Personal Work Group Evaluation Assignment Example | Topics and Well Written Essays - 750 words

Individual Work Group Evaluation - Assignment Example The objective of the gathering was to choose three subjects with which the program ought to be propelled and what number of inquiries ought to be posed in a solitary set. The gathering is additionally expected to recommend different highlights that would make this administration appealing for the understudies. The gathering enrollment comprises of four individuals: two school course book editors, an IT pro, and a fund master from a similar association. The gathering began on a constructive note since the individuals knew each other from the association at the same time, with the exception of the two editors, had not cooperated on a venture previously. During the framing stage, the individuals met in a meeting room and traded welcome. The IT pro presented the venture and requested that the reading material editors recommend which subjects ought to be chosen. The gathering at that point appeared to enter the raging stage as the editors differ on a few issues. One manager needed to choose secondary school science subjects while the other editorial manager worried on utilizing English inquiries for ESL and EFL understudies. They interfered with one another every now and again and eventually, the other two individuals had been kept separate from the conversation and it turned into a tussle between the two editors. In the end, the IT pro noticed that choices couldn't be made along these lines and that it was important to set up certain standards. He proposed that every part would share their thoughts alongside the upsides and downsides with the goal that a decent conversation can happen. He additionally proposed that at long last a vote would be taken to choose which subjects to choose. The individuals concurred that they required more opportunity to explore about the size of the market and the upsides and downsides of their decisions. The gathering was deferred on this note. During the procedures, the two editors occupied with turf assurance and in building up their position over one another. They shared thoughts however only to dismiss contending ones. The IT

process of specified project management

Question: Talk about the Processes for Managing Change to the Project ? Answer: Presentation Change the board is a procedure of indicated venture the board point of view that offers endurance arrangements from any new arrangement of specialized or instructive determination inside numerous specific associations or inside any predefined venture (Adeniji et al., 2013). Change the board procedure is useful in settling different issues and conditions required inside framework architectuco of any concerned undertakings or association. Each association and undertaking has distinctive change the board points of view concerning their feasible objectives and targets. Interestingly with these realities, this report is expounding about the significance of progress the executives procedure required inside any association or inside any predefined venture. Also, this report is thinking about California State University as chose association inside this report (Brighthub Project Management, 2017). Steady writing audit and authoritative setting of the chose association are being expounded inside this report so as to feature the significance of progress the executives rehearses inside the concerned association. Authoritative Context California State University is one of the well known state funded colleges of California. This college possesses 23 grounds just as eight off-grounds enlisted with 460,200 understudies with 24,405 resources and 23,102 staffs inside it (Brighthub Project Management, 2017). CSU is considered as biggest multi year state funded college framework. Interestingly with these realities, any IT framework engineering or educational changes that is made inside the system, should be flowed among these individuals from the college. Hence, already the association has one manual ITSM structure that works their different data and these are circled inside the individuals and houses required inside the grounds. This whole procedure was troublesome; hence, one car change the board framework concerning ITSM has been executed inside the system of the association. This change the board technique was considered for dealing with the framework engineering inside the association. California State University co nfronted such huge numbers of issues while working with this manual ITSM framework design inside their association (Brighthub Project Management, 2017). Their objective is to actualize and ad lib their data framework and system design concerning their inward and authoritative requests. The change the board point of view is included and presented inside the association for overseeing framework adequacy. Models, structures and procedures for how change is achieved or experienced in ventures As per Serra and Kunc (2015), the advantages acknowledgment the board procedure required inside task achievement and is reliant on the business techniques and procedures received by the activities. Conversely with this reality, this article is concentrating on the difference in procedures that are dealing with the authoritative turn of events and further advancement with deference different basic perspectives and realities. This article is thinking about the quantitative methodology of research technique for supporting perspectives identified with effect of progress the executives procedure inside any hierarchical setting. Examining strategy (Filling the Value Gap) is utilized for dealing with the steady proclamations. This article is for the most part reflecting about the need of acknowledgment the board and its impact on the venture achievement. As indicated by Hornstein (2015), the hierarchical change the executives inside data innovation and business change ventures is significant for change the board point of view. Interestingly with these realities, there are results are being expounded inside this article regarding remain solitary activities that centers around change of foundation and authoritative systems required inside any Information Technology ideas. This paper isn't focusing on a specific technique for examining their examination forms. This article is for the most part considering the how activities are affected by the authoritative change the executives forms required inside any association. The change the board idea is supported for this venture idea as there are odds of authoritative upgrades. As indicated by Santos and Quilliam (2015), the utilizations of programming in dealing with the progressions required inside the hierarchical procedures and advancement points of view is obligatory for authoritative turn of events. Also, the change the executives forms are considered concerning programming bundles and its use inside the association. This usage of programming bundles in associations likewise controls changes and dangers. The exploration technique considered inside this article is subjective however there are no elaborations about this; this is distinguished while basically assessing the paper. The points and targets of this paper is supported as this paper is expounding process that can supplant the specific change the executives procedure for overseeing authoritative turn of events. As indicated by Low et al. (2015), the use of progress in staff care rehearses for improving occupant results inside nursing homes significant for overseeing inhabitant results. Conversely with this reality, this article is featuring different issues and legitimization for embracing Change in Staff Care Practices for improving the occupant results. This article is thinking about subjective investigation for evaluating the staffs conduct and disposition towards changing the Resident Outcomes. This article is mirroring the way toward dealing with the worker conduct for evaluating the requirement for improving Resident Outcomes. This targets and objectives of this article is supported as for the hierarchical requests for overseeing change the executives procedure inside emergency clinics. Procedures for overseeing change to the undertaking As per Kuipers et al. (2014), the change the executives procedure are influencing the open associations. Conversely with these realities, the open associations are thinking about the change the executives viewpoints for dealing with the hierarchical improvement points of view. What's more, this article is thinking about subjective research strategy for gathering required information for the concerned research forms. What's more, the examination results and procedure are supported enough for dealing with the change points of view required inside the authoritative structure of open associations as for different basic viewpoints required inside hierarchical turn of events. As indicated by Adeniji, Osibanjo and Abiodun (2013), explicit venture of the Nigerian financial industry dependent on the hierarchical change and human asset the executives procedures and developments are significant for investigating the change the board in emergency clinics. Also, this paper is utilizing quantitative and subjective investigation for dealing with the gathering databases about the procedures of progress the board procedure required inside any authoritative culture. This article has defended elaboration about the change the executives forms required inside the authoritative structure of a particular association. As indicated by Jos et al. (2012) the expert disheartens and obstruction of changing procedures required inside any association includes change the board procedure for overseeing hierarchical viability. Conversely with these realities, there are different angles and explicit perspectives that are being expounded inside this article as for different viewpoints required inside any association. Conversely with these realities, this article is thinking about the subjective research system inside this exploration procedure. Conversely with these realities, the examination theme is compelling legitimized with deference authoritative objectives and targets concerning the change the executives procedures and financing. As per Pmibatonrouge.org (2017), this article is explaining about the empowering procedure of hierarchical change regarding vital activities inside any association. Conversely with these realities, this article is concentrating on the authoritative driving variables that are commandingly make changes towards the hierarchical improvement with the assistance of changing methodologies and functionalities. Conversely with these realities, examine techniques utilized inside this exploration article. The considered research subject is defended inside this article as this altogether dependent on change the board procedure for overseeing hierarchical turn of events. Correlations and Contrast Conversely with these realities, the idea of progress the executives is viably expounded regarding hierarchical setting. This can be expressed from the correlation of concerned writing that change the board has successfully executed inside authoritative setting as for different hierarchical destinations and objectives. The examination is expressing that the change the executives procedure is required for any sorts of authoritative or undertaking improvement as for different basic viewpoints relying upon the framework design and hierarchical requests venture required inside any association. In this part of the report it is dissected that change the executives forms are subject to the conditions required inside any task or association. Interestingly with these realities, it is discovered that however there are different determinations of various circumstance inside various undertakings, restrictive realities and conditions in overseeing change the executives are same as general feeling of the requirement for overseeing change the board inside any task or association. Interestingly with the realities and points of view required inside the framework design of any association or task, the viewpoint of progress the executives is carried inside the undertaking or association with deference different basic perspectives and ideas. Suggestions Th

Social Media to Influence Recruitment Decisions-Free-Samples

Question: Talk about the Social media to impact enlistment choices. Answer: Web based life to impact enrollment choices. Web based life enrolling has as of late developed as a widely drilled type of enlisting inside different associations (Gibbs, MacDonald Mackay, 2015). The appearance of online life through different stages like Facebook, Instagram and Twitter isn't obscure. These sites have given individuals a stage to transform themselves into a brand (Harris Rae, 2011). Individuals can share their thoughts, speak with others, voice their conclusion and examine driving issues perpetrating the present reality. Each individual has utilized the stage as it regards important for them. The issue for this situation isn't uncommon. During the time spent selecting people, an entrance to their online life stages gives the HR staff certain extra data that might possibly work for the applicant (Kaur Sharma, 2015). Yet, is it extremely moral to access a people individual life while choosing their expert profession? Additionally, are the subtleties shared by people on their online networking stages are extremely close to home? The main alternative of looking for consent from applicants before visiting their web based life stages is without a doubt a moral move. In any case, there two dangers presented in this procedure Firstly, both of the applicants have total position to deny the entrance. This would most likely give different up-and-comers an out of line advantage. Also, there is a solid chance that the up-and-comers alter, stow away or erase any subtleties that they regard may influence their odds of determination. In which case, the whole picture would get indistinct. The subsequent choice of survey the data that is open without illuminating the competitors appears to be reasonable on the grounds that each individual has an entrance to open data. In any case, there is no guarantee that the open data gave by the applicants on these stages genuinely thinks about their expert life. Likewise the three competitors might not have open data to a similar degree. All things considered, one up-and-comers profile is altogether available while different up-and-comers profile has security get to. Ultimately, after review the profile of both of these competitors, on the off chance that the HR finds any shared connections, at that point it might influence the HRs choice which thusly may prompt a one-sided judgment (Caers Castelyns, 2011). In conclusion, the alternative about not taking a gander at the web based life profile of both of the competitors leaves the association off guard of not having any extra insights concerning both of them. Later on the off chance that the chose up-and-comers open profile thinks about gravely the association, at that point it is hard to fix the issue. As I would see it social enlisting is without a doubt making progress however a specific line must be drawn. Each individual has an individual and an expert life and it might be uncalled for to pass judgment on anybody dependent on the other. Out of the numerous internet based life stages that exist, LinkedIn has been worked to empower and fabricate proficient connections. A people refreshed LinkedIn profile is the nearest to reflecting upon their expert life. The site likewise offers to show a people supports from their companions and seniors (Zide, Elman Shahani-Denning, 2014). These supports may be valuable for the association in assessing qualities of people which suit the hierarchical culture. When a worker is enrolled in the business, it is basic to prescribe the representatives to keep up a nice open profile on the entirety of their internet based life stages. This is explicitly valid for individuals who are recruited at a senior position. Trainings on viably overseeing online networking profiles could likewise be given by associations to every one of their representatives. In any case, it would be to a great extent out of line to pass judgment on a people candidature dependent on anything that they share in their own existence. Along these lines I suggest that the HR work force must welcome the contender for a second round of meeting or a psychometric test for better clearness on the issue. References Caers, R., Castelyns, V. (2011). LinkedIn and Facebook in Belgium: The impacts and predispositions of informal organization destinations in enrollment and determination procedures.Social Science Computer Review,29(4), 437-448. Gibbs, C., MacDonald, F., MacKay, K. (2015). Online networking use in lodging HR: enlistment, employing and communication.International Journal of Contemporary Hospitality Management,27(2), 170-184. Harris, L., Rae, A. (2011). Building an individual brand through social networking.Journal of Business Strategy,32(5), 14-21. Kaur, P., Sharma, S. (2015). Utilizing web based life for boss marking and ability the board: An experiential study.IUP Journal of Brand Management,12(2), 7. Zide, J., Elman, B. furthermore, Shahani-Denning, C., 2014. LinkedIn and enrollment: How profiles contrast across occupations.Employee Relations,36(5), pp.583-604.

Free Essay Writing Tips

<h1>Free Essay Writing Tips</h1><p>Are you pondering where to get great paper composing tips? You can locate a mess of them on the web. Some of them are free, while others you need to pay for.</p><p></p><p>If you're similar to me, you may not know precisely what you need to state in your paper. Regardless of whether you do, you're most likely not certain how to write it down. That is the reason I'm going to give you some free composing tips. Here they are:</p><p></p><p>o Good exposition composing tips are not just for understudies. You may not understand this however you additionally need to keep in touch with one for yourself at any rate once in your life. All things considered, you have a story to advise or a statement to utilize or a paper to compose. Along these lines, when you finish it, you'll certainly have gotten the hang of something important and will be better outfitted to manage whatever comes your way.< /p><p></p><p>o Writing doesn't need to be recorded. While composing for individuals may seem like a pleasant thought, the main time you can truly have some good times is the point at which you are communicating your emotions and you can communicate your contemplations plainly without being compelled to compose something in dark and white.</p><p></p><p>o Remember, you may not get everything right, except you can never be excessively cautious. In some cases it assists with flopping quicker than you can get it right.</p><p></p><p>o The principal thing to recall is that you may not generally get things right. The best activity when something turns out badly is to attempt once more. At times an increasingly definite thought can have the effect among progress and failure.</p><p></p><p>After perusing this article, you may think you've discovered some simple exposition composing tips. I wish you kar ma with composing your next paper.</p>

Create The Perfect Argumentative Essay Topic

<h1>Create The Perfect Argumentative Essay Topic</h1><p>Argumentative paper themes help to effectively express the idea in a reasonable and succinct way. In the event that you are searching for a particular contention, at that point composing your own article is the best approach. An elegantly composed contention can give you the favorable position over other paper subjects in a school article composing contest.</p><p></p><p>There are many configuration alternatives that can assist you with making a contentious exposition point that will be pertinent to your decision of exposition. Utilizing this paper subject format will make the way toward composing your article less tedious and progressively effective. It additionally makes it simpler to follow and will furnish you with an organized diagram of how you should function your article subjects into your essay.</p><p></p><p>The first activity when composing pugnacious pap er points is to conceptualize the theme. Consider what might have been your preferred liable to expound on when you were more youthful. This could be anything from a film, book, or music classification, to a particular leisure activity or interest.</p><p></p><p>Once you have your point as a main priority, the subsequent stage is to consider the zones where you believe you would fit best in your paper. Begin to envision yourself as the character in your article. How might you act in that circumstance? What might the perusers of your article expect of you?</p><p></p><p>Make sure that you submit to specific standards that will assist with keeping your exposition on target. Every one of your article themes ought to be novel and not the same as the last. Attempt to adhere to the subject of your choice.</p><p></p><p>Write a free-streaming story dependent on your insight into the subject. Try not to confine yourself to one point. On the off chance that you are going to explore the subject in a school exposition composing challenge, at that point you have to introduce the most ideal contention in your article. Ensure that you are building your contention in a sensible way that calls attention to the advantages of your picked topic.</p><p></p><p>Next, layout the principle thoughts that will be introduced in your article. Your diagram will fill in as the establishment for your whole contention. It will show how you are introducing the entirety of your contentions so as to back up your focuses. Continuously back up your focuses with an explanation that explains to the peruser why you accept the manner in which you do and what advantage your picked subject offers.</p><p></p><p>After you have arranged out your factious article points, record them. Ensure that you compose the subjects obviously and compactly. Attempt to hold every theme under 200 words lon g. This will assist with making the article progressively pleasant and simpler to peruse as well.</p>

Compare and Contrast Essay Topic Ideas For College

<h1>Compare and Contrast Essay Topic Ideas For College</h1><p>Compare and difference article theme thoughts for school ought to be pondered before any understudy acknowledges a proposal to go to a school. In the event that they don't plan to get a higher education, at that point there are choices accessible for them. Everybody ought to appreciate the school understanding. Individuals who figure this might be a risk to their quest for an advanced education should audit a portion of the focal points and detriments related with doing so.</p><p></p><p>Some undergrads will just apply to a school since they figure it will give them the most obvious opportunity to get into the college or school they pick. Others will do it for the opportunity to go to a specific school or college or to just have the hour of their life.</p><p></p><p>If an understudy is miserable at home and needs to make a few companions, at that point a schoo l offers some valid justification to visit and look at the grounds. Some will essentially need to mingle and arrange, however a couple will even go to an occasion or movement to praise some accomplishment. Some may conclude that they truly like the school they are joining in, yet the movement costs or cost of their lodging are a lot for them to have the option to afford.</p><p></p><p>It might be hard for an understudy to clarify what sort of intrigue and why the person in question wishes to go to their preferred school or college. The school article theme thought for school is an extraordinary method to communicate that interest.</p><p></p><p>A undergrad may feel as if the person in question has grown out of school life. This isn't generally the situation, be that as it may. A few understudies may find that they are exhausted or miserable in light of the fact that the remaining task at hand or the requests of the school every day have made them required their exercises to be postponed and can't put the things off.</p><p></p><p>A undergrad may feel that the individual can't bolster oneself that the individual in question has an excessive number of obligations. There are a lot of approaches to communicate this that are not out of the range of an understudy. They might have the option to support their folks, family, or companions figure out how to assist them with paying their tuition.</p><p></p><p>Students ought to likewise consider the way that the subject of their paper research might be fascinating and may make a decent story. A decent method to compose a persuading paper is to write to fulfill the entirety of the above requirements.</p>

Custom Term Paper For Sale - 4 Reasons to Order Custom Term Paper For Sale

<h1>Custom Term Paper For Sale - 4 Reasons to Order Custom Term Paper For Sale</h1><p>The thought of requesting custom research paper available to be purchased has for some time been a definitive method to put an extraordinary turn on your understudies' assignments. They are an incredible method to make enthusiasm for a topic that might be in any case overlooked by your understudies. As an educator, you can likewise utilize them to guarantee that your class is fascinating and that understudies are tested to thoroughly consider of the box.</p><p></p><p>When it comes to requesting custom research paper available to be purchased, there are a few alternatives accessible to you. One of the most widely recognized choices is the decision between utilizing pre-planned formats or making your own. There are favorable circumstances and drawbacks to both of these choices. This article will examine a portion of the favorable circumstances and impediments that will be introduced as you consider buying research project for sale.</p><p></p><p>First, when you buy pre-planned formats, you will probably profit by utilizing a similar subject all through the class. Research project available to be purchased is utilized in a wide range of settings. This can cause the task appear to be less special on the off chance that you to decide to make it with a format. Hence, when you buy a format, you should ensure that the design incorporates similar textual styles, styles, hues, shapes, and sizes all through. You ought to likewise discover a format that remembers your ideal wordings for every individual passage for each chapter.</p><p></p><p>Second, you should choose a text style for your article. Normally, you will need to pick a text style that is striking and that isn't too much enormous. The purpose behind this is the point at which you buy custom research paper available to be purchased, the text dimension is generally little and simple to peruse. On the off chance that you pick a textual style that is excessively huge, it will be hard to peruse and your understudies will think that its hard to concentrate on the essay.</p><p></p><p>Third, in the event that you need to make your paper one of a kind, you can make a format that joins hues that have never been utilized. On the off chance that you are going to utilize hues, you might need to be particularly mindful so as to pick hues that are anything but difficult to peruse. Pick a shading that is strong yet that doesn't stick out. You may likewise need to consider utilizing hues that stand out from the foundation of the paper. Utilizing hues that praise the foundation is a decent alternative as well.</p><p></p><p>Fourth, you might have the option to set aside cash by buying a format. The explanation behind this is you will purchase the paper in mass. You can buy a pack that inco rporates the entirety of the paper expected to finish one exposition. This alternative will generally cost more than the expense of utilizing a format, however it is certainly justified regardless of the extra cost to get a good deal on future paper purchases.</p><p></p><p>Finally, when you request custom research project available to be purchased, you will approach a wide assortment of text styles and hues. Since your understudies regularly blend various types, you may not generally approach similar hues or text styles. At the point when you buy a format, you can find that you are constrained regarding hues. At the point when you request custom research paper available to be purchased, you will have your preferred option to utilize the shade for every individual passage and chapter.</p><p></p><p>If you were thinking about buying custom research paper available to be purchased, you ought to consider the numerous advantages that you wil l get by requesting layouts as opposed to buying one from a book or from a school book shop. These kinds of procurement choices are energetically prescribed for instructors who like to purchase utilized things. You ought to likewise consider buying a one-page paper available to be purchased when you consider the manners in which that it very well may be utilized. At long last, custom paper available to be purchased can set aside you cash since you will have the option to purchase in bulk.</p>

Essay Writing - What Topics to Write About When Writing an Ivy League College Application Essay

<h1>Essay Writing - What Topics to Write About When Writing an Ivy League College Application Essay</h1><p>When composing an Ivy League school article, the most ideal approach to start is by first rattling off the entirety of the significant subtleties you need to remember for your exposition. You will at that point need to manage what to state about every theme as you compose your article. You might be astonished at how the points will come up all through the creative cycle and you should settle on the best choices you can on an individual level to use in your essay.</p><p></p><p>When composing an Ivy League school application paper, it is additionally essential to choose what the most significant components to incorporate are. It is a smart thought to incorporate the entirety of the significant data you feel is significant in your exposition so as to enable the peruser to comprehend what the paper is all about.</p><p></p>& lt;p>To start, there are three significant points that must be tended to when composing an Ivy League school application article. These three points incorporate affirmation measures, proposals, and utilization of the web. It is additionally critical to incorporate when a particular competitor has no recorded record of your own achievements.</p><p></p><p>A great tip when composing an Ivy League school application article is to list your most noteworthy scholastic accomplishment. You might be stressed this is simply excessively broad, yet when utilizing this as your beginning stage, you will have the option to discuss the achievements you feel are the most significant in your specific case. This won't be the most significant data you incorporate, however this will offer you a chance to discuss your secondary school and school experiences.</p><p></p><p>There are a few reasons why affirmations staff may get some information about confir mations norms. It could be on the grounds that they are keen on catching wind of any difficult work you did to get into school, or it could just be to perceive how far you've originated from when you initially selected. As you are talking about your accomplishments, make certain to expound on the best pieces of your life and consider your commitments to your own success.</p><p></p><p>Many individuals find that when composing an Ivy League school application exposition that their folks' expert certifications and suggestions will come up. This is a great idea to talk about when referencing guardians' names, however recollect to not cover everything about. Make certain to concentrate on a couple of the most significant things, however remember to be straightforward and notice how you utilized their administrations or exhortation. Incorporate your inspirations for applying and whatever other significant info that your folks have given you.</p><p></ p><p>When composing an Ivy League school application paper, the utilization of the web is significant. This ought to be recorded first when posting your scholarly accomplishments. You ought to likewise list on the off chance that you are recorded on any online grant databases that would incorporate your data. With this data included, you ought to have the option to list every one of your chances and urge your potential college to get in touch with you legitimately to examine the entirety of your options.</p><p></p><p>When composing an Ivy League school application article, there are numerous subjects that you will be approached to consider. Make sure to cover the entirety of the significant subtleties throughout your life and to incorporate the entirety of the significant variables you feel are significant so as to permit the peruser to completely comprehend your solicitation for confirmation. Make sure to deliberately list every one of your accompli shments and talk about any expert experience and recommendments from others in your life.</p>

Where to Find How to Write an Argumentative Essay Samples Outline

Creative Research Paper Topics For 20th Century Art

Creative Research Paper Topics For 20th Century ArtThere are many topic areas in the current academic study of twentieth century art, including paintings, sculptures, prints, figurative art, and other media. The topic is, for instance, sometimes called 'art for the people'the culture and history of art.' And yet, this context may not fit your research very well. Because, in an interview situation, you may need to present and defend your own perspective or the viewpoint of the profession.For instance, consider a journalist who is investigating a literary genre she thinks may have begun in the '70s. Perhaps she is particularly interested in the post-modern art movement. Or maybe it was the hippie movement. In an interview setting, she is eager to find out the dates of the art movements' start. What she finds instead is that the journalistic inquiry tends to shift with the industry.Cultural and social historians tend to focus on particular periods and even particular artists. Their view s tend to focus on the stories and fashions of a particular time in history. This is especially true in situations where the industry is much more modern than the specific subject. For instance, political theorists study current trends within nations or countries, while industrial sociologists study trends within specific industries. How can a reporter ask a question about something today, but only when there is someone to answer it?Artists today tend to make pieces that are more abstract and/or less specific in nature. There are lots of abstract art pieces. But these can often be very difficult to interpret, to the point where readers can get lost. Whereas traditional paintings have themes that follow specific themes, rather than abstract pieces that have no themes. They often have themes that are related to the content of a painting.Researcherscan study these themes and the media through research paper topics that look at specific artists. Topics may include: art installations, fi lms, installations, book covers, movie posters, paintings, videos, television programming, music videos, sculptures, and wall murals. These articles also consider the many sources of these media. How were they created? What are their stories?Because of the ubiquity of modern art, many of these concepts apply to artistic forms that were contemporary with these artists. The list includes collage, printmaking, digital arts, and typography. And then there is much more. And these are just the basics. This broad scope makes this a great starting point for studies of these specific media.In addition to creating articles with articles, students should keep a journal for studying interesting cultural references. Of course, this also helps to capture questions, and provides topics for future research. And you may want to create two or three different journals for these purposes. This allows you to study two or three different media from different angles and simultaneously.In short, when you a re creating research paper topics for an area of interest, try to get more information than what you think you will need. And do not be afraid to search beyond the popular media and genres. Look at why they have been popular, and why they are now important to a new generation. You may find something that's really new, but there's no substitute for experience.

Mean Reversion And Stochastic Volatility Finance Essay - Free Essay Example

Finance is one of the most rapidly changing and fastest growing areas in the business world and new mathematical models are essential to implement and price these new financial instruments. The classic Black-Scholes model, the Jump Diffusion model, Mean-Reversion Jump-Diffusion, Finite Differ ence Method and Monte-Carlo valuation arewidely used in option pricing. A meaningful number of empirical evidences in literature have tested the effectiveness of the models using historical data. This dissertation aims at analysing how do the prices of options react under mean reversion and stochastic volatility. Chapter 1 Introduction Anoptionisacontractwhichgivesits owner,therightbutnotthe obligation,tosellorbuy an underlying asset at a specified price which is known as the  ¿Ãƒâ€šÃ‚ ½strike price ¿Ãƒâ€šÃ‚ ½ before or at a fixed expiration date (at maturity). An important key to success in the market of options, is paying the the right price for an option, when an individual buys one, or when he writes (sells) an option. Nearly all operators on the option exchanges and most option professionals use modelsof option pricing, like  ¿Ãƒâ€šÃ‚ ½Option Master R . ¿Ãƒâ€šÃ‚ ½ every time, for the determination whether options are overpriced or underpriced according to such pricing models. Option pricing is so important because when an individual pays too much for an option, even if the commodity futures price or the underlying stock , on which the option is purchased moves in the right direction, the individual will most probably not get enough payment for his potential risk. On the contrary, when one buys an underpriced option, he will obtain more bene fits compared to the risk when the underlying stock, commodity or index moves in the right direction. Options can be written on a variety of things: Interest rates, stock indexes, stock price. Financial options can be useful in helping banks and corporations to be more effective and efficient in meeting their risk management objectives. When financial derivatives are 1 used without a plan or improperly used, they can cause serious damage by pulling the organization in a wrong direction so that it ¿Ãƒâ€šÃ‚ ½s ill prepared for the future, or by causing serious losses. Financial derivatives, make it possible for companies to initiate productive activities that might otherwise be done. There are many different forms of options, which is related to when you can exercise them. Among the different types of options, there ¿Ãƒâ€šÃ‚ ½s European option where the holder exercise only on the expiration date and is the most common type. There ¿Ãƒâ€šÃ‚ ½s American option where the holder can exercise at any time before the expiration date. The payoffof an option takesinto account what the holder will get at expiration, and not what they paid for the option. An option is known to be  ¿Ãƒâ€šÃ‚ ½in the money ¿Ãƒâ€šÃ‚ ½, if, it was able to be exercised immediately, so the payoff would be positive. An option is  ¿Ãƒâ€šÃ‚ ½at the money ¿Ãƒâ€šÃ‚ ½, if, the underlying asset is at the strike price and, an option is known to be  ¿Ãƒâ€šÃ‚ ½out of the money ¿Ãƒâ€šÃ‚ ½ if the holder wouldn ¿Ãƒâ€šÃ‚ ½t execute immediately. For both call and put options, the greater the volatility of the underlying stock, the more likely is the option to move into t he money. Financial options are important tools that help organizations to reach their specific risk management objectives. It is important that a user understands the type of the options ¿Ãƒâ€šÃ‚ ½s intended function and all the safety precautions to be taken before the option is used. Bachelier (1991) proposed a formula, in particular, for the price of an option, based on the idea that the fluctuations follow a Brownian process. Bachelier (1991) is the among the first person who made a contribution which was of utmost importance to the issue of valuing stock options was the Bachelier whose approach was based on one important assumption  ¿Ãƒâ€šÃ‚ ½the expected profit of investments in options is zero ¿Ãƒâ€šÃ‚ ½ Hjortsberg(2007). Option pricing was then workedby BlackScholes (1973), who introduced their sem 2 inal work on the theory of option pricing in 1973. Over the last decades, due to the work of BlackScholes(1973),opti on valuationproblemhasgainedalotof attention. SinceBlack and Scholes published their seminar, in 1973 on option pricing, there has been huge investigations, theoretically and empirically on option pricing. Over the past few decades, there has been the development of options to provide the basis for corporate hedging and for liability/asset management of financial institutions and corporate hedging. The theory of option pricing has a long history, only until Black and Scholes, in the year 1973, presented their first equilibrium option pricing model. Besides, Robert Merton extended the Black- Scholes model, in the same year, in several important ways. The Black-Scholes formula, since its invention, has been extensively used by traders for the purpose of determining the price of an option. Despite this, the famous formula was questioned after the crash in 1987. Following the Black and Scholes option pricing model in the year 1973, a numbers of other app roaches were developed, such as, the Jump Diffusion model which was suggestedby Merton (1976). CoxRubinstein (1979) introduced the numerical method FiniteDifferencesandCarlo(1964)forthepricingofthe derivativedirectedbysolvingthe underlying partial differential equation. Documentation from many empirical studies have shown that geometric Brownian motion (GBM) models are not adequate, in relation to the descriptive power, and even to the mispricing that they might generate. The contributions made to the actual volume deal with many generalizations of the basic geometric Brownian motion and here emphasisisplacedonthefactthatinterest ratesandreturnsof various assetpricesmay exhibita jumping behaviour. So possible superpositions of jump and diffusion processes have been studied, which is known as the jump-diffusion processes, the purpose of which is not the studyofaLevy drivingprocess,butto rather emphasizeonthe specific aspectsofthesubclass of Jump-di ffusion. Jump-diffusion models also have some intuitive appeal because they let interest rates and prices change continuously most of the time, however, they take 3 into account the fact that now and then, larger jumps may occur and these large jumps cannotbe suitably modeledby the pure diffusion-typeprocesses. Most of the financial theory is based upon random walk of asset prices, yields and returns. Still, over the last twenty years, the theory has been extended to have a closer view of the departures from a random walk. This theory was being used for various purposes which includes capital adequacy, investment strategy, and the pricing and hedging of options. However, the random walk is a special case for a wider range of models, which includes mean reversion. It is recommended that some classes of mean reversion models reducethe capital seeminglyrequiredformany classesof insurance business,relativelyto theresults obtainedfrom random walks. Thi s stimulatedthe actuarial interestinthe mean reversion class of models. My project will focus on Black Scholes, mean reversion and the mathematical implementation. In the next section we havePreliminaries anda deeper view on the Black Scholes (1973) model accompanied with its drawbacks. In the next chapter, mean reversion is discussed 4 Chapter 2 Preliminaries 2.1 Introduction to Functional Programming under the Mathematica. environment Functional programming is a programming model that treats calculations as the evaluation of mathematical functions and avoids mutable and state data. It places emphasis on the applications of functions, in cont rast with the style of imperative programming that places emphasis on state changes. The functional programming languages places emphasison pattern-matchingandrules.Itis particularlyusefulfor mathematicalwork,inwhich the notion of  ¿Ãƒâ€šÃ‚ ½function ¿Ãƒâ€šÃ‚ ½ is wel l a well established concept. Mathematicar,isapowerful computational softwareprogram thatis usedin engineering, scienti fic, and mathematical fields and other areas of technical computing. The person who firsthadtheideatocreatesucha softwareis StephenWolframandthis softwareis developed by the  ¿Ãƒâ€šÃ‚ ½Wolfram Research of Champaign, llinois ¿Ãƒâ€šÃ‚ ½. Nearly any workflow includes computing results, and this is what Mathematicar, does. That is, from building a hedge-fund trading website, to a devel oping embedded image-recognition algorithms or the teaching of calculus. Mathematicar,is well-known as the world ¿Ãƒâ€šÃ‚ ½s latest application for 5 2.1 Introduction to Functional Programming under the Mathematica. environment computations, but it ¿Ãƒâ€šÃ‚ ½s much more. It is the sole development platform that fully integrates computation into complete workflows, that moves you seamlessly from starting ideas, all the way to the arranged individual or enterprise solutions. Mathematica. is a computer language of high level, which has a large number of operations and in-built functions and this means that for various purposes, programming in the traditional senseisnot necessary. Alternatively,an individualjustusea seriesof built-in functions and this has various advantages of which the main advantage is the speed of programming, that is not having to do the basics over again, saves a lot of time. The built-in functions are very flexible and optimised and are also well joined with each other and their properties: numerics, derivatives, series and integrals, are included into the kernel. The proposed model, which will be implemented in Mathematica c . enables the prices of options to be consistent simultaneously consistent with the volatility smile and the observed future prices of the option market. The proposed model is suf ficiently flexible so that financial analyst can perform scheme analysis with it and is as follows: Figure 2.1: The proposed model where #(t)isthe deterministic functionthatrepresentsthe equilibriummeanlevelofthe volatility against time, the constant . isthe meanreversionspeedofthe asset,the function a(t) is the equilibrium mean level of the volatility against time, the constant b is the mean reversion speed of the volatility. #, a constant, is the volatility coefficient of the volatility 6 2.1 Introduction to Functional Programming under the Mathematica. environment model, the process vt is the volatility of the underlying asset, that follows the Heston (1993) model, and Wt are correlatedWienerprocesses witha correlation coefficient #. The proposedmo delreducestothe Heston(1993)modelifthe meanreversionspeed, . is equal to zero. With the dynamic of the underlying asset, it is possible to acquire the characteristic function for the log-asset value Xt, which is implemented in Mathematica c . as follows: Figure 2.2: The Characteristic Function implemented in Mathematica c . The equations of B1, C1, D1 and other equations which leads to the characteristic function will be shown in appendix. After founding the characteristic function, The European options can be valued using Fourier inversion. Carr and Madan argues the Fast FourierTransform (FFT) to compute the vanilla call and put options that arebased on the characteristic function of the log-asset value. The payoffof the plain vanilla call option is max(ST K, 0) where T is the option ¿Ãƒâ€šÃ‚ ½s maturity and K is the strike price. Let . denote the log of the strike price K and let qT (s) be the risk-neutral density of the log-asset price st = ln ST , and CT (#) be the desired valueofaT-maturity call option witha strike price exp(#). As per Carr (1999), CT (#), the modified call price is as follows: cT (#) = exp(##)CT (#) (2.1) for some constants # 0 7 Figure 2.3: FourierTransform .c and then call prices can numerically be obtained by using the inverse transform, implemented in Mathematica. itself: Figure 2.4: InverseTransform c . The values assigned to the various constants and variables to carry out the test in Mathematicar, is as follows: . = 0.25, n = 128, . = 1.5, a = 4.0339, . = 10, a = 0.5328, b = 3.33, . = 0.04, . = 0.9, r = 0.05, S0 = 1.3 v0 = 0.18, T = 1, and K, whichisthe strike price, variesandtheresultsofthe variationis showninthe table below: 8 2.1 Introduction to Functional Programming under the Mathematica. environment Table 2.1: Call option prices: FFT vs. Monte Carlo Strike Price FFT Monte Carlo %Difference 0.3747 1.06172 1.0658 -0.00408 0.4559 0.984477 0.9889 -0.00443 0.5549 0.890306 0.8958 -0.005494 0.6752 0.775873 0.7798 -0.003927 0.8217 0.636518 0.6413 -0.004782 1 0.466914 0.4719 -0.004986 1.2170 0.260889 0.2660 -0.005111 1.4810 0.0553335 0.0583 -0.0029665 1.8023 0.000997007 0.0011 -0.000102993 NIntegrate The function NIntegrate in Mathematica. is a general numerical integrator which can hand le a large range of one-dimensional and multidimensional integrals. NIntegrate[f, x, a, b] gives a numerical approximation to the integral . b f(x)dx (2.3) a Module Mathematica normally assumes that all the variables areglobal which means that every time an individual uses a name like x, Mathematica. will assume that the individual is referring to the same object. When one write programs, he might not want all the variables to be global. For example, he may want to use the name x as reference to two different variables in two different programs. So, in this case, he need the x in each program to be treated as a local variable. He can set up the local variables in Mathematica. using modules and within each module, the individual can give a list of the variables that are 9 2.2 Stochastic Calculus to be treated as local to the module. 2.2 Stochastic Calculus Stochastic calculusisacategoryof mathematicsthatisbasedon stochasticprocesseswhich comprises of a consistent theory of integration that are defined for integrals of stochastic processes withrespectto stochasticpr ocesses. Stochastic calculusis usedto model systems that have random behaviour. The main flavours. Stochastic calculus is the language of risk management and option pricing at fundamentally every big financial firm, and is also the strength of a large body of academic research on an corporate finance, asset pricing, and investor behaviour. 2.3 Call and Put Option Acall option is a financial contract between two parties which consists of the buyer and the sellerwherethebuyeroftheoptionhastherighttobuythe underlying assetatafixed price (strike price/exercise price) or at any time before the option expires. The buyer of a call option purchases the underlying asset with the hope that the price of the underlying asset will increase in the future while the seller of the call options expects that the price of the underlying asset is not more than the strike price. The buyer of the option makes a profit if the value of the asset is greater than the strike price,that is the buyer of the option buys the asset at the exercise price. On the other hand, if the value of the underlying asset is less than the strike price, then the option is not exercised and expires worthless. The diagram below shows the cash payoff on a call option at expiration. 10 2.3CallandPutOption Figure 2.5: Payoff on call option 11 2.3 Call and Put Option Aput option is a financial contract between two parties that gives the buyer of the option the right to sell an underlying asset at a fixed price or at any time before the option expires. Here, the buyer of the put option has the right, but not the obligation, to sell the underlying asset at the strike price and if he or she decides to sell at any time before the expiration date, the seller of the put option (put writer) is obliged to buy at the price stated by the buyer. The buyer of the put option makes an investment by purchasing the underlying asset with the hope that the price of the underly ing will decrease in the future when he will sell the underlying asset while the individual who is going to buy the put option expects that the price of the asset is less than the strike price. On the other hand, if the value of the underlying asset is more than the strike price, then the option is not exercised and expires worthless. The diagram below shows the cash payoff on a put option at expiration. Figure 2.6: Payoff on put option 12 2.3 Call and Put Option It has been seen that banks and other financial intermediaries reacted to new situation by developing many financial risk management products which were designed to control risk in a better way. The first product was the simple foreign-exchange forwards caused one party to sell, and the other party to buy, at an agreed date in the future, at a fixed amount of currency. Through entering in a foreign exchange forwardcontract, the customers could balance the risk that large changes in foreig n exchange rates would ruin the viability of the economy projects and hence options were first intended to be used to hedge certain risks effectively, and in fact, that was the key that opened up their explosive development. Options also help to ameliorate the market ef ficiency because risks can be classified and be sold to those parties who are willing to accept the risks at the least cost. The use of options breaks risks into pieces so that they be managed independently. Some corporations can keep risks that they can manage most comfortably and they have the choice to transfer those risks that they do not want to manage, to other companies that are willingto accept them.Fromamarket oriented perspective, financial optionsofferfree trading of financial risks. Similarly to the prices of most things, options ¿Ãƒâ€šÃ‚ ½s prices can be sensitive to market demand, but there are wise techniques one can use to determine the fair option value. There are many factors than can a ffect opt ion pricing. Before proceeding in the world of trading option, the investors should have a good knowledge of the different factors that are used to determine the value of an option. The factors are cash dividends, interest rates, volatility, time value or time to expiration, the intrinsic value and the current stock price. There are many options pricing models that use the mentioned parameters to determine the fair valueofanoption.Ofthe severaloptionpricingmodels,the Black-ScholesmodelBlack Scholes (1973) is the most widely used. In many different ways, options are just similar to any other investment in which one needs to understand what determines the fair price of an option in order to be able to use them. 13 2.4 Black-Scholes European optionsare securitiesthatgivethe ownertherighttobuyan indexorastock at a certain date at a certain price. It is an option that may only be exercised at the end of its life, at its maturity. An American is an option th at may be exercised at any time during its life. An American option allows the holder of the option to exercise the option at any time before and including the maturity date, which t his increases the value of the option to the holder of the option, as compared to a European option, where the option can only be exercised at maturity. ABermudaoptionisanoptionthatgivesthebuyertherighttoexerciseatasetnumber of times. This is an intermediate between a European option and an American option. 2.4 Black-Scholes TheBlackScholes(1973)formulaiswidelyusedto calculatepricesofoptions. However, in the year 1973, a turning point occurred in the evolution of trading of options, when the professors BlackScholes (1973) wrote the paper  ¿Ãƒâ€šÃ‚ ½Pricingof Options and Corporate Liabilities  ¿Ãƒâ€šÃ‚ ½. Based on the assumption that a risk-free rate of interest existed, this was viewed as a fundamental effort at expressing option pricing and corporate bonds. Thou it is us ed mostly in institutional portfolio management departments, it is still used today for predicting what options should be worth. Since its introduction in 1973, it has been been tested against the option miss-pricing, by many. The precision of the generated price of Black-Scholes depends very much on the accuracy of the parameter inputs, such as, time, interest rate, strike and exercise price are known precisely, and hence they are relatively accurate and easily determined. The  ¿Ãƒâ€šÃ‚ ½Black-Scholes ¿Ãƒâ€šÃ‚ ½ model which is also known as the  ¿Ãƒâ€šÃ‚ ½Black-Scholes-Merton ¿Ãƒâ€šÃ‚ ½ model 14 2.4 Black-Scholes is one of the most important concepts in modern financial theory which contains certain derivative investmentinstruments.Itwas developedbyBlackand Scholes.Itisoneofthe best ways of determining fair prices of options and it is still widely used today. The Black-Scholes formula is as follows: C = SN(d1) Ke..rtN(d2) (2.4) where: C=Theoretical call premium, S=Current Stock price, t=time until option expiration, K=option striking price, r=risk-free interest rate, N=Cumulative standardnormal distribution, e=exponential term(2.7183), s=standarddeviationof stockreturns, ln=natural log arithm. s 2 ln S K + r + 2 tv t (2.5) v d2 = d1 s t (2.6) Merton (1976) is an American economist who is known for his work on risk management and finance theory and especially for his contribution in assessing stock option ¿Ãƒâ€šÃ‚ ½s value and other derivatives. Merton ¿Ãƒâ€šÃ‚ ½s work on the valuation of option is perhaps the most influential even thou his research covers many areas of economics and finance theory. When Black and Scholes published their formula, prior to 1973, which determines the value of stock options, was very difficult and risky because of the nature of options, which essentially areagreementsthatgivethe investorstherighttoeithersellorbuyanassetatafixedtime in the future. The challenge of an option is to prognosticate its value at a distant time. Before the introduction of the Black-Scholes formula, those investors investing in options determined a risk premium in order to hedge against major financial losses. It was shown 15 2.4 Black-Scholes by the Black-Scholes formula that the risk premiums are not needful for investment in stock options because those premiums are already calculated in the prices of stocks. In order to generalize the Black-Scholes formula, Robert C. Merton used his knowledge in mathematics, by modifying certain assumptions and restrictions which was set by Black and Scholes, such as the unlikely assumption that no dividends will be not be paid by the stock. By modifying this formula, Merton permitted it to be applied to other financial issues, such as student loans and mortgages. Scholes, Black and Merton ¿Ãƒâ€šÃ‚ ½s assumptions are as follows: 1. There is no dividend during the life of the derivative 2. Options can be exercised only upon expiration 3. There are no arbitrage opportunities 4. The trading of security is continuous in time 5. There are no taxes or transaction costs, all securities are perfectly divisible 6. Stock returns are normally distributed and hence volatility is constant over time 7. Interest rates remain constant 2.4.1 Black-Scholes, Partial Differential Equation (PDE) A partial differential equation (PDE) is a differential equation that has unknown multi-variable functions and their partial derivatives. Partial differential equations are used for the formulation of problems that involve functions of various variables, and are either resolved by hand or used to create a relevant computer model. As stated above,The equationof BlackScholes (1973)isa partialdifferential equation, that describes the price of an option over time. An idea of upmost importance behind the equation is that an individual can perfectly hedge an option by selling and buying an 16 2.4 Black-Scholes underlying asset in the right way and hence eliminating risk. The hedging in turn means thatthereisonlyone correctpriceforanoption,asreturnedbytheBlackScholes(1973) formula which is as follows: @V 1 @V #2S2 @2V ++ rS rV =0 (2.7) @t 2 @S2 @S HereIpresent an analytical solution for the BlackScholes (1973) PDE, @V 1 @V #2S2 @2V rf =+ + rS ;V = V (t, S) (2.8) @t 2 @S2 @S over the domain 0 S 8, 0 = t = T , with a terminal condition V (T;S)= (S) , by the reduction of this parabolic PDE to the heat equation of physics. The substitution u = exp[-rt]f is made, which is stimulated by the fact that it is the portfolio value which discounted by the interest rate r that is a martingale. The product rule is used on V = exp rtu, and the the PDE that the function u should satisfy is derived: @u @u 1 u #2S2 @2 0= + rx + (2.9) @t @S 2 @S2 Now, we substitute y with log S, and s with T t. These changes of the variables can be stimulated by observing that:  ¿Ãƒâ€šÃ‚ ½ The underlyingprocess whichis describedby the variableSisa GBM (Geometric Brownian Motion), in order for log S describes a Brownian motion, with a possible drift. Then some sortof diffusion equation shouldbe satisfiedby log S.  ¿Ãƒâ€šÃ‚ ½ From the terminal state of the system, the evolution of the system is backwards. Actually, the boundary condition is given as the terminal state, and the coefficient of @u is positive in equation 2.6 and in order to get the heat equation, we have to make @t the useofa substitutiontoreverse time. Since 17 2.4 Black-Scholes @u @u @u @u dy 1 @u = , == , (2.10) @s @t @S @y dx [email  protected]/* */ and @2us 1 @u 1 @u 1 @2u =( )= + (2.11) @S2 @S [email  protected]/* */ S2 @y S2 @y2 we then substitute in equation 2.6, which results in: @u 1 @u 1 #2 @2u 0= +(r #2)+ (2.12) @s 2 @y 2 @y2 Withrespect toy, the first partial derivative does not cancel because we didn ¿Ãƒâ€šÃ‚ ½t take into account the drift of the Brownian motion. In order to cancel the drift, the use of substitutions is made: z = y +(r 1 #2)#;p = s. (2.13) 2 Under the new coordinate system (z, #), we have the relations amongst vector fields: s @z = s @y , s @T = -(r 1 2 #2) s @y + s @s , (2.14) which leads to the following transformation of the equation 2.9: 0 = @u @r (r 1 2 #2) @u @z + 1 2 #2 @2u @z2; (2.15) or @u @r = 1 2 #2 @2u @z2 , u = u(#, z) (2.16) which is one form of the diffusion equation. The domain is on -8 z 8 and 18 2.4 Black-Scholes 0 = p = T ;and the initial condition is: ..rT (e u(0;z)= e z) := u0(z) (2.17) The original function f can berecoveredby rt 1 f(t, x)= eu(T t, log x +(r #2)#) (2.18) 2 The fundamental solution of the PDE in equation 2.13 is: 1 Z G. (Z)= v exp (2.19) 2#2# 2##2p and the solution u with the initial condition u0 is given by the convolution: ..rT . e(Z #)2 u(#, Z)= u0 * G. (Z)= v (e #) exp(-)d#. (2.20) 2#2# 2##2p . in terms of the original function f which is as follows: e..r. . (log x +(r 1 #2)p #2 2 f(t, S)= v (e . ) exp -d#. (2.21) 2#2# 2##2p . 19 2.4 Black-Scholes Numerical Experiment on Option Pricing The Black-Scholes Option Price Calculator (BetaVersion) is used to generate the Call Price and the Put Price that will be figure in the table below. The volatility rate is taken as 50 0.5 for , the interest is taken as 0.5 for 50 and theTimeTo Expirationis taken as1year 100 100 Table 2.2: Prices of Call and Put Option under varying Stock and Price Stock Price Strike Price Call Price Put Price 50 50 21.264 1.590 90 70 48.423 0.880 100 90 47.397 1.985 60 110 9.423 16.142 80 130 16.261 15.110 170 150 82.098 3.077 90 170 13.191 26.301 100 190 14.455 29.696 20 2.4 Black-Scholes 2.4.2 Shortcomings of the Black Scholes (1973) model In the year 1973, a turning point occurred in the evolution of trading of options, when the professors BlackScholes (1973) wrote the paper  ¿Ãƒâ€šÃ‚ ½Pricingof Options and Corporate Liabilities  ¿Ãƒâ€šÃ‚ ½. Based on the assumption that a risk-free rate of interest existed, this was viewed as a fundamental effort at expressing option pricing and corporate bonds. Thou it is used mostly in institutional portfolio management departments, it is still used today for predicting what options should be worth. Since its introduction in 1973, it has been been tested against the option miss-pricing, by many. The precision of the generated price of Black-Scholes depends very much on the accuracy of the parameter inputs, such as, time, interest rate, strike and exercise price are known precisely, and hence they are relatively accurate and easily determine. But, most individual traders have since long recognized that there are flaws in the mode l, in the following stated ways: 1. It was first published in 1973, when the trading of the public options was in its early stages. Puts were not publicly traded and calls were only available on a handful of listed companies. Furthermore, the population of the trading of options was extremely limited, which means that the assumptions used for the Black-Scholes model cannot be applied in the more complex modern options industry. 2. Dividends are uncalculated in the equation, which means that no dividends are applicable. Option traders know that dividends play a big role in returns and cannot be ignored. In the modern industry, anyone trading in options has to consider the impact of the dividend yield on the overall return. In comparing values of two or more underlying assets, dividend yield is often the determining factor in making the decision of which one 21 2.4 Black-Scholes has a better value. 3. Some of the assumptions are questionable under today ¿Ãƒâ€šÃ‚ ½s market and economic conditions. For example, the model assumes that the income and valuation have to be compared to an assumed rate of risk free intere st. It is questionable if such a rate exists today. 4. The assumption of European style expiration (that is positions can be only exercised onthelast tradingday).Withafew exceptionsof some index options, mostofthe publicly traded options can be exercised at any time before expiration and thus this changes the calculation. So, the Black-Scholes model makes one assumption that is totally flawed. The assumption follows that the implied volatility on the analysis date willremain unchanged until expiration. Every trader knows that this is inaccurate. 5. Back then, the internet was not yet created. Without the ability to fragmentize numbers automatically and easily, the Black-Scholes model depended on calculations being made manually. With hundreds of more opt ions to trade and with more detailed, faster, and more w idely used formulas for the tracking of values, the entire options market is a different thing today than it was in 1973. Even the open levels of contracts have changed, increasing in billions since 1973 to a such a volume today that cannot be imagined in the past. This also directly affects valuation. The use of vega, gamma and delta are more reliable measurements of option pricing and implied volatility than the more unclear Black-Scholes model with its impractical vari ables. Black-Scholes contain numerous problems. Since its first publishment, more theories have been added to expand the Black-Scholes model in order to make it more applicable to the practical market conditions. Even so, its unlikely that an accurate market model will be produced in the near future. The Black-Scholes formula contains too many variables, and 22 2.5 The Jump Diffusion Model with the use of more variables, this results in a less reliable formula. These include the assumptio n of European expiration, lack of dividend, risk-free interest rate, and unchanging implied volatility. When one variable is used in the formula, it is rather troubling to a certain degree and when two or t hree variables are used, this increases the inaccuracy. The solution to the option pricing problem, should be restricted to studies of implied volatility and hence the volatility of the underlying , market conditions, and the closeness between strike of the option and current market value of stock, are the true factors that values an option. 2.5 The Jump Diffusion Model Afirst approach in further developing the basic Black and Scholes model adding the inclusion of jumps happens to be that of Merton (1976). The introduction of jumps in the Black and Scholes model has the implicatio n that the derivative prices are no longerresolvedby the principle of the absence of arbitrage only, this pricing problem was solved my Merton, following the assumption that t he jump risk was not systematic. But this was in later looked through a critical point of view, that such an assumption is equal to the the presenceofamarket portfolio,thatdoesnotpresentajumping behaviourandthat containsthe underlying asset.In empirical studies,itwas do cumentedthata combinationof stochastic volatility and jumps leads to better fits and enables to avoid the implied volatility skews. The models of stochastic volatility are teated elsewhere and hence the limitation is set to stochastic volatility on affiliation with the jump-diffusion modeling. This is also because empirical documentation gives the evidence for a jump-type action in the volatility and in the correlation between jumps in jumps in prices and volatility. Dueto some limitationsof the Black Scholes (1973) modelin the modelingof the 23 2.5 The Jump Diffusion Model distribution of logarithmic stock returns in considerable shorter periods, Merton (1976) improved the BlackScholes (1973) modelby adding the possibilityof jumps, which occurs correspondingtoaPoissonprocess, whichis independentfromtheBrownian motion. Dynamics of the stock prices in the Merton jump-diffusion model can be described by this equation: #t+#Bt+Xt St = S0 exp(2.22) where Xt isa compound Poissonprocess: Nt . Xt = Yi (2.23) i=1 Yi,i = 1,2,, are independent and identically distributed normal random variables. Nt is a Poisson process Processes Bt, Nt and variables Yt,i = 1,2, are independent. The assumption of the standardBlackScholes (1973) of log-normal stock diffusion with a constant volatility is, flawed, as we have already seen in the paper. In general, it is necessary to use different volatilities for different maturities (T) and option strikes (k), to equate the formula of BlackScholes (1973) with quoted prices of European calls and puts options. This event is most of the time referred to as the volatility smile or skew which depends on the figure of the mapping of implied volatility as of function ofTand K.Volatility skewispresentin all major stock index markets today. Usually, the steepness of the skew diminishes with the increasing option maturities. The presence of the skew is regularly associated with the fear of large downwardmarket movements. The extensions o f the Black Scholes (1973) model that capture the presence of the volatility smiles, can be grouped into three approaches. Firstly, it can be grouped in the stochastic volatility approach (Hull White 1987), in which, the stock ¿Ãƒâ€šÃ‚ ½s volatility is as 24 2.5 The Jump Diffusion Model sumedtobea meanrevertingdiffusionprocess, whichis usually correlatedwiththe stock process itself. A variety of volatility smiles and skews can be generated in this model, depending on the parameters and the correlation of the volatility process. It has been shown by empirical evidence from time-series analysis which confirms the presence of stochastic volatility in stock prices. However, often, unrealistically high negative correlation between volatilityand stock indexisrequired,inorderto cause implied Black-Scholes volatility skews in a stochastic volati lity that are uniform with those that are observed in traded options. Furthermore, while looking from a computationa l perspective, the models of stochastic volatility are not easy to handle as they are multi-factor models, which means that, an individual would nor mally need multi-dimensional lattices to evaluate, for example, American options. It has been noticed that stochastic volatility do not allow for absolute option hedging by dynamic positions in the money market account and in the stock. Firstly generated by Merton (1976), a new approach, generates volatility smiles and skews by adding discontinuous jumps to the Black-Scholes diffusion dynamics. By appropriately choosing the parameters of the jump process, a multitude of volatility skews and smiles can be generated by this s o-called jump diffusion model. Similar to stochastic volatility models, the jump-diffusion models are very challenging to handle numerically and this results in bonds and stocks forming an incomplete market. 2.5.1 The Black-Scholes vs the Jump-Diffusion model During the year 2011,in i ts second half,it was seen that financial marketsreacted strongly to the signs of the intensifying Eurozone crisis. During this disruptive time, the stock prices acted in a very volatile way, as a reaction to the inflow of all the information on the problems of debts of the various Eurozone members and even to the emergency solutions applied. In order to capture such a quick, sometimes discontinuous movements of 25 2.6 Finite Difference the stock prices, The jump-diffusion model of stock prices seems to be appropriate. The jump-diffusion model, on contrary to the classical Black-Scholes model, doesn ¿Ãƒâ€šÃ‚ ½t solve the normalityofthe logarithmicreturnsanditisusuallyfitsthe distributionofthe logarithmic returns in shorter periods, in a better way, compared to the Black-Scholes model. 2.6 Finite Difference The methods of finite difference (also known as finite element methods), for option pricing, are numerical methods that are used in mathematical finance to valuate options. Schwartz (1977) first applied Finite difference methods to option pricing. It is known that finite differences methods may solve problems of derivative pricing, and in general, as complex as those problems solved by the tree app roaches. Finite difference methods are usually applied only when other approaches proves to be inappropriate but at the same time, in terms of the number of underlying variables, this approach is limited, and as for problems with multiple dimensions, the Monte Carlo methods areusually preferred, for option pricing. Finite difference methods are used for the pricing of options by the approximation of the (continuous time) differential equations which describes the evolution of an option price through time by a set of (discrete-time) differential equations. To calculate the price of the option, the discrete differential equation can then be solved iteratively. The method of finite difference value a derivative by solving a differential equation that is satisfied by the derivative. The differential equation is first converter into a set of difference equations, and then the difference equations are solved iteratively (Hull White 1987) It is started from the Black-Scholes partial differential equation: @[email  protected]/* */ +1=2#2 (2.24) 26 2.6 Finite Difference The methodsof finitedifferencecreatea mathematicalrelationshipthat connects every point on the solution domain, like a chain. From the first links, that are boundary conditions, it is discovered what every other point in the domain has to be. The most popular methods of finite difference used in computational finance are: Implicit Euler, Explicit Euler, and the Crank-Nicolson method. The use of these three methods has both advantages and disadvantages and easiest of which to implement is the Explicit Euler method. Crank-Nicolson and Implicit Euler are implicit methods, that generallyrequirea linear equation s system to be solved at each step of time, that can be intensive, computationally, on a fine network.Adisadvantage for the useof Explicit Euleris the unstability for certain choices of domain discretisation. The equation of BlackScholes (1973) has been used as the standardpricing formula fordifferenttypesofoptions.The assumptionsusedintheBlackScholes(1973) formula do not eternally hold and the original equation has been generalised to receive different types of options, which means that an exact solution for the BlackScholes (1973) equation cannot be always found and one must consequently resort to approximate methods, which are: 1. Monte Carlo and quasi Monte Carlo methods (Boyle 1977) 2. Reductionofthe equationtoasimplerform(Wilmott1994) 3. Reductionto other formsby meansof Fourier transforms(Carr1999) 4. Binomial and trinomial methods by Cox in 1985 Since each of mentioned methods contain disadvantages and advantages, one prefers to approximate the PDE that mod els the option prices by means of finite difference methods. The advantages of using finite difference methods are: 1. They have a sound mathematical and theoretical basis 2. They are easily programmed on a digital computer 27 2.7 Geometric Brownian Motion (GBM) 3. The different methods have a long history which goes back till the eighteenth century 4. The various finite difference methods are flexible and may be applied to many types of problems of pricing. The disadvantages of using the finite difference methods are: 1. Under certain circumstances, higher-order partial differential equations can degenerate into lower-order partial differential equations (with exponentially decaying volatilities or large factorsof driftin the B lackScholes (1973) equation) 2. It has been seen that a number of finite difference schemes are too complicated or elaborate. The numerical schemes that, in nature, are non-linear, while the equivalent partial dif ferential equations are linear and they deserve special attention since they can be  ¿Ãƒâ€šÃ‚ ½overkill ¿Ãƒâ€šÃ‚ ½ in one ¿Ãƒâ€šÃ‚ ½s opinion. 3. New techniques are needed for proving the stability of schemes that approximate the partial differential equations that have nonlinear, non-constant and discontinuous coefficients. 2.7 Geometric Brownian Motion (GBM) AGeometric Brownian Motion which is also known as the exponential brownian motion is a time-continuous stochastic process where the logarithm of the quantity varying randomly follows a Brownian motion which is also called the  ¿Ãƒâ€šÃ‚ ½Wiener process ¿Ãƒâ€šÃ‚ ½ with drift. It is an important example of the stochastic processes satisfying the stochastic differential equation (SDE) and in particular, its use in mathematical finance is to model stock prices in the BlackScholes (1973) model. Many current engineering economic analyses have depended on an explicit or an im 28 2.7 Geometric Brownian Motion (GBM) plicit assumption that some quantity that changes with time with uncertainties that follow a GBM process. The GBM process, which also sometimes called a lognormal growth pro-cess,hasbeen acceptedbymanyasavalid modelforthegrowthinthepriceof stocks over time. Under this model, the formulaof BlackScholes (1973) for the pricingof European call and put options, even as their variations for some of the morecomplicated derivatives, provide simple analytical evaluation of asymmetric risks. Many recent examples of the GBM models have appeared in the analysis of real options, where the value of some underlying asset is assumed to be evolving like a stock price. The GBM assumption is stated exp licitly, in some cases. In other cases, when optionsare evaluatedbythe BlackScholes (1973) formula,theGBMis implicitly evaluated St,astochasticprocess,issaidto followaGBMifit satisfiesthe followingSDE (Stochastic Differential Equati on): dSt = #Stdt + #StdWt (2.25) where Wt isa ¿Ãƒâ€šÃ‚ ½Wienerprocess ¿Ãƒâ€šÃ‚ ½orBrownian motionand #, the percentage drift and #, the percentage volatility, are both constants. Hypergeometric Function A generalized hypergeometric function pFq(a1, :::, ap; b1, :::, bq; x) is a function which may be defined in terms of a hypergeometric series, which means, a series for which the ratio of successive terms may be written as: Ck+1=Ck = P (k)=Q(k) = ((k+a1)(k+a2):::(k+ap)=(k+b1)(k+b2):::(k+bq)(k+1))x (2.26) In-the-money, At-the-money and Out-of-the-money 29 2.7 Geometric Brownian Motion (GBM) If the strike price is more than the market price of the underlying asset, a put option is in-the-money. Acall option is  ¿Ãƒâ€šÃ‚ ½in-the-money ¿Ãƒâ€šÃ‚ ½ if the market price of the underlying asset is greater than the strike price. An option is at-the-money when the strike price is equal to the price of the underlying security. If the market price of the underlying asset is less than the strike price, a call option is out-of-the-money. Aput option is out-of-the-money if the market price of the underlying security is greater than the strike price. Derivative A derivative is a security and its price depends on or derived from one or more underlying assets. The derivative is itself a mere contract between two or more parties. The valueof the derivativeis obtainedby fluctuatio nsin the underlying asset. Some examples of underlying assets are: bonds, stocks, commodities, market indices, interest rates and currencies. Most derivatives are characterized by high leverage. Security Asecurityisaninstrumentthatrepresents ownership (stocks),therightsto ownership (derivatives) ora debt agreement (bonds) Underlying asset An underlying asset is a term from derivatives trading. For example, in Microsoft stock option, Microsoft stock is the underlying asset. In case of gold options, gold is the underlying asset. Price movements of the underlying assets determine the price movement of options. Stochastic Volatility 30 2.7 Geometric Brownian Motion (GBM) Models of stochastic volatility areused in mathematical finance field for the evaluation of derivative securities, such as options. The name stochastic volatility is derives from the model ¿Ãƒâ€šÃ‚ ½s treatment of the volatility of the underlying security as a random proc ess, which is governedby state variablessuchasthe tendencyof volatilitytorevertto somelong-run mean value, the price level ofthe underlying security, and the variance of the process of the volatility itself. Standardized futures contract Afutures contract is known to be a standardized contract between two parties to exchangea speci fied assetof standardized quantityand standardized qualityforapricethat is agreed today. Log-Return The advantage of looking at log return is that one can see relative changes in the variable and comparedirectly with other variables whose values may have very different base values Strike Price In options, the strike price, which is also known as the exercise price, is the fixed price at which the owner of the option can purchase, in case of a call option, or sell, in case of a put option, the underlying security or commodity. Characteristic Function The characteristic functionofareal-valued random variable definesits probabilitydensity function.Ifa random variableintroduceaprobability density function,thenthechar acteristic functionis the FourierTransformof theprobability density function. Wiener Process 31 2.7 Geometric Brownian Motion (GBM) Inthe worldof mathematics,theWienerprocessifatime-continuous stochasticprocess which is named in honor of NorbertWiener. It is often called the Brownian motion. Is is one of the best known stochastic process and occurs often in applied and pure mathematics, quantitative finance, economics and physics. Levy process Intheprobabilitytheory,theLevyprocess, whichis named afterthePaulLevy,aFrench mathematician, is a stochastic process with independent and stationary increments. A Levy process represents the motion of a point where its successive displacements are independent and random, and statistically identical over different time intervals of the same length. Systematic Risk In economics and finance, a system atic risk is the vulnerability to events which affects the aggregate outcomes such as total economy-wide resource holdings, market returns or aggregate income. Interest rates , wars and recession all represent sources of systematic risk because they affect the entire market and cannot be diversified to avoid the risk. The Poisson Process In the theory of probability, a Poisson process is a stochastic process that counts the numberof eventsandthe timesthat these events occurinagiven intervalof time.Theduration between each pair of consecutive events is known to have an exponential distribu tion with . as parameter and each of these inter-arrival times is assumed to be independent of the other inter-arrival times. Volatility Smile or Skew In the world of finance, a volatility smile is the pattern in which out-of-and in-themoney options areobserved to have bigger implied volatilities than at-the-money options. 32 2.7 Geometric Brownian Motion (GBM) Agraph of the strike price vs. the implied volatility for a given expiry will form an upturned curve, just like the shape of a smile. Figure 2.7:Volatility Smile 20. The Deterministic Model In mathematics,a deterministic modelisasystemin whichno randomnessis involved in the process of the development of the future states of the system and will thus always produce the same output from a given initial state or starting condition. Map Function (Mathematica) Mathematica.chas many powerful functions for working with lists. It is frequently desirable to map a function into each individual element in a list. While listable functions do this by default, one can use Map to do this with functions that cannot be listed. Discretisation In mathematics, discretisation infers the process of transferring continuous equations and models into discrete counterparts. 33 2.7 Geometric Brownian Motion (GBM) Explicit Euler, Implicit Euler, Crank-Nicolson method Explicit Euler involves the calculation of the state of a system at a later point in time, from the state of the system at the actual time. Implicit euler, derivesa solution,by solving an equation that involves both the current state of the system and the later one. The Crank-Nicolson methodisafinitedifference methodthatisusedforthe numerical solving of partial differential equations and it s second order in time, and is numerically stable. The Log-normal distribution Alog-normal distributionisacontinuousprobability distributionofarandom variable whe reits logarithm is normally distributed. For example, if X isarandom variable following a normal distribution, then Y = exp(X) contains a log-normal distribution. Similarly, if now Y is a random variable following a normal distribution, then X = log(Y ) has a normal distribution. Therefore, the log-normal distribution, is a distribution of random variable which takes only real values that are positive. 34 Chapter 3 Mean Reversion Model In the history of commodity pricing, the most common way is to model the logarithmic price through a mean-reverting process (Buhler 2009). Similar to the model of BlackScholes-Merton, the process of mean-reversion is based on the exponential treatment of the stochastic spot price (Merton 1976) (BlackScholes 1973). If for example, these models are used for electricity, they may catch the mean-reversion of electricity prices, but they will not be able to account for non-negligible and huge observed spikes in the market. It is necessaryto extendtheBlackScholes(1973)modelbyajump component,tobeableto catch the behavior of spikes of the electricity spot price dynamics. This model was applied to the English andWelsh electricity marketby CarteaFigueroa (2005) and findsit gives aprope r adjustments to the abnormalities of the electricity markets. RoncoroniGeman (2006) has discussed and tried to try to fix the deficiency of this model, that is to say that it uses only one unrealistically high mean-reversion rate, both for the jump process and for the diffusion. Nevertheless, a single rate of mean-reversion for these two visible features only has limited use because the price of elasticity does not exhibit classical jumps but instead exhibit spikes and these spikes have the tendency to revert fast, which leads ti a high rate of mean-reversion following a spike . The mean-reversion rate is in fact much lower, during 35 normal times. As a consequence, the use of a single mean-reversion factor causes a too slowremovalof intenseprice movements (spikes)andalsoatoorapidreturntoa seasonal trend of periods without intense events. Asolution exists. This problem can be solved by the separation of the mean-reversion factors for the  ¿Ãƒâ€šÃ‚ ½no rmal ¿Ãƒâ€šÃ‚ ½ and the  ¿Ãƒâ€šÃ‚ ½extreme ¿Ãƒâ€šÃ‚ ½ process. An applicable approach for this purpose was described by Benth (2005) where he predicts the parameters for the diffusion process taken from historical data and considers a constant volatility over time. But this approach resulted in several drawbacks. Firstly, the spike process ¿Ãƒâ€šÃ‚ ½s parameters are not estimated from the time series, but are based on the opinions of expert. Secondly, this approach has neglected the fact that the volatility in the electricity markets is stochastic over time. Deng (2006) then compares Merton (1976) Jump-Diffusion model with the stochastic volatility and constant and derives prices for the dissimilar energy derivatives using Fourier transform and hence shows that the stochastic volatility is important. Escribano et al. (Villaplana 2003) supply huge empirical tests on an ample range of markets and then makes the conclusion that it is important to include stochastic volatility and jumps. The model of an asset follows a mean reversion process if the prices of assets tend to fall after hitting a maximum. Similarly, the price will rise after hitting a minimum. Let us consider a deterministic model, in which, cash and world stocks must have identical returns and the cash rates must deterministically follow the existing current path of forward interest rates. Everyone can know that stockreturns or interest rates havereachedalow or high point without breaching the unimportant market and arbitrage free conditions. This could be consequently viewed as a form of mean reversion, if the current forwardinterest rate curve is smoothly downward or upwardsloping. Mean-Reverting Stochastic Process dSt = a(L St)dt +stochasticterm ¿Ãƒâ€šÃ‚ ½ (3.1) 36 The mean-reversion stochastic process has a drift term that takes the variable being modeled back to the equilibrium level. The result at the end is that the variable will have the tendency to oscillate around this equilibrium point and each time the  ¿Ãƒâ€šÃ‚ ½stochastic term ¿Ãƒâ€šÃ‚ ½ pushes the variable away from the equilibrium level, the deterministic term will act in a way such that the variable will go back to the equilibrium level. The stochastic term is #t St dWt where #t is a volatility (standarddeviation), # 0, the constantListhe  ¿Ãƒâ€šÃ‚ ½long-termmean ¿Ãƒâ€šÃ‚ ½oftheprocess,towhichitrevertsinsometime, a 0 is a measurementof the strengthof meanreversion and Wt isa standardWienerprocess, v Wt = N(0;t) (3.2) Lets consideranaturalgascalloptionthatfallsand becomes worthlessincasetheprice of natural gas increases above a certain price at anytime during the lifetime of the option. The use of the process of a Geometric Brownian Motion (GBM) price to model natural gas prices gives such price paths that as a result, gives a much higher probability of ending up to the barrier level during the life of the option, than a mean reverting process does. The option pricing models that involves the use of a mean reverting process ensures that prices is drawn towards the mean reversion level, which assigns less chance of touching the barrier during the option life. The graphs below shows simulated paths of price and a resulting histogram for for example, natural gas, using Geometric Brownian Motion vs. Geometric Brownian Motion with mean reversion. The greater prices that are produced by the GBM method can be seen, clearly. In this example, both proce sses of price produce the same result for a European exercise option, but awfully dissimilar option prices for a barrier option. 37 Figure 3.1: Geometric Brownian Motion: Sample Price Paths Figure 3.2: GeometricBrownian MotionWith Mean Reversion: Sample Price Paths 38 Drawback of the Mean-Reverting Model 1. L, the long-term mean, stays fixed over time: It needs to be readjusted on a continuous basis, to ensure that the curves resulting are mar

Literary Essay Samples For Fourth Grade

<h1>Literary Essay Samples For Fourth Grade</h1><p>There are numerous abstract article tests accessible on the web, however which ones are acceptable? In this article I will go over a couple of the most well-known decisions, and afterward offer some guidance for settling on the correct decision for your own composing needs.</p><p></p><p>The first choice is the fifth grade paper or comparative. These works are extremely simple and are ideal for use in writing or craftsmanship class. Here's why.</p><p></p><p>They give some understanding into an essayist's capacity to compose and portray essential plot lines. Since they are genuinely standard, they are likewise composed by journalists who need to get their work out there. This is frequently the situation, and in the event that you look cautiously, you may discover something that interests to you.</p><p></p><p>If you need something increasingly out of the experience, notwithstanding, it's critical to utilize these assets to pass on some close to home encounters. All things considered, on the off chance that you have composing experience, you ought to have the option to portray individual occasions also. These examples, similar to the past alternatives, are genuinely standard in scope.</p><p></p><p>The next choice is more assorted and is bound to speak to scholars keen on expounding on increasingly savvy points, for example, social investigations. There are two fundamental alternatives here, so how about we start with the less obvious.</p><p></p><p>Of course, with regards to expositions that expect you to look into a point, the well known decision is the fourth grade reading material. Since this kind of article expects you to utilize explicit jargon and furthermore to make a couple of general associations between the different sections, they are compelling. Here's why.</p>&l t;p></p><p>They're standard, and in light of the fact that they are more often than not founded on look into effectively done in a homeroom, they are a decent beginning stage for a scholarly paper. You can even utilize the definitions for explicit bits of writing and melodic scores that are incorporated. There's nothing amiss with doing this, since practically the entirety of the definitions are from long-standing works.</p><p></p><p>The last two artistic article tests are for authors who are searching for more profundity. This is where you'll need to invest some exertion, and I'd suggest working with a secondary school understudy. They can be more yearning, and they're continually ready to invest the additional energy necessary.</p>

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