finishes processing the item (22), the blocking of further reception from B2 is lifted (23).

The situation of whether M2 is blocked with regards to receiving further items from M1 is registered regardless of whether the number of items in B2 reaches 3 (24). This situation is communicated to M1 (25) and affects the decision to send items from M1 to M2 (26 and 13).

States as TM events

An event in a TM is a subdiagram of the static model (called the region of the event) plus a time machine. It can endure, or it may be instantaneous. The five generic actions form five generic events. They may be compound, forming "higher order events" consisting of generic events fitting together as subthimacs to produce larger thimacs. Fig. 4 shows the event An Item has arrived to M1 and added to B1. For simplicity, we represent an event by its region.

According to Steward [START_REF] Steward | The Ontology of Mind Events, Processes, and States[END_REF], events happen whereas states obtain (or do not obtain). From the TM point of view, obtaining is achieved as the result of the create event. In TM, the so-called states are treated as TM events. To support this thesis about states (model focus only), in such a discussion, we appeal to an indication of that (states as TM events) in some aspects of "world"-based ontology.

First, we claim that states are results of generic events (generic actions plus time). Therefore, states are compound events of generic events. For example, temperature and pressure states in gases are triggered by movements (changes in positiongeneric actions) of the molecules of the gases. Thus, the events of movements of the molecules trigger the events of changes in temperature and pressure. Both types of events involve dynamism: generic actions of movement and the increase/decrease (process) in values. Even states of equilibrium in certain chemical systems might involve the constant passage of molecules between their liquids. Gaseous states [START_REF] Steward | The Ontology of Mind Events, Processes, and States[END_REF] involve the generic events (generic actions plus time), create, process, release, transfer, and receive, which implies that the so-called states involve changes.

Returning to Huang's [START_REF] Huang | Discrete Event System Modeling Using Sysml and Model Transformation[END_REF] example (Fig. 3), let us locate the assembly line states in the TM model as events. Fig. 5 shows such a model built, according to Huang [START_REF] Huang | Discrete Event System Modeling Using Sysml and Model Transformation[END_REF], of states, using the static model in Fig. 3. Note that in a TM, the so-called objects (continuants) are a type of event (have at least the create action) (e.g., 0, 1, 2, and 3 in Fig. 5). Even though "continuants" persist through time and exist as wholes at every moment of their existence (starting from create), in a TM, such an existence is divided into a sequence of events that involves processing, releasing, transferring, and receiving. Analogously, water's being at 90°C, for instance, seems to be a state which exists, as it were, in full at all times at which the water is at that temperature [START_REF] Steward | The Ontology of Mind Events, Processes, and States[END_REF]. In a TM, this state is modeled as heat flows (actions) in water, and processing that heat triggers an increase in the water temperature to 90°C. The whole event (being at 90°C) is a composite event of generic events of continuous supplying of heat. The state is incomplete whenever one of the generic events (transfer/receipt of heat) is incomplete, just as a football match is incomplete when some actions in the game stop.

No. of items equest

Region

To elaborate on such an issue, consider knowledge (classical mental state) as an event when the knowledge is permanent or long-lasting over a person's life. Fig. 6 shows the TM model of this situation, in which the involved events include receiving information, processing it, storing it as knowledge, and processing stored knowledge to be released. In a TM, creating knowledge is the event of starting the existence of the knowledge as a thing in the memory. This existence reflects knowledge as a being in the model, in contrast to the general notion of existence in the "world." It is an event (occurrence) in the sense that it is an announcement of the availability of knowledge to participate in the dynamism of the world through other events. It is an event that creates a thing.

The knowledge, after its creation, becomes a thing. This thing is now not a continuous creation of knowledge but a finished event. After the creation, the event-ization finishes and thingness starts. Therefore, knowledge is not a lifelong event; rather, it is a thing that participates in many events whose regions include knowledge. The creation event of a thing is a unique event that produces a thing that "is there"; however, create-ness has finished as an event. A thing "being there" is not a long-lasting event; rather, the thing continued being inside other events.

Suppose that a person is born (created) but never participates or appears in any further event (in the model). We claim that this is not possible because a TM's existence implies eventi-zation.

There is not a second in a person's life that does not involve participating in some events (e.g., after birth, sucking milk, growing, crying). Suppose that a thing, E1, is created, but it was not involved in any event for a certain period, and then it appears in an event, E2. We claim that in this "disappearance" period, the time slice between E1 and E2, contains all the events that are not detected because they are not of interest/capability to/of the modeler (observer).

TM existence of a thing after the creation event is its participation in a sequence of events.

Consider a very static object, such as Cleopatra's Needle on the Victoria Embankment in London. According to Whitehead [START_REF] Whitehead | Adventures of Ideas[END_REF] Cleopatra's Needle "is a series of events. It is actively happening. It never remains the same. A physicist who looks at that part of the life of nature as a dance of electrons will tell you that it loses molecules and gains others daily. Even the plain man can see that it gets dirtier and is occasionally washed" [START_REF] Whitehead | Adventures of Ideas[END_REF]. Even knowledge in the mental domain, after it is acquired, goes through continuous mental events that diminish, update, or modify it.

Accordingly, we claim that there is a unique type of event, create, that results in existence. We are developing this thesis in the context of modeling where existence refers to being (appearance) in the model. Therefore, a TM's existence begs the question "What is modeled?" A dead person (a thing) can exist in a TM if it participates in an event such as, say, celebrating his/her memory. Also, a round square "exists" if it is created (appears) in the TM description (e.g., it is talked about). For example, There being round squares is impossible [START_REF] Chisholm | Person and Object: A Metaphysical Study[END_REF] is modeled as the creation of a box for impossibility and, inside it, creating a box for a round square. To make such a subdiagram an event, it is complemented with a thimac of "all times" (create and process time).

Fig. 6 Events of getting information, processing it, storing it as knowledge, and processing stored knowledge to be released.

Idle

Blocked Busy

Fig. 5 The given states superimposed on the TM description (for simplicity, we removed some boxes).

Additionally, such a state (of an affair) as Abdul-Jabbar's being more than seven feet tall is a creation event that occurs at a certain point in Abdul-Jabbar's life. Many events occur for him during his life. However, these events are not things that hang from him as leaves attached to a tree; rather, whenever his height participates in a future event, it flows (transfer, receive) from the creation event to the new event. The new event "constructs" Abdul-Jabbar by importing continuing "attributes" from previous events.

We will apply this claim that the state is a type of event in Huang's [START_REF] Huang | Discrete Event System Modeling Using Sysml and Model Transformation[END_REF] assembly line example.

Projection States as events

Returning to Fig. 5, which shows states projected [START_REF] Huang | Discrete Event System Modeling Using Sysml and Model Transformation[END_REF] over the TM model, we note that the diagram is missing many TM details. In a state machine, more details can be specified as triggers of the states. Fig. 7 identifies possible triggers of states.

The resultant diagram of the whole approach of identifying states and triggers seems to produce an unsystematic way to specify the assembly line's behavior. It seems that there is no reason for this top-down process that starts from compound events given as states and triggers. A simpler approach is to construct the TM diagram from scratch, identify a suitable set of events, and specify the behavior as a chronology of these events.

To illustrate such an approach, consider the simple state machine of a door given by Sparx [START_REF]UML 2 Tutorial -State Machine Diagram[END_REF] and shown in Fig. 8. First, the door is described in terms of a situation (state) with an initial state and create trigger. Then Close/[doorWay->is Empty] and Open/ are used to described the changes in the door's position. Alternatively, Fig. 9 shows the TM static description of a door.

The diagram reflects the intuitive idea of events as the door moves between two positions; therefore, as Fig. 10 shows, there are two events, closed and open, and two changing events, opening and closing. Fig. 11 shows the corresponding behavioral model. The TM model seems simpler than the state diagram, but it is richer in semantics.

Fig. 7 The given states with triggers.

TM model of the assembly line

4.1 The static model Fig. 12 shows a fresh start to model the assembly line example.

First, an item is received in M1 (number 1). The number of items in B1 is incremented [START_REF] Buede | The Engineering Design of Systems: Models and Methods[END_REF]. The number of items in B1 is checked.

-If it is 3, then no additional item is accepted in M1 ( 3). -If it is greater than 0 (4), then an item is released to M1 (this happens automatically because M1 is initially idle; later, M1 will trigger this action), the number of items in the queue is decremented [START_REF] Obota | Modeling & Control of Event Based Behaviour Using State Machines[END_REF], and accepting an item from the outside is activated ( 6). Accordingly, an item moves from B1 to M1 (7 and 8) and processed [START_REF] Baldawa | Writing Maintainable Code Using State Machines in Python[END_REF]. If M1 is not blocked, [START_REF] Al-Fedaghi | Change in Conceptual Modeling and Systems Reconfiguration[END_REF] flows to M2 [START_REF] Al-Fedaghi | Conceptual Modeling of Events Based on One-Category Ontology[END_REF]. The number of items in B2 is incremented [START_REF] Huang | Discrete Event System Modeling Using Sysml and Model Transformation[END_REF]. If the number of items in B2 is 3, then M2 is blocked [START_REF] Whitehead | Adventures of Ideas[END_REF]; otherwise, it is unblocked. If the number of items in B2 is >0 [START_REF]UML 2 Tutorial -State Machine Diagram[END_REF], then an item is sent to M2 (15) and the number of items in B2 is decremented (16). The received item is processed and leaves M2 (17).

The Events model

Accordingly, Fig. 13 shows the set of events superimposed on the static model as follows: E1: A new item enters M1, and the number of items is incremented in B1. E2: The number of item in B1 reaches 3, so further acceptance of items is blocked. E2: If the number of items in B1 is greater than zero, send an item to M1. E4: Decrement the number of items and accept further items in B1. E5: An item flows from B1 to M1. E6: An item is processed in M1. E7: M2 is not blocked. E8: If M2 is not blocked, then send the processed item from M1 to M2. E9: Increment the number of items in B2. E10: If the number of items in B2 is 3, then block M2 from accepting an item from M1; otherwise, unblock it. E11: If the number of items in B2 is greater than zero, then decrement the number of items. E12: Send an item from B2 to M2. E13: Process an item in T2 and release it when finished processing. Fig. 14 shows the behavioral model of the assembly line. The model represents the chronology of events and stands as the controlling module of the assembly line system. Suppose that S1 stands for an input item to the assembly line. Fig. 15 shows a sample sequence of events, in which -it is assumed, initially, that B1 and B2 contain zero items and M1 and M2 are unblocked and idle, -bookkeeping events (e.g., E3 and E4) are shown only when they are applicable, -all events have the same time duration, and -we assume priority of events from left to right.

Another example of TM Modeling

We noted previously that that the TM model includes more details than are necessary to produce an accurate basic model. Of course, in general, such details come with the cost of having a more complex model system. However, we think that TM modeling has appropriate level of detail because it is constructed from one category: the thimac. The thimac includes five actions, flows and trigger that are applied repeatedly while a high level abstraction is preserved. Additionally, the TM modeling seems more complete than a state machine because it starts with a static model, then includes an events model, and last, the behavioral model To emphasize such a thesis, we analyze in this section a more complex system than the one in the previous section.

Consider the state machine given in the classical book Object-Oriented Modeling and Design with UML [START_REF] Blaha | Object-Oriented Modeling and Design with UML[END_REF] in Fig. 16. The state diagram is for a telephone line. According to Blaha and Rumbaugh [START_REF] Blaha | Object-Oriented Modeling and Design with UML[END_REF], the diagram concerns a phone line and not the caller or the callee. It contains sequences associated with normal calls as well as some abnormal sequences, such as timing out while dialing or getting a busy signal. According to Blaha and Rumbaugh [START_REF] Blaha | Object-Oriented Modeling and Design with UML[END_REF], "At the start of a call, the telephone line is idle. When the phone is removed from the hook, it emits a dial tone and can accept the dialing of digits. Upon entry of a valid number, the phone system tries to connect the call and route it to the proper destination. The connection can fail if the number or trunk are busy. If the connection is successful, the called phone begins ringing. If the called party answers the phone, a conversation can occur. When the called party hangs up, the phone disconnects and reverts to idle when put on hook again" [START_REF] Blaha | Object-Oriented Modeling and Design with UML[END_REF].

TM Static Model

Fig. 17 shows the TM model of this phone line. When the phone is lifted from the hook (1), a signal is created (2) and omitted (for simplification, we do not show the flow of this signal). This signal triggers the following initialization; -A dial tone (3) that indicates acceptance of the dialing of digits. -The number of digits is initializing to zero (4).

-The dialed number is initialized to blank (5).

-Timing is set on (6). Accordingly, with each created digit (7) by the user, -the number of dialed digits is incremented (8), -the timing is reset (9), and -the input digit to construct the dialed number ( 10) is sent. The number of dialed digits is examined [START_REF] Al-Fedaghi | Conceptual Modeling of Events Based on One-Category Ontology[END_REF], and -if it is less than n, then this triggers expecting additional digit input (13); -if it is = n, then the input number is processed [START_REF]UML 2 Tutorial -State Machine Diagram[END_REF] -If the number is not valid (17), a recorded message is output (18) and the system is initialized again to accept the dialing of digits (see initialization above). The other side of the line sends a response (20).

-If it is a connected response (21), it is sent to the speaker (22). -If it is a ringing tone (23), a ringing tone is produced on the speaker (24). -If it is busy or disconnected, a busy sound (25) is produced and sent to the speaker (26).

Meanwhile, during the connection, the user sound is captured by the microphone and sent to the other side. Additionally, when time expires (27), a warning time is output.

Projecting States over the TM model

If we project Blaha and Rumbaugh's [START_REF] Blaha | Object-Oriented Modeling and Design with UML[END_REF] states over the TM representation of the phone line, we end up with many details the state diagram does not include. To save space, Fig. 18 shows projecting the state of a dial tone, which includes many initializations when the phone is removed from the hook. Applying triggers would produce very a messy diagram.

Events model

Fig. 19 shows the events model of the phone line. To save space, we will not list these events here. Some of Blaha and Rumbaugh's [START_REF] Blaha | Object-Oriented Modeling and Design with UML[END_REF] states coincide with the selected events in Fig. 19. Fig. 20 shows the behavioral model.

Conclusion

We have critically analyzed the notion of a state in conceptual modeling utilizing the thinging machine model. In a TM, a state is defined as a type of an event. In two case studies, we contrast the state-based method with the corresponding TM model. The TM model is more complete than the state-based model because it starts with a static model, followed by an events model to construct a behavioral model. The TM notion of events is well defined in comparison to the state-based model because -things/machines form thimacs; -machines have the five-action structure; -machines create, process, release, transfer, and receive things; -things are what can be created, processed, released, transferred, and/or received; -things denote the wholeness of the thimac; -events are static thimacs with time; -behavior is the chronology of events; and -a single static TM model leads to an event model, which in turn results in a behavioral model using the same modeling notations. Contrasting this with the numerous notations of state charts and other UML heterogeneous multigraphs, we observe that integrating state machines with other types of UML diagrams is difficult.

Nevertheless, this paper is a work in progress that requires further investigation, specifically experimentation with more state machines of various types. This investigation could also be supplemented with philosophical issues related to states, events, facts, and propositions, which we will do in future research.

Dial tone

ReceiveFigFig. 2

 2 Fig. 1 Thing machine.

Fig. 3 Fig. 4

 34 Fig.3The TM static model of the assembly line.

Fig. 8 Fig. 9 Fig. 11

 8911 Fig.8An example of a state machine (partial, from[START_REF]UML 2 Tutorial -State Machine Diagram[END_REF]).

Fig. 13 The

 13 Fig.[START_REF] Whitehead | Adventures of Ideas[END_REF] The TM events model of the assembly line.

Fig. 12 The

 12 Fig.12The TM static model of the assembly line.

Fig. 14 The

 14 Fig.14The TM behavioral model of the assembly line.

Fig. 15

 15 Fig. 15 Sample partial run of the behavioral model.

Fig. 16

 16 Fig. 16 State diagram for phone line with activities (From [6]).

Fig. 17

 17 Fig.17The static TM representation of the phone line.

Fig. 19 Fig. 20

 1920 Fig.19The events TM representation of the phone line.

not blocked Queue equest Transfer Process If =3 then Process Increment equest Decremen t equest Release No. items Queue equest Transfer Process If >0 equest Process Increment equest Transfer No. items Receive Blocked

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		Create	Release Transfer

  Fig. 18 Partial view of the static model of the phone line showing the part that represents the state Dial tone. IJCSNS International Journal of Computer Science and Network Security, VOL.22 No. 5, May-II 2022 (Author's version)

												11
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Behzad Alimoradian

Karim Barigou

Anne Eyraud-Loisel

Derivatives under market impact: Disentangling cost and information

Keywords: Option pricing, market impact, illiquid markets, transactions costs, stochastic optimal control

This paper studies the role of market impact in option pricing theory through the nonrestrictive case study of the existence of a large trader. Under market impact, the standard Black-Scholes model and unique risk-neutral pricing theory are no more applicable. We postulate then that there are two sides to the story: i) the transaction costs and ii) the information of the large trader about its own activity. To disentangle these two factors, we define a hypothetical trader called the insider trader who has the same level of information as the large trader but does not bear any transaction costs. We show that there exist a set of probability measures, which we call information-neutral probabilities under which the discounted asset is a martingale for the insider trader. We then derive the optimal cost hedging program for the large trader and emphasize the importance of avoiding any market manipulation in the optimal hedge problem so that it does not become inherent to the model. We conclude with pricing models and numerical examples for both the large and insider trader.

Introduction

Standard option pricing theory is based on the assumptions of a frictionless and liquid financial market. However, having large notional derivatives on illiquid assets with market impact, one may face difficulties hedging its position. When the size of the hedge is substantial, executing shares consumes the market's liquidity, and consequently impacts the underlying spot price. In such circumstances, the standard framework of the Black-Scholes model does not stand anymore. This paper presents a mathematical framework for hedging derivatives under such liquidity issues.

The literature predominantly refers to two types of market impact: the permanent impact, and the temporary impact which is also known as slippage or transaction cost. 1 One of the first models to take into account the temporary impact (transaction costs) in derivatives was introduced by [START_REF] Leland | Option pricing and replication with transactions costs[END_REF]. [START_REF] Liu | Option pricing with an illiquid underlying asset market[END_REF], on the other hand, integrated the permanent impact in their extension of the Black Scholes model and came up with a parabolic partial differential equation which can be interpreted as multiplying the volatility of the underlying asset by a sum of a series (see [START_REF] Glover | On nonlinear models of markets with finite liquidity: some cautionary notes[END_REF], [START_REF] Said | Market Impact in Systematic Trading and Option Pricing[END_REF]). Recent papers on the permanent impact in derivatives are also [START_REF] Loeper | Option pricing with linear market impact and nonlinear blackscholes equations[END_REF], [START_REF] Said | How Option Hedging Shapes Market Impact[END_REF], [START_REF] Bordag | Pricing options in illiquid markets: symmetry reductions and exact solutions[END_REF] and [START_REF] Abergel | Option pricing and hedging with liquidity costs and market impact[END_REF].

While being in the lineage with the past literature, our paper broadens the academic debate by suggesting to consider the permanent and temporary impacts as two unrelated phenomena, different in essence and nature; the permanent impact as information and the temporary impact as costs. We do so by defining an hypothetical insider trader that does not bear any transaction costs but is aware of the permanent impact.

In this paper, we conceptualize the permanent price impact by the means of the auxiliary concept of information, meant to represent the information about trading activities of one or many traders. Knowing that such trading activities may result in liquidity (im)balance, any entity being aware of them will have a different opinion about the spot price. Such entity will apply a correction to the price. We regard the permanent impact as this correction to the price judged by the informed individual. Once the trade is executed, such information is shared among the whole market and the price impact becomes a reality; it contributes to the filtration explaining asset variation in the real world. It is worth noting that the role of information in asset pricing has been studied by [START_REF] Eyraud-Loisel | Backward stochastic differential equations with enlarged filtration: Option hedging of an insider trader in a financial market with jumps[END_REF] and [START_REF] Eyraud-Loisel | Quadratic hedging in an incomplete market derived by an influential informed investor[END_REF] on their contribution to understanding the filtration enlargement due to some private information. [START_REF] Cetin | Pricing options in an extended Black Scholes economy with illiquidity: Theory and empirical evidence[END_REF] and [START_REF] Cetin | Liquidity risk and arbitrage pricing theory[END_REF] have also interpreted the permanent market impact through the concept of asset's supply curve. Another recent work by [START_REF] Brigo | Price Impact on Term Structure[END_REF] studies the arbitrage pricing theory for the bond market in the presence of market impact. Our imperative standpoint is that the price impact of some information in the market is more pertinent than the nature of the information itself. In fact, information is not even measurable, however the market impact of the information is. For this reason, practitioners refer to the concept of impact factor when tackling information. Therefore, any theory of information results into a theory of market impact. With this imperative standpoint, there is no easily discernible example of information other than an information about one large trader's activity which is the subject of study in this paper.

One other main distinction of this paper is that our permanent impact process is stochastic and L 2 -integrable, while the past literature has been using L 1 -integrable price impact processes.

L2 -integrability is more pertinent because, in such case, the market impact will have the same order of finite moments than the brownian motion variation, which is the same finite order of moments than the volatility. This is an important property of a price impact model because it

gives the model the potential to justify/explain the market volatility. One other consideration with derivatives is the hedging policy; in general, there are two trends: the first one suggests a full hedge policy as in [START_REF] Liu | Option pricing with an illiquid underlying asset market[END_REF] and the second one suggests partial hedging, e.g. [START_REF] Jonsson | Partial hedging in a stochastic volatility environment[END_REF]. The full hedge approach generally results in option prices that are too expensive in illiquid markets since the cost of rebalancing the delta of the option can be considerable when getting closer to the strike and expiry date. A full hedge might be also not feasible because the market liquidity may be insufficient. For this reason, this paper's main focus is on partial hedging.

In the case of a partial hedge, a predominant approach consists of optimizing a gain function for the large trader, e.g. [START_REF] Guéant | Option pricing and hedging with execution costs and market impact[END_REF]. However, with this approach the solutions may lead to the large trader manipulating the market in its own benefit as defined by its utility function.

Therefore, it is important to correctly define the problem so that a market abuse does not become inherent to the model and in that regard, this paper is one of the first to introduce the issue of market manipulation in derivatives hedging. It provides a discussion on the model risk of market abuse in derivatives and suggests an approach to address it.

Lastly, it should be mentioned that even though at first sight the case study presented in this paper may seem too restrictive, it is not. This case example is quite common when dealing with illiquid markets. 2 To give more clarity about its practicality, it is worth mentioning that this research project originated from a real practical experience when a trader consulted one of the authors of the paper being concerned about the incapacity of existing models to hedge and price of a large derivative transaction. The state-of the-art derivatives models are generally calibrated to the implied volatility surface. However if the market is blind about the existence of some new transactions it will lack a critical information, and therefore the implied volatility surface, as it stands, is biased and needs to be corrected.

Also, even though here we assume the existence of one large trader only and in reality a market may include several large traders, we should mention that it happens quite often that in the market most large traders have in general the same directional strategies, especially in activities that may results in market impact, this phenomenon has been pointed out by [START_REF] Cont | Fire sales, indirect contagion and systemic stress testing[END_REF] in their analysis of fire sale of large traders contingent to some critical market environment. Lastly, it should be mentioned that the results obtained in this paper could be generalized for any information process that generates market impact; to this point, we also refer to the work of [START_REF] Lee | Insiders' hedging in a jump diffusion model[END_REF]. This paper is organized as follows: Section 2 presents the market ecosystem, with one large and one insider trader. Section 3 describes the stock dynamics under the market impact. Section 4 derives the pricing equation for the insider trader and explains the existence of a set of probability measures called information-neutral probabilities under which the asset is a martingale. In Section 5, we derive an optimal hedging strategy for the large trader and discuss the important issue of market manipulation in the stochastic control problem. We present a finite difference method for solving the optimal shares of the large trader and provide numerical results that compare, in particular, strategies for both the large and insider trader in Section 6.

Market Ecosystem Description

We consider a market with many small active traders, but with limited liquidity. A large trader decides to enter this market with a trading policy {χ t } t≥0 , where χ t is the amount of shares to be executed at time t. The large trader hides its order size from the market. Therefore the market will not know about (χ t ) until it is executed3 . However, we assume that there is one insider trader who is informed about the trading policy of the large trader. Hence, we have three types of traders in this market:

• The large trader who has market impact and is aware of its own market impact.

• The insider trader who knows as much as the large trader, but has no market impact.

• The average trader who is too small to have a market impact and is unaware of the large trader activities. This trader is not the subject of study in this paper.

It should be noted that while the conclusions of this paper can be extended to any trading policy, here we assume that the large trade strategy is to hedge a derivative exposure.

The Market Impact Model

Let's assume that the large trader would like to execute χ t shares at any time t. The size of χ t being too big, there may be not enough liquidity for this order. Considering that the depth of the market varies over time and there is liquidity in some days compared to others, one can not predict in advance the impact of that execution. We assume that within an infinitesimal period [t, t + dt), there would be either enough market depth or not. If there is enough market depth, then there is no market impact. On the contrary, if there is not enough depth, then there is a market impact of size Λ(χ t ). We assume that the chances of being in a no depth market during the period [t, t + dt) are ϕdt, with ϕ a positive constant. Taking the above into consideration, we define π t as the process for the depth of the market and we assume that it follows a Poisson process with intensity ϕdt. 4 The dynamics of the stock price are then given by the following extended Black-Scholes formulation:

dS t S t = µdt + σdW t + Λ(χ t )dπ t , dπ t ∼ Poisson(ϕdt), (3.1)
where W t is a Brownian motion, µ is the drift and σ is the volatility of the asset. Λ(•) represents the market impact defined as a non-decreasing function of executed shares with Λ(0) = 0. Note that the market impact could be positive or negative depending on whether the shares are bought or sold. In the case of no trading activity by the large trader, Equation (3.1) reduces to the standard Black-Scholes dynamics.

For the sake of completeness, we also assume that the holder of the stock benefits from a dividend yield noted as q, and that there exists a securities lending market and a future market for the underlying stock and the holder of the stock bears a cost of carry of b. The cost b is generally the adjustment rate between the future market and the value of the stock as of today.

Although these markets may be subject to stringent liquidity issues, they allow to enter into short selling positions.5

Insider Trader's Martingale Measure

The insider trader has access to the same level of information than the large trader, it knows about the trading policy {χ t } t≥0 and its potential market impact. Let {F t } t≥0 be the natural filtration generated by {W t } t≥0 and {π t } t≥0 , and let {r t } t≥0 be a deterministic interest rate process, the following theorem explains that there exists a set of probability measures under which the asset is a martingale.

Theorem 1. There exists a set of probability measures I where for any measure I ∈ I the returns of the discounted asset are martingales under this measure:

S t = E I e -T t rsds e (q-b)(T -t) S T F t . (4.1)
An option O bought/sold by the insider trader is also a martingale under this probability:

O t = E I e -T t rsds O T F t .
We call this set of probabilities information-neutral probabilities.

very comprehensive model, but layout a justified mathematical framework for studying the role of information on market impact and arbitrage pricing theory.

Note that Theorem 1 infers that the market is not complete. In fact, it is known in the literature, see Tankov & Voltchkova (2009), that in the presence of jumps in addition to a Brownian motion, the market becomes incomplete because one cannot perfectly hedge both sources of variability with a standard hedging portfolio of the underlying asset and a risk free instrument. This also implies that there exists an infinity of martingale measures. With this consideration, the proof of Theorem 1 lays within the general framework of Martingale Pricing Theory and detailed discussions about the set of probability measures under an incomplete market can be found in Tankov (2003).

We called I the set of information-neutral measures because it corresponds to the probability measure for those who hold the information and get over it. To clarify this, an example would be when χ t is always positive as formulated in Equation (4.2); this Equation suggests that the asset is being sold at a discount. However the insider individual will still view the asset as a martingale under I and gets over the discounted misprice of the asset. Despite the fact that it could benefit from the mispriced asset, it will project the asset value as a martingale in the information-neutral probability; such proposition is the basis of Equation (4.1).

∀u ≥ 0 :

χ u ≥ 0 ⇒Λ(χ u ) ≥ 0 ⇒ Expected Discounted Asset Returns = E Q e T 0 -rudu e (q-b)T S T = S 0 E Q e σW T -1 2 σ 2 2 T π T i=0 (1 + Λ(χ ti )) = S 0 E Q π T i=0 (1 + Λ(χ ti )) ≥ S 0 , (4.2)
where pi T is the number of market impact jumps occured up to time T , and

{t i } {0≤i≤π T }
represents the times of the market impact jumps. The Equation (4.2) shows that the asset returns form a sub-martingale under the risk-neutral measure Q for a filtration generated by {W t } t≥0 only.

Out of the set of information-neutral measures, there is one that reduces the quadratic variance between the hedging and the option portfolio, we call this measure the minimal variance information-neutral probability and Theorem 2 describes such measure.

Theorem 2. Let's assume the regulatory condition formulated in Equation 4.3:

Λ(χ t )(µ + (q -b) -r t ) < σ 2 , (4.3)
there exists a probability measure I * that minimizes the quadratic variance of the difference between the hedging portfolio and option value, and the discounted return of the asset is a martingale:

dS t S t = (r t -(q -b) -Λ(χ t )ω t ) dt + σdW I * t + Λ(χ t )dθ t . (4.4)
Here, θ t is a compound Poisson process with intensity ω t , defined as

ω t = ϕ σ 2 -Λ(χ t ) (µ + (q -b) -r t ) σ 2 + ϕΛ(χ t ) 2 .
The option price O can be formulated via the following partial differential equation:

r t O(t, S t ) = ∂O(t, S t ) ∂t + (r t -(q -b) -Λ(χ t )ω t ) S t ∂O(t, S t ) ∂S t + 1 2 ∂ 2 O(t, S t ) ∂S 2 t σ 2 S 2 t + (O (t, S t + S t Λ(χ t )) -O (t, S t )) ω t (4.5)
A detailed proof of this theorem is available in Appendix A and is based on quadratic hedging and local risk-minimization methods. It is worth mentioning that within the proof of Theorem 2, Girsanov Theorem has been applied on both the jump process and the Brownian process. The consequence of applying a change of probability on the Brownian process is a change of drift from µ to r t -(q -b), the consequence of applying the change of probability to the Poisson process is a change to the jump intensity.6 Note that if χ t = 0, or ϕ = 0, then Equation (2) reduces to the standard Black & Scholes partial differential equation. We remark that [START_REF] Lee | Insiders' hedging in a jump diffusion model[END_REF] also derived local risk-minimizing strategies for insider traders who have more information than the general public. However, the insider information is modelled via another diffusion process, which models a sequence of firm-specific information, and is only observed by insiders. We also note that the normality condition in Equation ( 4.3) is necessary to guarantee that the intensity process ω is positive.

Option Hedging for Large Trader

We define an option with a terminal payoff of cash and shares to be delivered physically7 . If the large trader is, for instance, the seller of the option, the number of shares component, noted as Q(S T ) would be negative and the terminal cash of the transaction H(S T ) would be positive.

Here, T denotes the expiry date of the option. When hedging its position, the large trader has the amount of H t in cash -usually negative, borrowed -and V t in stock at time t. Contrary to the standard Black-Scholes framework, V t is not necessarily the standard Delta (derivative of the option value to the spot price). Considering that the cost of hedging may be too high, the large trader may decide to leave some of its positions un-hedged with the hope that the market will go on its desired direction and the un-hedged portion will not become problematic, therefore it sets a delta-hedging shares trading policy {V t } 0≤t≤T .

The dynamics of the portfolio are described in Equation (5.1). As presented in the equation, we assume that dV t assets are purchased/sold at the total cost of (S t + dS t )dV t + dC t , where dCt dVt is a transaction cost and S t + dS t is the realized price at the end of the execution period. We assumed that the total cost depends on the price at the end of the execution period simply because of the execution time; because of the size of dV t it will take some time for it to be executed and this won't be instantaneous. Also there is additional transaction costs noted as one dC t , this is the cost of purchasing dV t shares during the interval [t, t + dt). We will parametrize dC t further in Section 6.1 but at this stage the reader shall be contempt to know that dC t would depend on the size of the execution request dV t .

V t+dt = V t + dV t H t+dt = H t (1 + r t dt) + V t S t (q -b)dt -(S t + dS t ) dV t -dC t (5.1)
It is worth noticing that the hedging strategy is not self-financing; there may be cash flow injection whenever it is necessary to finance any cash shortage. Taking the integral of Equation (5.1), the accounting value of hedges can be reformulated as in Equation (5.2), where H s + S s V s represents the "accounting" value of the hedging portfolio as the sum of the shares account V s S s and the cash account H s 8 :

H T + S T V T = e T t rsds (H t + V t S t ) -e T t rsds
T t e -u t rsds S u dC u + P &L trading ,

P &L trading = e T t rsds T t V u d e -u t rsds S u -(q -b)e -u t rsds S u du , (5.2)
where P &L trading in Equation (5.2) is the product-sum of V u , the number of shares owned at time u, and the discounted assets return over the time frame du. As a sum, it could be interpreted as a profit-and-loss made from the delta rebalancing trading activities. Under a riskneutral Black & Scholes framework, the expectation value of P &L trading is zero. The same is true for the information-neutral measure under the filtration F t (in mathematical terms, E I [P &L trading |F t ] = 0) since the discounted asset returns are martingale under such probability measure and filtration. However, this is not true under the historical measure P (in mathematical terms, E P [P &L trading |F t ] ̸ = 0). It will be explained further, one has to carefully consider this component as it may result in a squeezing activities that are considered as a market abuse.

The terminal profit and loss (P &L) is defined by Equation (5.3) and has three components:

the payoff, the accounting value of hedges and the terminal cost of executing the missing shares between the time of expiry and the settlement date of the derivative transactions. With a physical delivery option, any missing shares have to be executed by the time between the expiry T and the settlement date of the option denoted by T + τ . Due to the size of the shares there would be 8 See Appendix B for the proof.

some transaction costs, noted as C (Q (S T ) + V T , τ ) in Equation (5.3) 9 .

P &L = Payoff H (S T ) + S T Q(S T ) +
Accounting value of hedges

H T + S T V T -
Terminal execution cost of missing shares

S T C (Q (S T ) + V T , τ ) .
(5.3)

Combining Equations (5.2) and ( 5.3), one can obtain the following formulation for the terminal P &L: (5.4)

P &L = Payoff H (S T ) + S T Q(S T ) -

Market Abuse Constraints and Problem Reduction

The large trader has to be careful about manipulating the payoff through its hedging strategy. In essence, the trader's interest is to optimize the P&L of the Equation (5.4) but (s)he is bounded by two regulatory statements:

• Manipulating the payoff : The large trader shall not manipulate the payoff of the option through the market impact of its hedging activities. Otherwise, it will be considered as a The prohibition of market manipulation should capture these inter-linkages ... 10

• Abusive squeeze: Equation (5.4) has a P &L trading component. An expected positive value for this component means that the large trader makes profits on the valuation of its shares owned, as a result of its hedging activities. By regulation this is considered as a 9 The amount of shares to be delivered/received is Q (S T ). C (Q (S T ) + V T , τ ) i a deterministic quantity at time T . At this time, we assume that we know exactly the cost of executing shares between T and T + τ . Please note the sum of Q (S T ) + V T is the large trader shares account as the terminal date; to be a hedge V T and Q (S T ) do not have the same sign, in other words if Q (S T ) is positive, V T shall be negative, and vice versa. 10 The statement continues with ...a trader holds a short position that will show a profit if a particular financial instrument , which is currently a component of an index, falls out of that index ... He places a large sell order in this financial instrument just before the close of trading. His purpose is to position the price of the financial instrument at a false, misleading, abnormal or artificial level so that the financial instrument will drop out of the index so as to make a profit ... market abuse as stated in section 1.6.4 of the Market Abuse directive of the FCA handbook ( Financial Conduct Authority 2005):

...an abusive squeeze -that is, a situation in which a person ... has a significant influence over the supply of, or demand for, or delivery mechanisms for a qualifying investment or related investment or the underlying product of a derivative contract... Considering the above, the optimization problem needs to be reduced firstly by excluding the payoff component of Equation ( 5.4) to address the manipulating the payoff constraint, and secondly by defining the stochastic control problem under one of the information neutral measures so that the expected value of the P &L trading becomes zero and an abusive squeeze is not inherent to the model. Considering that the only information neutral measure we have specified in this paper is the minimum variance information neutral measure, we will use this measure for the stochastic control problem. Therefore the problem formulation is reduced to Equation (5.5), with function J as it solution.

J(t, V t , S t ) = inf {Vs} s≥t E I * e -T t rsds S T C (Q (S T ) + V T , τ ) + T t e -s t rudu S s dC s F t (5.5)
The first component of Equation (5.5) is the transaction cost of executing missing shares between the expiry date T and the settlement date. The second part of Equation (5.5) is the transaction cost of executing shares before the expiry date. These two terms act against each other and the trader has to make a compromise between i) buying shares too early and getting closer to a full delta hedge with less missing shares to recover in the future, but paying extra costs because of executing the shares too quickly, ii) or leaving its position partially unhedged. If lucky the market will go on its direction and the partial hedge will be closer to the full hedge. Otherwise it will have some time between the expiry date T and the settlement date T + τ to execute the missing shares. Note that at the expiry date, the number of shares that is required is fixed and is no more uncertain.

Theorem 3. The cost function J can be deduced from the following partial differential equation:

∂J * ∂t + rS ∂J * ∂S + 1 2 ∂ 2 J * ∂S 2 σ 2 S 2 + J * -J dt (t,V,S)
= rJ (t, V, S) , (5.6)

with the terminal condition

J (T, V, S) = S T C (Q (S T ) -V T , τ ) , (5.7)
where J * is defined as follows:

J χ (t, V, S) = J (t, V + χ, S + SΛ(χ)) ω t dt + J (t, V + χ, S)(1 -ω t dt) + SdC t (χ), J * (t, V, S) = inf χ {J χ (t, V, S)} , (5.8)
and χ * t is the optimal number of shares to be executed during the period dt and obtained from

χ * = arg inf χ {J χ (t, V, S)} .
(5.9)

Proof is available in Appendix D. Note that ω t also depends on χ. Also note that in case of no market impact, Λ = 0 and Equation (5.6) reduces to the standard Black-Scholes equation.

6 Numerical Results

Transaction cost parametrization

Over the infinitesimal period dt, we define the infinitesimal cost of execution dC t (χ t ) of the trading policy χ t by Equation (6.1) where A is the (annualized) average volume, η is an impact factor and α < 1 is the exponential factor. Except for some differences on the denominator, this formulation corroborates with some of the past literature, e.g. [START_REF] Grinold | Active portfolio management[END_REF] and Zarinelli et al.

(2015) where α is often set to 1/2, and corresponds to a square root formula.

dC t (χ t ) = η χ t χ t + Adt α χ t (6.1)
For a longer period τ between the expiry and settlement date, one approach can be to apply an optimal purchase/liquidation algorithms which is well studied in the literature, e.g. [START_REF] Almgren | Optimal liquidation[END_REF]. But here we stick into a purchase constant rate program. Then in that case the expectation of function C(x, τ ) -the terminal cost of executing x at the time τ between the expiry and settlement-can be formulated as in Equation (6.2). For a formal proof, we refer to Appendix C.

E [C(x, τ ) |F T ] = η × ( x τ ) α ( x τ +A) α x (6.2)

Solving the partial differential equation

Given the two-dimensional partial differential Equation (5.6), numerical implementation can be done via the dynamic programming approach described in Algorithm 1.

Algorithm 1 Methodology for solving the large trader's cost function

1: J (T, S, V ) ← C(Q(S) -V, τ );
2: Discretize the time to △t intervals;

3: for t = T ; t ← t + ∆t; t = 0 do 4:

Using J(t + ∆t, ., .) apply equation (5.9) and solve χ ⋆ ;

5:

Using χ * , apply Equation (5.8) and calculate J * (t + ∆t, ., .) ; 6:

Using J * (t + ∆t, ., .), solve the Equation (5.6) and obtain J (t, ., V ), the finite difference scheme is explained in Appendix E;

7:

for iter = 0; iter ← iter + 1; iter < numiter do 8:

Using J(t, ., .) apply equation (5.9) and solve χ ⋆ ;

9:

Using χ * , apply Equation (5.8) and recalculate J * (t + ∆t, ., .) ; 10:

Using J * (t + ∆t, ., .), solve Equation (5.6) and obtain J (t, ., V );

11:
end for 12:

Having χ * , solve insider's trader option price O using a finite difference scheme of the partial differential Equation (2);

13: end for 14: Obtain J (0, S 0 , 0) and O (0, S 0 )

Numerical Results

Here we assume that the market follows the parameters described in Table 1 and that the large trader is the seller of the call option with parameters given in Table 2. Our overall parameter assumptions on the market impact are in line with the empirical study of [START_REF] Jarrow | An empirical investigation of large trader market manipulation in derivatives markets[END_REF].

The choice of parameters for the call option are also in line with one could expect from over-thecounter ("OTC") derivatives.

µ σ η ϕ λ S 0 A r α q b
0.02 0.25 0.1 20.0 2.0 100 5e+6 0.01 2.0 0.0 0.0 Table 2: Large Trader's option Figure 6.1 shows the terminal cost function J for the call option. For example, the back corner of the graph shows situations when the product is in the money. The number of shares V T is then exactly equal to the shares to be delivered and the cost function is 0. The left corner shows when V T is 0 and the product is in the money. The large trader needs, then, to purchase the notional amount of shares before the settlement date. In such case, the cost will be high since our model for the cost, see Equation (5.7), is linear in the spot price. As it can be viewed from The graphs show that because a full hedge strategy is too costly, the optimal χ * are much lower than the the Black-Scholes Delta. The difference in Delta are also presented in Figure 6.5; this graph suggests to purchase only few percent of χ * at time 0 while the Black Scholes Delta is 100% for the same spot levels (right side of the graph). This is significantly lower than the Black Scholes Delta, because the large trader has time before expiry to execute the full hedge, hence it will not make a big execution at time 0, Nevertheless, the large trader strategy χ * gets closer to the Black-Scholes Delta when getting closer to expiry. This is shown in Figure 6.6 illustrating the hedging strategies χ * and Black-Scholes Delta just before the expiry date. We observe that the number of shares to be purchased by the large trader is still smaller than the Black Scholes Delta but reaches 50% of the B-S Delta for high spot prices.

Near the expiry there is less time left for share execution and the large trader has to overcome significant stock shortage in a limited time frame compared to the inception date. Considering that large trader is hedging the call position defined in Table 3, it would be interesting to study how it will impact the insider trader's option price. Figure 6.7 compares the insider's trader price with the Black and Scholes model for the same call option. It shows that the option has more convexity than the Black & Scholes model around the money, it also shows that the prices are higher than the Black & Scholes when the option is in the money, and even higher than the intrinsic value. This is because of the directional market impact of the large trader. If the spot price is in-the-money, the large trader will purchase shares and therefore will make the spot price go deeper in-the-money.

T

Call strike 120 days 110

Table 3: Small Trader's option We would like also to highlight the fact that the insider option price is sensitive to parameters ϕ, η , λ as shown in Table 4. Economically speaking, an increase in the market impact parameters ϕ and λ results in a higher call option price for the insider trader (for the option defined in Table 2).

Conclusion

In this paper, we presented a model framework for martingale pricing theory under market impact.

We considered the case of existence of a large trader. We observed that the large trader, if negligent, can cause market manipulation by the means of its hedging activities. We emphasized that the mathematical model should not inherit a market manipulative strategy, and we proposed a hedging policy for the large trader that could deal with this issue 12 . We consider the market manipulation in derivatives as an important subject for research as the problem is quite present in the industry even though there is a lack of practical models, and this paper is one of the first in the literature that addresses this issue.

We also proposed a pricing model for an insider trader that is knowledgeable of the large trader's trading policy. We derived the pricing equation for such trader and observed that there is a set of measures called information-neutral measures in which the return of the assets are a martingale for the insider. While the presented model assumes that the information the insiders holds is the large trader's share trading activity, the scope of this paper goes beyond that; in fact, the Martingale Pricing Theory that we have presented is valid for any type of insider information as long as it generates market impact. The conclusion of this article can be then extended to these cases.

We concluded the paper with numerical results for the optimal delta-hedging strategy of an out-of-the-money Call option. While the payoff of the option has significant influence on our analysis, we observed that the model suggests a more relaxed delta hedging strategy than the Black and Scholes model, and the price of the call option for the insider trader is higher under market impact than the Black and Scholes. On that note, this paper provides a practical model that could be easily implemented for pricing derivatives under market impact.

impact. However, it is impossible due to the jump component of the market impact. One can only find a strategy that minimizes the quadratic variation but it will not neutralize it. This means that the market is incomplete and one cannot perfectly hedge a call option with the underlying stock only, in the presence of market impact. The quadratic variance of O t -P t reads as follows:

d ⟨P t -O(t, S t )⟩ = ∂O ∂S -δ t 2 S 2 σ 2 dt + δ t - O (t, S + SΛ(χ t )) -O (t, S) SΛ(χ t ) 2 ϕS 2 Λ(χ t ) 2 dt.
The insider trader minimizing the quadratic variance will obtain the following formula:

δ t S t = σ 2 σ 2 + ϕΛ(χ t ) 2 ∂O ∂S S + ϕΛ(χ t ) σ 2 + ϕΛ(χ t ) 2 (O (t, S + SΛ(χ t )) -O (t, S))
Having the expectation of dP t -dO t to be equal to 0 knowing F t . We can derive the following partial differential equation:

E [dPt] dt = δtSt (µ + q -b -rt + Λ(χt)ϕ) + rtE [Pt] = rtPt + (µ + q -b -rt + Λ(χt)ϕ) σ 2 σ 2 + ϕΛ(χt) 2 ∂O ∂S S + ϕΛ(χt) (O (t, S + SΛ(χt)) -O (t, S)) σ 2 + ϕΛ(χ) 2
On the other hand, we have

E [dO t ] dt = ∂O ∂t + ∂O ∂S S t µ + 1 2 ∂ 2 O ∂S 2 σ 2 S 2 + [O (t, S + SΛ(χ t )) -O (t, S)] ϕ
We conclude that:

r t P t = ∂O ∂t + µ - σ 2 (µ + q -b -r t + Λ(χ t )ϕ) σ 2 + ϕΛ(χ t ) 2 S t ∂O ∂S + 1 2 ∂ 2 O ∂S 2 σ 2 S 2 + ϕ 1 - Λ(χ t ) σ 2 + ϕΛ(χ t ) 2 (µ + q -b -r t + Λ(χ t )ϕ) (O (t, S + SΛ(χ t )) -O (t, S)) (A.1)
We simplify Equation (A.1) by defining ω t as

ω t = ϕ σ 2 -Λ(χ t ) (µ + q -b -r t ) σ 2 + ϕΛ(χ t ) 2 . (A.2)
One can easily verify that based on the definition of ω t and based on the assumption that we have made on the value of the hedging portfolio being cash injected/depleted to satisfy P t = O t , we obtain:

r t O = ∂O ∂t + (r t -q + b -Λ(χ t )ω t ) S t ∂O ∂S + 1 2 ∂ 2 O ∂S 2 σ 2 S 2 + ω t (O (t, S + SΛ(χ t )) -O (t, S))
This is the partial differential equation for insider trader. We now use the Feynman-Kac formula, the equivalent stochastic equation for the spot price S t under a probability measure that we call minimal-variance information neutral probability and note it as I * :

dS t S t = (r + b -q -χ t Λ(χ t )ω t ) dt + σdW t + Λ(χ t )dθ t ,
where dθ t is a compensated jump process with intensity equal to ω t . Also e -t 0 rsds S t and e -t 0 rsds O are martingales under the I * probability probability:

O (t, S t ) = E I * e -T t rsds O (T, S T ) F t , S t = E I * e -T t rsds S T F t .

B The P&L formula

Let b u = e u t -rsds denote the discount factor or the zero coupon bond price, Hu the discounted cash Hu = b u H u , and Su = b u S u the discounted asset. Therefore:

d Hu = -r u Hu du + b u dH u = V u Su (q -b)du -Su dV u -b u dS u dV u -b u S u dC u (χ u ) d Su = -r u Sdu + b u dS u (B.1)
On the other hand:

d(V u Su ) = -r u Su dV u du + V u d Su + Su dV u = b u dV u dS u + V u d Su + Su dV u (B.2)
Therefore we have:

d V u Su + Hu = (V u + dV u )d Su + V u Su (q -b)du -b u dC u (χ u ) (B.3)
If we are in one of the information neutral probability spaces, we can deduce that d Su -Su (q-b)du is a martingale. In the case of the minimum variance probability I * , we can deduce that:

d Su -Su (q -b)du = Su σdW u + Su Λ(χ u )θ u -Su Λ(χ u )ω u du (B.4)
Now taking the integration we obtain:

e -T t rsds (S T V T + H T ) = H t + S t V t - T t V u d e -u t rsds S u -(q -b)e -u t rsds S u du - T t e -u t rsds dC u (χ u )
C Transaction cost at option expiry For a longer purchase period, the optimal purchase/liquidation algorithms problem is a well studied topic, e.g. [START_REF] Almgren | Optimal liquidation[END_REF], but here we intend to consider a constant rate program. There are good reasons why we should apply a constant purchase program rather than a more complicated one since most derivatives desks do not optimise their cost of execution and they are more worried about the volatility. Therefore even though optimal algorithms exist, it will not be applied in derivatives. Here, we do not intend to make a suggestion on how the purchase should be done, but how things are behaving in the most practicable scenario. This is why we will use a constant rate program. In the case of having the Equation 6.1 for the short term cost, and with the strong assumptions that all shares requested will be executed, the long term cost of execution of X shares in τ time is

C(X, τ ) = τ 0 S t dC t X τ .
Note that in our case, X τ is the constant execution rate. In case we have Equation (6.1) for the short term cost, the long term cost of execution is as follows:

D Hamilton-Jacobi-Bellman Equation for the Large Trader

The trader wants to follow a delta hedge program that minimizes its cost. Under the information neutral measure the large trader wants to optimize the following equation: This equation can also be written as follows:

J (t
J (t, V, S) + rJ (t, V, S) dt = inf dV E (dV t )J (t + dt, V + dV, S + dS) + SdC t (dV t ) (D.1)

Let's define J χ as the solution to the following equation:

J χ (t, V, S) = J(t, V + χ, S(1 + Λ(χ)dπ t )) + dC(χ)

Then applying Itô formula with jumps leads to the following equation where dS C represent the non-jump part of the asset return:

J χ t + dt, V, S + dS C = J χ + ∂J χ ∂t dt + ∂J χ ∂S dS C + 1 2 ∂ 2 J χ ∂S 2 dS C 2 (t,V,S) (D.2)
And let's also define the optimal χ * such that:

χ * = arginf χ E[J(t, V + χ, S(1 + Λ(χ)dπ t )) + dC(χ)]

In such case, the equation (D.1) becomes as such:

rJdt = J χ * -J + ∂J χ * ∂t dt + ∂J χ * ∂S E[dS C ] + 1 2 ∂ 2 J χ ∂S 2 E[ dS C 2 ]
(t,V,S)

(D.3)
Hence, expanding the expectation of (D.3), we obtain the partial differential equations as formulated in (5.6) and (5.7).

E Numerical implementation of the large trader PDE

In terms of log returns, and assuming that χ * is already obtained, Equation (5.6) can be written as x = log(S), Υ (t, x, V ) = J(t, V, S), Υ * (t, x, V ) = J * (t, V, S), Λ(.) = ln(1 + Λ(.)),

rJ = ∂Υ * ∂t + r -(q -b) - 1 2 σ 2 ∂Υ * ∂x + 1 2 ∂ 2 Υ * ∂x 2 σ 2 + Υ * t, V, x + Λ(χ ⋆ ) -Υ (t, V, x) (χ ⋆ ) 1 dt
We define the boundary levels x min , x max for the log spot and d min , d max for the shares. We define the discretization {(t i , x l , d k , )} with (0, 0, 0) ≤ (i, l, k) ≤ (I, L, K) , (0, x min , d min ) ≤ (0, x k , d l ) ≤ (T, x max , d max ) .

Here we assume a uniform grid, with intervals of triplet (△t, △x, △d). At the maturity level, we define the following equation:

Υ I,l.k = C (d k -Q (e x l ) , τ )
At t i+1 , we first calculate the optimal value in discretize terms. Assuming that we have the optimal χ ⋆ we define ι * = χ ⋆ ∆d and L ⋆ i,l.k as below equation for the gridpoint (t i , x l , d k ).

L ⋆ i,l.k = Υ * i+1,l,k+ι * -Υ i+1,l,k (1 -ω * ∆t) + Υ * i+1,l+ Λ(ι * ∆d) △x

,k+ι * -Υ i+1,l,k ω * ∆t + e x l dC (ι∆d) (E.1)

We apply a fully implicit scheme and the discretization will be as of below:

1 + r△t + △t (△x) 2 σ 2 Υ i,l,k + (r -(q -b) -1 2 σ 2 ) △t 2△x -1 2 △t (△x) 2 σ 2 Υ i,l-1,k + -(r -(q -b) -1 2 σ 2 ) △t 2△x -1 2 △t (△x) 2 σ 2 Υ i,l+1,k = Υ * i+1,l,k + L ⋆ i,l.k
With Neumann boundary condition, we have:

Υ i,l,k = Υ i,0,k = C (N T -Q (e xmin ) , τ ) ≈ -C (d k -Q (e xmin ) , τ + T -t i ) Υ i,L-1,k = Υ i,L,k ≈ C (d k -Q (e xmax ) , τ + T -t i )

  market abuse as prescribed in article 12.1.a.ii of the EU market manipulation regulation (European Parliament and of the Council 2014) and the FCA handbook example of Market abuse ( Financial Conduct Authority 2005) (MAR 1.6.15):... entering into a transaction, placing an order to trade or any other behaviour which: ii) secures, or is likely to secure, the price of one or several financial instruments ... at an abnormal or artificial level ... ... manipulative strategies can also extend across spot and derivatives markets. Trading in financial instruments, including commodity derivatives, can be used to manipulate related spot commodity contracts and spot commodity contracts can be used to manipulate related financial instruments.

Figure 6

 6 Figure 6.1, even though the cost function has different values on each corners of the graph, cost function gradually flattens when we go backward in time as it can be observed in Figure 6.2.

Figure 6 . 1 :

 61 Figure 6.1: The terminal cost function J for a hedged option. V T shall be read as a multiple of the notional.

Figure 6 .

 6 Figure 6.4 shows the amount of shares to be executed (χ * ) at time 0, and Figure 6.3 shows this amount just before the expiry. 11 The graphs show that because a full hedge strategy is too

Figure 6

 6 Figure 6.2: The cost function J at time 0 for a hedged call option.

Figure 6 . 3 :Figure 6 . 5 :Figure 6

 63656 Figure 6.3: The optimal shares strategy function at time T -dt for a call option.

Figure 6

 6 Figure 6.7: Comparison between Black and Scholes price and insider trader price for a call option.

  rsds S u dC u + S T C (Q (S T ) + V T , τ ) + P &L trading + e

		Total transaction cost of missing shares
	e	T t rsds	T	e -u t
		t

T t rsds (H t + V t S t ) .

Table 1 :

 1 Market parameters

	T	τ	Call strike Notional
	120 days 15 days	110	1e+6

Table 4 :

 4 Call option price for insider trader at t = 0

  , V, S t ) = e -rdt inf

	{Vs} s≥t

E [ J (t + dt, V + dV, S t+dt ) + S t dC t | F t ]

The transaction cost and temporary impact have been used interchangeably in some of the literature, see[START_REF] Alfonsi | Order book resilience, price manipulation, and the positive portfolio problem[END_REF], or for example[START_REF] Huberman | Price manipulation and quasi-arbitrage[END_REF] who use the concept of temporary market impact as the slippage cost. Some other studies, such as[START_REF] Gatheral | No-dynamic-arbitrage and market impact[END_REF] and[START_REF] Bouchaud | Fluctuations and response in financial markets: the subtle nature ofrandom'price changes[END_REF] consider a temporal price impact decaying over time.

The reader is referred to the introduction section of[START_REF] Almgren | Option hedging with smooth market impact[END_REF] for a lucid description that such case is common.

While this goes beyond the scope of this paper, there are many algorithmic methods to hide the order size, the most simple one is a price taker. Another one may be iceberg type limit orders. See[START_REF] Christensen | Prediction of Hidden Liquidity in theLimit Order Book of GLOBEX Futures[END_REF] for more details.

It is worth mentioning that for the sake of simplicity, we assume that the impact rate of the order (ϕ) is independent from its size (χt). A relationship between ϕ and χt is less studied in the empirical literature, because most academic literature so far investigate the average price impact rather than its variations. We make the point that while our model can be generalized by integrating this relationship, our intention is not to define a

Even though our model does not consider this, in illiquid markets, the borrowing spread can have a very stochastic behaviour and change drastically over time. For example, if the market experiences a short squeeze situation in some period, the borrowing rate will increase drastically. An example was the QuantumSpace Corp on beginning of January 2021 where its lending rate reached more that 100% per year because of a lack of supply in the lending market.

Note that even though our model has the capacity to include collateral or margin, for simplicity we assumed that there is no requirement. We refer the reader to[START_REF] Piterbarg | Funding beyond discounting: collateral agreements and derivatives pricing[END_REF] for more details on the impact of collateral agreements and funding on derivatives pricing.

Physical delivery is the most common type of transactions on stock options, See for example Cboe Inc. (2022).

The graph shows that the number of shares to be executed in the z-axis are layered. This is mainly due to the methodology that we use in our algorithm where we determine the χ * within the discretization points of (V ) in the finite difference scheme and therefore the optimal points will be discrete. This ensures a better stability of the finite difference scheme.

Acknowledgements

Karim Barigou acknowledges the financial support of the Joint Research Initiative on "Mortality Modeling and Surveillance" funded by AXA Research Fund.

Disclosure statement

The authors report there are no competing interests to declare. Tankov, Peter. 2003. Financial modelling with jump processes. Chapman and Hall/CRC. Tankov, Peter, & Voltchkova, Ekaterina. 2009. Jump-diffusion models: a practitioner's guide.

Banque et Marchés, 99(1), 24.

Zarinelli, Elia, Treccani, Michele, Farmer, J Doyne, & Lillo, Fabrizio. 2015. Beyond the square root: Evidence for logarithmic dependence of market impact on size and participation rate.

Market Microstructure and Liquidity, 1(02), 1550004.

Appendix

A Differential Equation for insider trader

The insider trader is aware of the large trader position. It knows the amount λχ t and can be hedged against it. We assume that the insider trader is aware of the hedges χ t . Let's assume that the insider trader has entered into a transaction with a specific payoff O (S T ), we call O (t, S), the price of this option at time t and spot level S. The insider trader portfolio at time t is given by

The insider trader has δ t shares and P t -δ t S t in cash amount. Due to the fact that the notional is small, the insider trader, contrary to the large trader, can hedge its position without additional cost of execution. The dynamics of the portfolio P t are as follows:

In our incomplete market, the option cannot be perfectly hedged by a self-financing portfolio.

Therefore, we search for a hedging strategy that satisfies the following properties:

• P t = O t : As this is not a self financing portfolio, at each time step there will be cash injection/depletion so that the hedging portfolio value is equal to the theoretical value of the option.

• E[dP t ] = E[dO t ]: The average ( first order) option price movement is fully hedged by the portfolio.

• δ t = arg inf δ d ⟨P t -O(t, S t )⟩ : The second order option price movement is reduced and optimally minimized by the hedging portfolio.

On the other hand, by Itô formula, the option dynamics are given by If the market was complete, it is possible to find δ t such that the quadratic variation d < P -O > t is always zero. This would have been the case for the Black-Scholes model without any market