Pythagorean Theorm Essays - Triangles, Triangle Geometry

Pythagorean Theorm The Pythagorean Theorem is a geometrical articulation utilized frequently in math and material science. It used to 2 locate the obscure side of a correct triangle. The exponential type of this hypothesis a + b = c . That is the condition you use when you are searching for the obscure side of a correct triangle, and it is the thing that I'll show on the connected display. The topsy turvy capital L in the base of the left hand corner shows that sides An and B are the legs of the triangle. Since we know side A = 5 inches and B = 3 inches we may fill that in to 2 or on the other hand condition for stage one. (1) 5 + 3 = c What the hypothesis will assist us with finding is the c side of this triangle. 2. 25 + 9 = c All we do is disperse 5 to the subsequent force and 3 to the subsequent force as observed is stage two. Next, we add these two numbers together to get 34, 25+9=34, in sync three. 3. 25+9=34 Then, in sync four we locate the square foundation of 34. 4. 34 In sync five we see that 5.83 is the obscure side of the correct triangle. 5. c= 5.83 We discovered this answer by utilizing the Pythagorean Theorem as instructed in geometrical structure. This hypothesis may likewise be summarized by saying that the territory of the square on the hypotenuse, or inverse side of the correct point, of a correct triangle is equivalent to total of the territories of the squared on the legs. The Pythagorean Theorem was a concentrated by numerous individuals and gatherings. One of those individuals being Euclid. Now and then the Pythagorean Theorem is likewise alluded to as the 47th Problem of Euclid. It is called this since it is incorporated by Euclid in a book of numbered geometric issues. In the difficult Euclid contemplated he would consistently utilize 3, 4, and 5 as the sides of the correct triangle. He did this since 5 x 5 = 3 x 3 + 4 x 4. The edge inverse the side of the legs was the correct edge, it had a length of 5. The 3:4:5 in the correct triangle was known as a Pythagorean triple or a three digits that could be placed in a correct triangle effectively. These three numbers were likewise entire numbers and were utilized in the Egyptian string stunt, which I will discuss later. This Pythagorean triple, 3:4:5, are the littlest whole number arrangement to have been shaped, and the main back to back numbers in that bunch that is significant. These numbers can be, and frequently were, concentrated from a philosophical viewpoint. The representative implications of the 3:4:5 triple told by current scholars, for example, Manly P. Lobby say 3 represents soul, 4 represents matter, and 5 represents man. Utilizing Hall's investigation the imagery of this course of action is as per the following: ?Matter? (4) lays upon the plane of Earth and ?Spirit? (3) comes to up to the Heaven and they are associated by ?Man? (5) who takes in the two characteristics. A procedure like that of Euclid's 47th Problem was the Egyptian string stunt. Egyptians were said to have designed the word geometry (geo = earth, metry = estimating.) The Egyptians utilized the 3:4:5 right triangle to make right triangles when estimating there fields after the Nile floods cleaned out there old limit markers. The Egyptians utilized a similar hypothesis of Euclid, 5 x 5 = 3 x 3 + 4 x 4, to arrive limits stamped accurately. Despite the fact that Euclid and the Ancient Egyptians considered the hypothesis, the genuine innovator of it ( or the individual a great many people accepted designed it first ) was Pythagoras of Samos and his gathering the Pythagoreans. Pythagoras was a man conceived in 580 B.C. on the island of Samos, in the Aegean Sea. It is said Pythagoras was a man that went through his time on earth venturing to the far corners of the planet looking for intelligence. This quest for knowledge drove him to settle in Corona, a Greek province in southern Italy, in around 530 B.C. Here Pythagoras increased well known status for his gathering known as the Brotherhood of Pythagoreans. This gathering committed there lives

The Role of External Auditing in Public Sector free essay sample

As a matter of first importance I need to stretch out my essential commitment to the Lord Almighty for the quality He offered me to continue on with this task as to be sure â€Å"the going got tough† however inside Him nothing is outlandish. I might likewise want to recognize each one of the individuals who helped me make this venture a triumph, much obliged to my administrator Mr. A. Mpofu . much thanks for your significant direction and your difficult work you were persistent with me from the initiation of this venture to its ebb and flow state I wouldn’t have created this artful culmination without your help, Thank You!!!!!Special Appreciation goes to each one of those experts who pushed me with my essential research exceptional notice to the Deloitte Bulawayo Audit group your help was vital to the achievement of this undertaking. I might likewise want to offer my earnest thanks to my family ,without the your money related help and empowering bolster I wouldn’t have gotten this far. We will compose a custom exposition test on The Role of External Auditing in Public Sector or then again any comparative point explicitly for you Don't WasteYour Time Recruit WRITER Just 13.90/page A unique note of gratefulness likewise goes out to the accompanying people for their immovable help all through the planning of this task Mr. P. T . Nyamuvhurudza Mr. T Mukono Takudzwa Nyuke Tapiwa Magunda Mildred PepukaiYou gave me invaluable help folks May the Lord keep on gift you and your families. We are living in a powerful world described by unconstrained changes occurring in the twinkle of the eye realized chiefly by changes in Information Technology. Data Technology is gradually changing the way of our every day living and our occupations. This exploration was decided to recognize the progressions that Information Technology has realized explicitly on the reviewing calling. The examination was provoked principally by the perceptions that the analyst saw during their mechanical connection experience inside a review firm.The decision to concentrate on the suggestions on the inspecting calling was because of the way that the specialist has enthusiasm for the progressions that are happening in the review calling in the previous decade because of Information Technology. A survey of writing with data relating to changes that have happened to the bookkeeping procedure which influence examining was done broadly this helped to recognize the general pattern in the progressions occurring in the calling. The discoveries featured that they was an extreme change in the way wherein examiners execute their obligations because of Information technology.Information was accumulated from both essential and optional sources and it was investigated so as to give the analyst a comprehension of what precisely is going on the ground I. e. inside the business. The examination showed that the Information Technology brought such a large number of preferences to the calling contrasted with the conventional way of completing reviews. Part one of this examination venture is a short rundown featuring the reason for the investigation just as the methodology taken by the analyst to address the exploration questions, it likewise incorporates the point by point meaning of terms t hat were utilized in the exploration venture. Part two is the writing survey which features what different writers have composed relating to Information Technology and evaluating. The data assembled was gathered from a wide scope of creators, the perceptions that these creators made are featured in the subsequent section. Section three shows the examination technique and how the data utilized was gathered; it likewise features the downsides looked by the different strategies for gathering data that were utilized. An examination of the data that was gathered was then done in Chapter 4.

6 Books About Net Neutrality An Unfortunately Necessary Reading List

6 Books About Net Neutrality An Unfortunately Necessary Reading List If youve been paying attention, you know that on December 14, the FCC (under Ajit Pais leadership) dismantled net neutrality regulations that prohibited internet service providers (ISPs) from altering the way consumers connect to the internet. Net neutrality is the way your internet has always worked. In as simple terms as possible, net neutrality regulations prohibit companies (like Verizon or ATT) from blocking or slowing down your access to certain websites, or forcing companies to pay to stay competitive. Ill be linking to further net neutrality resources and an overview video at the end of this post, but as a book lover, I know that I can most fully understand the world through books. Books about net neutrality help me better grasp the very real issues, its nuances, and the freedoms at stake with this regulatory change. To that end, Ive compiled a list of six books about net neutralityâ€"a starting reading list. These books all come to the discussion from a different angle and with different view points, with some tackling our current governance of the internet, how the internet works to begin with, how big telecom companies formed, and predictions for the future of the internet in terms of social change. But, I also know that our current world moves faster than publishing, and net neutrality books can only go so far. Ive also included more information about each of the authors, along with links to their Twitter accounts where youll find more up-to-date information and developments about net neutrality changes and challenges. *Note: Book descriptions come from publishers Amazon descriptions. Captive Audience: The Telecom Industry and Monopoly Power in the New Gilded Age  by Susan Crawford Ten years ago, the United States stood at the forefront of the Internet revolution. With some of the fastest speeds and lowest prices in the world for high-speed Internet access,  the nation was poised to be the global leader in the new knowledge-based economy. Today that global competitive advantage has all but vanished  because of a series of government decisions and resulting monopolies that have allowed dozens of countries, including Japan and South Korea, to pass us  in both speed and price of broadband. This steady slide backward not only deprives consumers of vital services needed in a competitive employment and business marketâ€"it also threatens the economic future of the nation. Susan P. Crawford  is the John A. Reilly Clinical Professor of Law at  Harvard Law School. She served as President  Barack Obamas Special Assistant for Science, Technology, and Innovation Policy (2009) and is a columnist for WIRED. She is a former Board Member of  ICANN, the founder of  OneWebDay, and a legal scholar. | Twitter Code: And Other Laws of Cyberspace, Version 2.0  by Lawrence Lessig Under the influence of commerce, cyberspace is becoming a highly regulable space, where behavior is much more tightly controlled than in real space. But thats not inevitable either. We can-we must-choose what kind of cyberspace we want and what freedoms we will guarantee. These choices are all about architecture: about what kind of code will govern cyberspace, and who will control it. In this realm, code is the most significant form of law, and it is up to lawyers, policymakers, and especially citizens to decide what values that code embodies. Lawrence Lessig is the Roy L. Furman Professor of Law and Leadership at Harvard Law School.  He was also a professor at Stanford Law School, where he founded the school’s Center for Internet and Society, and at the University of Chicago. | Twitter Consent of the Networked: The Worldwide Struggle For Internet Freedom  by Rebecca MacKinnon A clarion call to action,  Consent of the Networked  shows that it is time to stop arguing over whether the Internet empowers people, and address the urgent question of how technology should be governed to support the rights and liberties of users around the world. Rebecca MacKinnon is a Bernard L. Schwartz Senior Fellow at the New America Foundation, and cofounder of Global Voices, an international citizen media network. She also serves on the Boards of Directors of the Committee to Protect Journalists and the Global Network Initiative. | Twitter The Master Switch: The Rise and Fall of Information Empires by Tim Wu It is easy to forget that every development in the history of the American information industryâ€"from the telephone to radio to filmâ€"once existed in an open and chaotic marketplace inhabited by entrepreneurs and utopians, just as the Internet does today. Each of these, however, grew to be dominated by a monopolist or cartel. In this pathbreaking book, Tim Wu asks: will the Internet follow the same fate? Could the Webâ€"the entire flow of American informationâ€"come to be ruled by a corporate leviathan in possession of the master switch? Here, Tim Wu  shows how a battle royale for Internet’s future is brewing, and this is one war we dare not tune out. See also Tim Wus book co-written with Jack Goldsmith,  Who Controls the Internet: Illusions of a Borderless World.   Tim Wu is a professor at Columbia Law School and the director of the Poliak Center for the First Amendment at the Columbia Journalism School. | Twitter Twitter and Tear Gas: The Power and Fragility of Networked Protest  by Zeynep Tufekci Zeynep Tufekci explains in this accessible and compelling book the nuanced trajectories of modern protestsâ€"how they form, how they operate differently from past protests, and why they have difficulty persisting in their long-term quests for change.  Tufekci speaks from direct experience, combining on-the-ground interviews with insightful analysis. She describes how the internet helped the Zapatista uprisings in Mexico, the necessity of remote Twitter users to organize medical supplies during Arab Spring, the refusal to use bullhorns in the Occupy Movement that started in New York, and the empowering effect of tear gas in Istanbul’s Gezi Park. These details from life inside social movements complete a moving investigation of authority, technology, and cultureâ€"and offer essential insights into the future of governance. Tufekci is an associate  professor  at the School of Information and Library Science at the  University of North Carolina  and a faculty associate at the  Berkman Center for Internet and Society  at  Harvard University.  | Twitter Tubes: A Journey to the Center of the Internet  by Andrew Blum Takes us on an engaging, utterly fascinating tour behind the scenes of our everyday lives and reveals the dark beating heart of the Internet itself. A remarkable journey through the brave new technological world we live in,  Tubes  is to the early twenty-first century what  Soul of a New Machineâ€"Tracy Kidder’s classic story of the creation of a new computerâ€"was to the late twentieth. Since 1999, Blums articles and essays have appeared in  Popular Science,  Metropolis,  Vanity Fair,  Wired,  The New York Times,  The New Yorker,  Bloomberg Business Week,  Gizmodo,  The Atlantic Online,  Architectural Record,  Slate, and many more. | Twitter Reading these books about net neutrality can only go so far. As were finding over and over again in 2017 (and as we journey into 2018), collective action is an important component of being an engaged and responsible citizen today. The following video gives a one-minute overview of net neutrality, and why its so important. Organizations like Free Press have committed themselves to challenging the most recent dismantling of regulations. Find ways to get involved (either by donating to their action fund or calling your government representatives) here. Net neutrality regulations protect everyoneâ€"from classrooms, to libraries, to small businesses. Read these books about net neutrality, gear up, and then find a way to take action to protect continued, unfettered access to the internet.

Conclusion And Evaluation Of Osmosis Potato Lab - 889 Words

Saam Baharmand Period 5 11 October 2014 Conclusion and Evaluation of Osmosis Potato Lab Hypothesis: I believe that the O Molar (M) solution of sugar and distilled water will be isotonic because there is no sugar in the solution. This Hypothesis was not supported by the results of the lab. 1) The line graph of the data collected shows the least amount of change in mass (1.8%) in the potato soaked in the 0.2M sugar solution. This means that the potatoes soaked in the 0.2M solution were the most isotonic of all the potatoes. 2) The beginning mass of the potato soaked in 0.2M solution was 10.6g and the ending mass was 10.8g. This data collection proves that there was no significant loss or gain of mass in the potato (soaked in 0.2M sol.) thus making it isotonic. 3) The percent (%) change in mass of the potato in the 0.0M sugar solution was recorded on the graph at 31.8%. This means that there was a significant increase in mass of the potato thus making the solution hypertonic and the potato cells hypotonic. Variables Independent: 5 interval sugar solutions of 0.0M, 0.2M, 0.4M, 0.6M, and 0.8M (mixed with distilled water) Dependent: % Mass Change of potato after 24 hours in solution Quantitative Relationship: When the intervals of sugar solution change, the % mass of the potato changes as a result. The 0.2M solution proved to the most isotonic while the 0.0M and 0.6M potatos proved to be the most hypotonic and hypetonic respectively. The potato of the 0.2M sol.Show MoreRelatedPotato Cubes Mass Lab Report1117 Words   |  5 Pages 2. Research Question: How will a potato cube’s mass change if put in a mixture of Glucose and Salient together vs a potato cube just being put into a solution with Glucose or Salient. 3. Background Information: Diffusion is where particles move from areas of higher concentration to areas of lower concentration. Osmosis is the diffusion of water. Glucose (sugar) will not diffuse as much as Salient because there is already sugar inside of the potato so not as much water be affected by diffusionRead MoreEffect of Different Concentrations of Salt on Potato Cell Mass1461 Words   |  6 Pages Aim To investigate the effects of increasing salinity on potato cell mass. Background Information This experiment is based upon osmosis. Osmosis can be defined as the net movement of water molecules from a region with high concentration to a region with low concentration. This movement must take place across a partially permeable membrane such as a cell wall, which lets smaller molecules such as water through but does not allow bigger molecules to pass through. This processRead MoreMicroscopy, Cell Structure And Function1208 Words   |  5 Pagescell structure and function TASK 3 Write a scientific report on osmosis experiment showing the: Aim, Introduction, and Hypothesis. Risk assessment, Procedure, Results, Analysis and conclusion, Evaluation, Sources or error, Anomalous results, Biological or industrial significance, Health and safety. A.C.2.2 AIM: Investigating the effect of Different concentration of Sucrose on Osmosis in potato chips. INTRODUCTION: First of all, Osmosis is the net movement of water molecules from an area of high waterRead MoreThe Effect of Solution Concentration on Osmosis1491 Words   |  6 PagesThe Effect of Solution Concentration on Osmosis The aim of my experiment is to find out the effect of sucrose solution concentration on osmosis in potato cylinders. To do this I will conduct an experiment. Introduction To carry out my experiment, I will place the potato tubes into a solution containing part sucrose and part water. The potato tubes will all be cut out of the potato using a cork borer and will all be cut to the length of 25mm. This will remainRead More Investigate the Osmosis of Potato Cells in Various Salt Solutions2525 Words   |  11 PagesInvestigate the Osmosis of Potato Cells in Various Salt Solutions Introduction I have been asked to investigate the effect of changing the concentration of a solution on the movement of water into and out of potato cells. I will be able to change the input of my experiment. The input variable will be the concentration of the solution. The 100% solution is sodium chloride dissolved in water (salt water). This will be referred to as the 100% solution from now on. But the type of solution is notRead MoreOsmosis Internal Assessment - Biology Higher Level3984 Words   |  16 Pages| The effect of salinity on osmosis of solanum tuberosum L.(potatoes) | Biology HL Internal Assessment – Year 10 | | Teresa Nguyen | | Table of Contents 1 DESIGN 2 1.1 Defining the problem 2 FOCUS QUESTION 2 HYPOTHESIS 2 BACKGROUND INFORMATION 2 INVESTIGATION VARIABLES 3 1.2 Controlling Variables 3 TREATMENT OF THE CONTROLLED VARIABLES 3 CONTROL EXPERIMENT 4 1.3 Experimental Method 4 MATERIALS 4 RISK ASSESSMENT 5 METHOD 5 2 DATA COLLECTION and PROCESSING 7 2.1 Recording RawRead MoreOsmosis in Different Concentrations2449 Words   |  10 PagesI am going to investigate osmosis when potato is placed in different   concentrations of sucrose. I am aiming to witness osmosis in 5   different concentrations of sucrose. I will use 5 varying   concentrations so that I have a wider spread to compare the results,   and check that I don’t have any anomalies   Prediction   Osmosis  is the process of diffusion of water molecules from a weaker   solution into a stronger solution, through a semi  permeable membrane.   The tiny pores in the membraneRead MoreThe Four Main Types Of Tissues2505 Words   |  11 Pagescell structure and function TASK 3 Write a scientific report on osmosis experiment showing the: Aim, Introduction, and Hypothesis. Risk assessment, Procedure, Results, Analysis and conclusion, Evaluation, Sources or error, Anomalous results, Biological or industrial significance, Health and safety. A.C.2.2 AIM: Investigating the effect of Different concentration of Sucrose on Osmosis in potato chips. INTRODUCTION: First of all, Osmosis is the net movement of water molecules from an area of high waterRead MoreDiffusion And Osmosis And The Cell Membrane2124 Words   |  9 Pagessolute concentrations on both sides of the membrane are equal. The diffusion of free water across a selectively permeable membrane, whether artificial or cellular, is called osmosis. The movement of water across cell membranes and the balance of water between the cell and its environment are crucial to organisms. (Diffusion And Osmosis - Difference And Comparison | Diffen). A semi-permeable membrane known as the cell membrane surrounds the living cells of both plants and animals. Both solute concentrationRead More The effect of osmosis in potato cells with different concentrations of sucrose solution1887 Words   |  8 PagesThe effect of osmosis in potato cells with different concentrations of sucrose solution Aim: To test the effect of different concentrations of sucrose solution to osmosis in a potato cells by putting potato pieces in test tubes of water containing different concentrations of sucrose solution. Scientific Theory: Osmosis is defined as the movement of water molecules across a partially permeable membrane from a region of high water concentration to a region of low water concentration

Diabetes in Developing Countries Free Essays

Diabetes in developing countries Deaths from diabetes, which has two primary forms including type1 and type2 diabetes, have become a significant problem in the world. Nowadays, diabetes is still a disease not having precise method to cure. As a result of surplus blood sugar, it has a negative effect on the human body and leads to several complications, such as vision problems, kidney damage, nerve damage and heart and circulation problems (Pollock, 2006). We will write a custom essay sample on Diabetes in Developing Countries or any similar topic only for you Order Now Consequently, the increased risk of these diseases makes it become one of the major causes of deaths. For example, according to the WHO (2011), more than 346 million people were diagnosed with it worldwide and between 50% and 80% of them died from CVD. With the development of health care, the mortality in developed countries was decrease, while the situation in developing countries is so serious that 80% of diabetes deaths exist in low and middle income countries (WHO, 2011). For instance, such countries in The Middle East, Pacific Islands and Southeast Asia had 115million diabetic patients in 2000 and the WHO (2011) predicts that the number will double between 2005 and 2030. To mitigate the effects of diabetes, the causes of it need to be detected. Type 1 diabetes, which is known by lacking insulin production, results from several causes and possible factors. First, genes attribute mainly to it. More than 18 genetic locations related to it have been discovered by researchers and they have found that people with an especially HLA complex which means human leukocyte antigen, are more likely to develop it. A good illustration of it is other autoimmune disorders may caused by such complexes, such as rheumatoid arthritis, ankylosing spondylitis, or juvenile rheumatoid arthritis (Smith, 2010). The second factor is a viral infection which may affects the disease by attacking immune system. For instance, Kamiah (2010) states that a series of diseases from gastrointestinal problems to myocarditis can created by the coxsackie B virus. In addition, there are some special conditions which may attribute to it. For example, certain drugs including corticosteroids, beta blockers, and phenytoin, rare genetic disorders such as Klinefelter syndrome and Wolfram syndrome, and hormonal disorders such as acromegaly and hyperthyroidism all raise the possibility of it (Simon, 2009). It has been one of the most increased diseases worldwide, however, type 2 diabetes is more common. Unlike type 1 diabetes, causes of type 2 diabetes, which results from the ineffective use of insulin (WHO,2011), usually are multifactorial. First, being overweight or obese is a primary reason for it. The increased risk of it may bring several complications including heart disease, stroke and some cancers. A good illustration of this is 82% of people with it are caused by overweight or obese and such complications (Vann,2009). The second is genetic factors which have been found more than 10 genic material associated with it. For example, there are more possibilities for people to get it if they have close relatives having it, such as parents and siblings. Thirdly, ethnic origin also plays a part in it. For instance, NHS (2010) points out that people with it from South Asian, African, and Middle Eastern are six times likely than people in the UK. In addition, incorrect living styles such as poor eating habits, too much TV time and physical inactivity also have a negative effect on it. It is often not a single factor but two or more causes above combined to lead to it. According to the CDC (2010), such combinations give rise to approximately 95% of it in the U. S. As can be seen from data, diabetes in developing countries has become a huge problem and the mortality from it has a continued increase worldwide. Not only government, but people should change their attitudes and aware the importance in order to prevent it. How to cite Diabetes in Developing Countries, Essay examples

Reconstruction to Industrialization in the Western United States free essay sample

The Freedmans Bureau bill, helped with recognizing free labor, schools for the newly freed persons were being highly oversee, making ere that the newly free were being treated with justice and having the same rights as anyone else. When President Lincoln proposed a plan in this Reconstruction time, this plan was the plan. This meant that about 10% of those who voted would have to pledge to the Union their loyalty. Of course this plan was not much of a success and the radicals (Republicans) did not like this. Soon after there was a bill created by congress that was called the Wade Davis Bill.This bill made many (majority) of the southern states take an oath hat would have them say they never had any support towards the confederacy. This bill never really made it out because Lincoln vetoed the bill, and soon after also did his assassination (Franklin, 1970). If Lincoln was not assassinated, I think the process of Reconstruction would have been smother and with much more success. We will write a custom essay sample on Reconstruction to Industrialization in the Western United States or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page He would have pushed for more plans and bills that would help the freed people and help them get recognized for their rights.For example, once President Johnson had power his Black Codes was not beneficial to all freed people. If Lincoln was still alive, he Black Codes would probably not have even been thought of, not right away any. Ways or even vetoed. The Black codes were a way to restrict freed people of many justices, something Lincoln would not have agreed on. This bump in the road during Reconstruction would have never happened, and if it wouldnt have, perhaps this period would have been much more successful for the former slaves.Industrialization and arbitration affected the life of the average working Americans in many ways. Arbitration at these times had its ups and down (positives and negatives). The affect arbitration had on the average American in the urban areas was overcrowding disease and crime (Stuntman, 1973). For those in the new metropolitan centers, the problems began with pollution and sewage. With all these negative impacts, help was needed from local leaders in order to live correctly and fix these problems, improvements were much needed.Just like arbitration, industrialization had its pros and cons. On the pros list would lay railroads in the west. This would not only allow for goods and services to be transported quicker, but this allowed people to go on further journeys in a shorter amount of time then they would before. Since everyone now had the freedom to open their own business, there was now much more options in order to buy products. Since people were now looking for the best place to buy their products, competition and working to have the best product, service and price in town was very important.On the other end when industrialization happened, the classes between the rich and poor changed. There was now added types of class, like the middle working class for example. New jobs were created for this middle lass and the new upper class (rich) were not leading their lives in a different way. For example they were making sure that their money and class would stay the same and therefore would now classify arrange marriages for their children in order to have this happen (Stuntman, 1973). As described before, when President Johnson went into power, he created the Black Codes.These codes did not allow former slaves the right to vote, it limited the right to testify against whites, or sit in jury. Once the 14th amendment was approved in 1867, this gave the blacks the right to vote. Even though non whites were given more rights, there were still much more injustices going on for them. Once they became free, their former owners would try to keep them in order to pay the minimal amount for the cheapest labor. Some nonwhites would agree to this because they did not know any better and others would not go for that deal and go off to find their families (if they were not already together). Many of these newly freed slaves were illiterate and had a very low set of education. It was not convenient for the whites to allow nonwhites and immigrants to have more education than what they had because it would allow the nonwhites to realize what was really going on. They would realize that there is much more injustice than what they thought and being able to be a little more education would allow them to realize and process what they could do in order to fix that. Thanks to the Freedmans Bureau education was possible for those who are nonwhite, but unfortunately lasted for a few years.During the gap of reconstruction to industrialization in western United States, there have been many changes politically and economically (Franklin, 1970). Politically, major changes were really obvious during the reconstruction period. Politically when Lincoln was assassinated, president Johnson tried to change what was around him. He tried changed the thoughts that Lincoln had, such as wanted the nonwhites to stay without rights. Ultimately although many wanted him impeached, it came down to two votes and at the end, he was not impeached.Economically during the time of industrialization, farmers had hard times making money, having their rights in place and had trouble getting good eels. Farmers were not making as much money as they used to. The most trouble they had was during the winter season, nothing was growing and not much was selling. Farmers had to think of what to do quickly in order to survive. Therefore, farmers would have to work twice as hard during all other season before/after winter season in order to make up the profits they missed (Augustan, 1973). Overall the period between 1865 and 1 900, were very intense and had a lot of important events going on.

Design Trends in the 21st Century

Design Trends in the 21st Century The most exciting thing about the course Among the variety of courses offered to the students, it is hard to define the one that excites the most because each subject has its own positive and negative aspects. However, talking this particular course, it is possible to identify several aspects which turn out to be rather interesting and education. First of all, this course helps students understand that the role of history is integral indeed, and the way of how people perceive their backgrounds and their roots defines their present and future. In addition, with the help of this course, students realize how to learn their past and what spheres become more important for people.Advertising We will write a custom essay sample on Design Trends in the 21st Century specifically for you for only $16.05 $11/page Learn More Still, the most exciting thing that has been learnt so far in this course is all about the works of art, their diversity, and importance to society . One the one hand, all buildings and other pieces of art may be considered as the required things for living. However, on the other hand, each building helps to comprehend the history of the whole community. For example, Frieze, Tempietto that is San Pietro in Montorio, Rome was created by Bramante between 1502 and 1503. Bramante’s main purpose was to create a kind of fusion of humanism and Christianity essentials. It is not the only another rotunda building, it is an attempt to help people improve their lives and beliefs. Looking at the Hardwick Hall that is Derbyshire, England, people could not help but think about the power of English will. This magnificent building was created by Robert Smythson in later 16th century. This hall is a perfect union of stability, order, and aristocratism that were inherent to English people during that period. This is what excites the most about the subject: the possibility to learn more about the nature of other people, their interests, an d preferences. Three important trends in the 21st century How does each trend relate to a specific historical design trend? The peculiar feature of all the 21st design trends is that all of them are based on a variety of technological innovations, flexibility, profuseness of colors, and lighting (Browne 7). It is not always that easy to identify the most important trends in designs which are inherent to the 21st century, still, it is always possible to think about the most captivating and influential values which change human perception of the reality.Advertising Looking for essay on art and design? Let's see if we can help you! Get your first paper with 15% OFF Learn More Taking into consideration personal preferences and the number of works analyzed, the three most important trends of the 21st century are involving people to art works by means of new technologies like optics which define new perspectives of environment, encouragement of movement and continu ity between the outside and the inside, and virtual and hyperspace repetitions that serve as a bright evidence of a technological revolution. Each trend relates to some historical design trends which were popular during the period of Renaissance and the 17th-18th centuries. For example, the use of optics was not inherent to 16th-18th centuries, and this trend may be regarded as the new one due to the development of the technologies (Figure 1). The trend to involve people to the scenarios offered by the designer seems to be a new achievement that has to be recognized within a short period of time. New technologies help not only to present a work to public but also to make them become a part of the idea. The work by Jeppe Hein in Bristol is one of the best examples of how optics may influence human perception of the reality. However, similar traits and effects on people are observed in the works by Etienne-Louis Boullee (Design for a National Library in 1784). This designer supported the picturesque mode and the idea of sensation in architecture that will involve public to his works. The Rococo period was characterized by the appearance of new technologies which helped to create huge buildings and organize each detail of the building in a unique way, this is why it seems to be possible to relate the works by Hein and Boullee as those where the role of new technologies regarding the century influence public’s perception of the art work. The encouragement of movement and continuity related to the interior and exterior is another trend to be identified. It relates to the Renaissance trend of vertical openings and movements supported by Andre le Notre. Shigeru Ban (Figure 3) introduced his Centre Pompidou-Metzs (2004 – 2010) to prove how it is possible to relate the outside and the inside. His work looks like the work of an artist who put the lines slightly and playfully taking into consideration the things around. His attempt is similar to the histori cal design trend offered by Le Notre in 1657 where an emphasis was made on dramatic vistas. This historical trend shows that it is possible to define the sequence of spaces that may move from public to private sectors (Figure 8). Finally, the trend to benefit from repetitions in white colors (Figure 6) has to be mentioned. The similarities to this trend may be observed in the works by John Soane Home (Office to the Dome Exhibition Space).Advertising We will write a custom essay sample on Design Trends in the 21st Century specifically for you for only $16.05 $11/page Learn More The designer made an attempt to repeat the elements of the room in order to prove that the chosen art work was worth attention and recognition. It was not enough to enter the room but follow a kind of map offered by the designer in the form of constantly repeated huge columns which were inherent to the style of Rococo (Figure 9). The pros of the trends identified The point is that the trends identified have a number of pros which make many designers choose them and improve by means of the technology available. For example, the work by Electric Dreams, Pleasant Bar that is in Stockholm, Sweden from 2007 (Figure 2) shows how several optic elements may change the reality and involve people into a new world, full of mystery and pleasure. It is not enough to create some visual elements but define each element in a proper way to introduce a true masterpiece. Another pros of the trend based on movement and continuity is that the designers are free to use the space available to its full extend. For example, the bathroom (Figure 4) in the Spanish Hotel Puerta America proves that it is possible to use several massive elements of furniture in a small room and create comfortable apartments. Continuity is not always easy to create, still, if the designer succeeds in the chosen activity, the results may be amazing. The trend of repetition is the key point of many exhibitions around the whole world due its main pros – compactness. The designers find this trend rather beneficial for meeting their purposes: they repeat their thoughts to involve the public into their worlds and their ideas. Though it is not an easy task to repeat the ideas and objects and remain to be logical and comprehensive. This is why some misunderstandings may take place. Still, the designers’ main task is to choose the most appropriate ways and achieve success. The cons of the trends identified Talking about the cons of the trends identified, it is necessary to admit that not all these trends may be understood by the public. For example, the idea to use optics in design is quite new, and many people are at loss when the time to observe the creation comes. So, the main con of the three trends discussed in this paper is designers’ inability to interpret their ideas and intentions to all people in a proper way. In spite of the fact that the trends relate to the hist orical design trends in some ways, failures to meet public’s expectations may take place. Conclusion In general, each trend identified in this paper is worth attention and recognition. Repetition and white objects, encouragement of movement and continuity, and choice of optical technologies have their own pros and cons in the sphere of design. The works chosen for this papers show that the designers are able to use their skills, ideas, and technologies available to attract more people to art.Advertising Looking for essay on art and design? Let's see if we can help you! Get your first paper with 15% OFF Learn More They are successful and they are unique for the 21st century. This is why the chosen trends may be regarded as ones of the most important ideas which are supported and developed in a variety of ways in the 21st century. Browne, Beth. 21st Century Interiors. Mulgrave, Victoria: Images Publishing, 2010. Appendixes Figure 1 Follow Me. Jeppe Hein. Bristol, England. 2008. Figure 2 Pleasant Bar. Stockholm, Sweden. Electric Dreams. 2007. Figure 3 Centre Pompidou-Metz. Metz, France. Shigeru Ban. 2004-2010. Figure 4 Bathroom. 7th Floor. Hotel Puerta America. Madrid, Spain. Ron Arad. 2002 – 2005. Figure 5 Non-Standard Architecture Exhibition. Pompidou Center. Paris, France. 2004. Figure 6 Ascension of Polka Dots. Yayoi Kusama. Singapore Biennale Exhibit. Singapore, 2006. Figure 7 Design for a National Library. Etienne-Louis Boullee. France. C. 1784. Figure 8 Gardens. Chateau Vaux-le-Vicomte. Andre le Notre. Begun 1657. Figure 9 Office to the Dome Exhibition Space. John Soane Home. Lincoln’s Inn Fields, London. Soane. 1812-1835.

Class of 2017 New SAT or Old SAT (Updated)

Class of 2017 New SAT or Old SAT (Updated) SAT / ACT Prep Online Guides and Tips The Class of 2017 is in a tough spot on the SAT. Should you take the new SAT or old SAT? If you take the new SAT, you would be the guinea pigs for a totally new system; if you take the old one, you have a tighter schedule. How does it break down? Important Note: This article has been updated as of January 25th 2016. If you are reading this, thenat least for official testings, it is too late to choose the old SAT. This article was for historical SAT test-takers registering before January 2016. However, the advice below is still useful for any future test version changes, whether SAT or ACT. Aside: Not in the Class of 2017? Find out whether you should take the new or old SAT here! The answer, according to multiple experts, is to definitely take the old SAT (though there are a few exceptions, see below). Fred Zhang, cofounder of PrepScholar, went through the last SAT transition in 2005, and saw tremendous advantages for the takers of the old SAT. Admittedly, the schedule is less than optimal. The final chance you have to take the SAT will be January 2016, according to the College Board. This is only the middle of your junior year, which gives you less slack than your older classmates, but has huge advantages. You know what the current SAT looks like. There is almost a decade of history with the current SAT. Everyone knows what the old SAT looks like, how to effectively prepare for it, and what's on it. You can do real practice tests that have actually been given in the past. This advantage cannot be overstated. If you are the studying type, taking a test with no history will greatly lower your relative advantage. Here are all of the extra resources you get access to with the old SAT: You can ask older classmates for their best tips and strategies. SAT prep companies and study guides will have perfected prepping for the old SAT. More than 20 real practice SATshave been released by the College Board, and practicing using real tests is so important. You can take a real administration early on to get a feel for timing and the tests. Almost all free advice you can find online about the SAT applies best to the old SAT. They may still hold for the new SAT, but there's a chance the diametric opposite is true. The January 2016 Deadline Is Not That Bad Yes, it does restrict some backup options for you, but really, you don't want to bestudying junior summer, or worse, senior fall. If you study for the SAT too late, it will interfere with your extracurriculars and college application process. Ideally, you want junior summer, and certainly senior fall, to be free from the distraction of taking SATs. Also, if you plan correctly, you'll actually have more opportunties to take the SAT with a schedule that puts your test in January or before, rather than March or later. The January 2016 deadline is really not that bad with just a bit of earlier study. You Get to Take Two Bites at the Apple You follow the old SAT timeline. Suppose you do great on the old SAT, blowing it out of the water great, you don't need to take the SAT anymore! Now assume the reverse: you royally mess up on the old SAT, its style isn't for you. Is it over? Not if you're taking the old SAT you get to pick from the best of two tests! Just take the New SAT, and if it's a better fit for you, you're golden! But the reverse scenario? If you plan to take the new SAT, and then bomb it and figure out you're actually better at the old SAT? Sorry pal, but you can't turn back time you're stuck! Avoid Being a Guinea Pig In first few administrations of the new SAT, you'll be a guinea pig for the College Board. They don't yet have experience in designing flawless problems, calibrating their scale, and perfecting the proctoring instructions. I personally would not be surprised at more than a few hiccups. Exceptions: If you just want to wing the SAT There are a few exceptions to the advice that the Class of 2017 should take the old SAT. The primary oneis if you're going to wing the SAT. For the same reason that studiers benefit from the old SAT, if you're going to totally wing it anyway, the new SAT is better for you since studiers won't have as large of a relative advantage. What if you don't have enough time to study for the January 2016 SAT? Say it's three weeks before, is it still worth signing up? I would say if you are a serious SAT taker yes for sure! Even if you don't study much, if you're naturally better at the old SAT, you could do better on the old SAT naturally compared to the new SAT. If you've got an extra four hours, and don't mind retaking the new SAT again, it is definitely worth it to strategically just try the old SAT even with minimal study in case you're naturally better at it! What to Do Next: Now that you know which SAT to take, you'll probably want to know how to study for it. We've researched hundreds of student stories and academic studies and found the 5 principles you need to follow to improve your score. Click the link below and enter your email address to get the best SAT prep advice you can get anywhere. Follow these 5 strategies to improve your SAT score by 160 points or more. Long story short: if you can manage it, aiming for the old SAT would be advantageous. Don't wait for the new SAT to take it! Also, check back here for a complete timeline guide for the Class of 2017 taking the SAT. Other Posts You May Like: Not in the Class of 2017? Should you take the new or old SAT? What should your SAT target be? Get Started Improving Your SAT Score Today: Have friends who also need help with test prep? Share this article! Tweet Dr. Fred Zhang About the Author Fred is co-founder of PrepScholar. He scored a perfect score on the SAT and is passionate about sharing information with aspiring students. Fred graduated from Harvard University with a Bachelor's in Mathematics and a PhD in Economics. Get Free Guides to Boost Your SAT/ACT Get FREE EXCLUSIVE insider tips on how to ACE THE SAT/ACT. 100% Privacy. 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Major project Research Paper Example | Topics and Well Written Essays - 3750 words

Major project - Research Paper Example Both these organizations advocate rules and policies favoring workers in the US, and play an active political role, mostly favoring the Democratic Party. The recent Economic Crisis of 2008 that shook the economic basis of the US found the Labor Unions a burden. According to some people, the three major automakers of US found it difficult to cope with the changing situations and to compete with their global opponents as a result of the costly labor agreements including pension and health plan. As a result of this poor economic condition, states, especially the ones led by Republican Party like Wisconsin, Ohio, Tennessee, Michigan, and Illinois have introduced bills that would result in salary cuts, and some are planning to restrict the collective bargaining rights the workers enjoyed in the US for about a century. Thus, it becomes evident that the labor unions, despite decades of struggle and violence, seemed to have gained little as the new developments take away most of the freedom they gained over a century. Though polls reveal that most people are against this governmental step, the Democratic Party seems strong in its decision. ... However, it faced eventual collapse due to poor organization. Though there were labor protests even before 1800s, they were poorly coordinated and localized. The first recorded labor union strike took place in 1786 in Philadelphia. It was conducted by printers opposing a wage cut. However, the first major strike that brought the idea of the never ending conflict between workers and business owners was the railway strike in 1877. It was against the Baltimore and Ohio railroad and it spread to the whole of northeast. To curb the violence, troops were called out and using considerable amount of force, the strike was crushed. Throughout their history, the labor unions used ideologies ranging from intimidation, threats, vandalism, and violence, and their history was never peaceful. Another important incident (cited in Foner, 27-37) that deserves to be mentioned is the Haymarket Riot that eventually led to the development of AFL. On 4 May 1886, a labor rally was conducted at the Haymarket Square in Chicago in support of the eight-hour workday. However, the rally turned violent leading to bloody clashes between the agitators and police. The courts had a rough time dealing with the legal status of labor unions. The question was and still is whether these cartels of laborers lawful or not? Do they amount to criminal conspiracies against trade? Are the strikes an expression of the individual right to bargain for lawful employment? One can find a lot of disparity in the issue from the very beginning. To illustrate, as Tomlins (p.128) reports, in the 1806 Commonwealth v. Pullis, the jury convicted the accused unionists of criminal conspiracy. However, as Bennett, Heard and Holland (235) write, in Commonwealth v. Hunt (1842), the Massachusetts

Competitors Analysis Critical Review Essay Example | Topics and Well Written Essays - 2000 words

Competitors Analysis Critical Review - Essay Example A way in which this can be done is through competitor analysis. This review attempts to find the common ground or differences in approach between the authors and checks them against the background of the strategic needs of organisations for sustaining competitive advantage. maintaining and improving profitability targets. In the face of fierce competition, the market place is merciless in changing the fortunes of companies. If firms ignore or neglect strategic moves by competitors, they can find their market share and profitability disappear in quick time. Competition can be in many forms such as new entrants, new products / services, more economical processes that offer cheaper alternatives to customers, new marketing or distribution channels, newer strategic associations or markets, etc. While the challenge of keeping track of so many variables may be daunting, it is comforting to remember that generally each business has its own limited set of rivals. Largely, this depends upon its own profile of size & operations. It is this universe of potential rivals that is important to a firm’s competitive position. According to Porter, â€Å"Strategy is an internally consistent configuration of activities that distinguishes a firm from its rivals† (Porter, M E. 2004, p.vi). Porter’s theory identifies a firm as one based on activities and points out that it is these activities that create opportunities for a firm to offer value to a customer (Porter, M E. 2004). To this extent, analysis of a firm’s own or its rival’s activities can be considered as essential as strategic to a firm’s competitive advantage. The question then is ‘when or how often should one conduct such an analysis?’ and the answer should be that it must be an on-going process in view of the constantly emerging competitors’ challenges. Competitor analysis is the management

Essay --

Describe the role and functions of the PCI security standards council â€Å"The PCI Security Standards Council is an organization created by the major credit card companies in an effort to better protect credit card holder data.† (Rouse, 2012) The council was formed in response to the increase in data security breaches that not only affected customers but also credit card companies cost. With PCI Security Standards Council being a open global forum, The five founding credit card companies – American Express, Discover Financial Services, JCB International, MasterCard Worldwide and Visa Inc. – are responsible for carrying out the organization’s work. Functions of the council include coming up with a framework of specifications, measurements, and support resources to help organizations ensure the safe handling of cardholder information at every step. This is done by managing the Payment Card Industry Security Standard (PCI DSS) and the Payment Application Data Security Standard. Identify/describe key requirements for data security standards The key requirements for the Data Security ...

Accrual Swaps

ACCRUAL SWAPS AND RANGE NOTES PATRICK S. HAGAN BLOOMBERG LP 499 PARK AVENUE NEW YORK, NY 10022 [email  protected] NET 212-893-4231 Abstract. Here we present the standard methodology for pricing accrual swaps, range notes, and callable accrual swaps and range notes. Key words. range notes, time swaps, accrual notes 1. Introduction. 1. 1. Notation. In our notation today is always t = 0, and (1. 1a) D(T ) = today’s discount factor for maturity T. For any date t in the future, let Z(t; T ) be the value of $1 to be delivered at a later date T : (1. 1b) Z(t; T ) = zero coupon bond, maturity T , as seen at t. These discount factors and zero coupon bonds are the ones obtained from the currency’s swap curve. Clearly D(T ) = Z(0; T ). We use distinct notation for discount factors and zero coupon bonds to remind ourselves that discount factors D(T ) are not random; we can always obtain the current discount factors from the stripper. Zero coupon bonds Z(t; T ) are random, at least until time catches up to date t. Let (1. 2a) (1. 2b) These are de? ned via (1. 2c) D(T ) = e? T 0 f0 (T ) = today’s instantaneous forward rate for date T, f (t; T ) = instantaneous forward rate for date T , as seen at t. f0 (T 0 )dT 0 Z(t; T ) = e? T t f (t,T 0 )dT 0 . 1. 2. Accrual swaps (? xed). ?j t0 t1 t2 †¦ tj-1 tj †¦ tn-1 tn period j Coupon leg schedule Fixed coupon accrual swaps (aka time swaps) consist of a coupon leg swapped against a funding leg. Suppose that the agreed upon reference rate is, say, k month Libor. Let (1. 3) t0 < t1 < t2  ·  ·  · < tn? 1 < tn 1 Rfix Rmin Rmax L( ? ) Fig. 1. 1. Daily coupon rate be the schedule of the coupon leg, and let the nominal ? xed rate be Rf ix . Also let L(? st ) represent the k month Libor rate ? xed for the interval starting at ? st and ending at ? end (? st ) = ? t + k months. Then the coupon paid for period j is (1. 4a) where (1. 4b) and (1. 4c) ? j = #days ? st in the interval with Rmin ? L(? st ) ? Rmax . Mj ? j = cvg(tj? 1 , tj ) = day count fraction for tj? 1 to tj , Cj = ? j Rf ix ? j paid at tj , Here Mj is the total number of days in interval j, and Rmin ? L(? st ) ? Rmax is the agreed-upon accrual range. Said another way, each day ? st in the j th period contibutes the amount ? ?j Rf ix 1 if Rmin ? L(? st ) ? Rmax (1. 5) 0 otherwise Mj to the coupon paid on date tj . For a standard deal, the leg’s schedule is contructed like a standard swap schedule. The theoretical dates (aka nominal dates) are constructed monthly, quarterly, semi-annually, or annually (depending on the contract terms) backwards from the â€Å"theoretical end date. † Any odd coupon is a stub (short period) at the front, unless the contract explicitly states long ? rst, short last, or long last. The modi? ed following business day convention is used to obtain the actual dates tj from the theoretical dates. The coverage (day count fraction) is adjusted, that is, the day count fraction for period j is calculated from the actual dates tj? 1 and tj , not the theoretical dates. Also, L(? t ) is the ? xing that pertains to periods starting on date ? st , regardless of whether ? st is a good business day or not. I. e. , the rate L(? st ) set for a Friday start also pertains for the following Saturday and Sunday. Like all ? xed legs, there are many variants of these coupon legs. The only variations that do not make sense for coupon legs are â€Å"set-in-arrearsâ €  and â€Å"compounded. † There are three variants that occur relatively frequently: Floating rate accrual swaps. Minimal coupon accrual swaps. Floating rate accrual swaps are like ordinary accrual swaps except that at the start of each period, a ? ating rate is set, and this rate plus a margin is 2 used in place of the ? xed rate Rf ix . Minimal coupon accrual swaps receive one rate each day Libor sets within the range and a second, usually lower rate, when Libor sets outside the range ? j Mj ? Rf ix Rf loor if Rmin ? L(? st ) ? Rmax . otherwise (A standard accrual swap has Rf loor = 0. These deals are analyzed in Appendix B. Range notes. In the above deals, the funding leg is a standard ?oating leg plus a margin. A range note is a bond which pays the coupon leg on top of the principle repayments; there is no ? oating leg. For these deals, the counterparty’s credit-worthiness is a key concern. To determine the correct value of a range note, one needs to use an option adjusted spread (OAS) to re? ect the extra discounting re? ecting the counterparty’s credit spread, bond liquidity, etc. See section 3. Other indices. CMS and CMT accrual swaps. Accrual swaps are most commonly written using 1m, 3m, 6m, or 12m Libor for the reference rate L(? st ). However, some accrual swaps use swap or treasury rates, such as the 10y swap rate or the 10y treasury rate, for the reference rate L(? st ). These CMS or CMT accrual swaps are not analyzed here (yet). There is also no reason why the coupon cannot set on other widely published indices, such as 3m BMA rates, the FF index, or the OIN rates. These too will not be analyzed here. 2. Valuation. We value the coupon leg by replicating the payo? in terms of vanilla caps and ? oors. Consider the j th period of a coupon leg, and suppose the underlying indice is k-month Libor. Let L(? st ) be the k-month Libor rate which is ? xed for the period starting on date ? st and ending on ? end (? st ) = ? st +k months. The Libor rate will be ? xed on a date ? f ix , which is on or a few days before ? st , depending on currency. On this date, the value of the contibution from day ? st is clearly ? ? ? j Rf ix V (? f ix ; ? st ) = payo? = Z(? f ix ; tj ) Mj ? 0 if Rmin ? L(? st ) ? Rmax otherwise (2. 1) , where ? f ix the ? xing date for ? st . We value coupon j by replicating each day’s contribution in terms of vanilla caplets/? oorlets, and then summing over all days ? st in the period. Let Fdig (t; ? st , K) be the value at date t of a digital ? oorlet on the ? oating rate L(? st ) with strike K. If the Libor rate L(? st ) is at or below the strike K, the digital ? oorlet pays 1 unit of currency on the end date ? end (? st ) of the k-month interval. Otherwise the digital pays nothing. So on the ? xing date ? f ix the payo? is known to be ? 1 if L(? st ) ? K , (2. 2) Fdig (? f ix ; ? st , K) = Z(? f ix ; ? end ) 0 otherwise We can replicate the range note’s payo? for date ? st by going long and short digitals struck at Rmax and Rmin . This yields, (2. 3) (2. 4) ? j Rf ix [Fdig (? f ix ; ? st , Rmax ) ? Fdig (? f ix ; ? st , Rmin )] Mj ? ?j Rf ix 1 = Z(? f ix ; ? end ) 0 Mj 3 if Rmin ? L(? st ) ? Rmax . otherwise This is the same payo? as the range note, except that the digitals pay o? on ? end (? st ) instead of tj . 2. 1. Hedging considerations. Before ? ing the date mismatch, we note that digital ? oorlets are considered vanilla instruments. This is because they can be replicated to arbitrary accuracy by a bullish spread of ? oorlets. Let F (t, ? st , K) be the value on date t of a standard ? oorlet with strike K on the ? oating + rate L(? st ). This ? oorlet pays ? [K ? L(? st )] on the end date ? end (? st ) of the k-m onth interval. So on the ? xing date, the payo? is known to be (2. 5a) F (? f ix ; ? st , K) = ? [K ? L(? st )] Z(? f ix ; ? end ). + Here, ? is the day count fraction of the period ? st to ? end , (2. 5b) ? = cvg(? st , ? end ). 1 ? oors struck at K + 1 ? nd short the same number struck 2 The bullish spread is constructed by going long at K ? 1 ?. This yields the payo? 2 (2. 6) which goes to the digital payo? as ? > 0. Clearly the value of the digital ? oorlet is the limit as ? > 0 of (2. 7a) Fcen (t; ? st , K, ? ) = ? 1  © F (t; ? st , K + 1 ? ) ? F (t; ? st , K ? 1 ? ) . 2 2 ? 1  © F (? f ix ; ? st , K + 1 ? ) ? F (? f ix ; ? st , K ? 1 ? ) 2 2 ? ? ? ? 1 ? 1 = Z(? f ix ; ? end ) K + 1 ? ? L(? st ) 2 ? ? ? 0 if K ? 1 ? < L(? st ) < K + 1 ? , 2 2 if K + 1 ? < L(? st ) 2 if L(? st ) < K ? 1 ? 2 Thus the bullish spread, and its limit, the digitial ? orlet, are directly determined by the market prices of vanilla ? oors on L(? st ). Digital ? oorlets may develop an unbounded ? - risk as the ? xing date is approached. To avoid this di? culty, most ? rms book, price, and hedge digital options as bullish ? oorlet spreads. I. e. , they book and hedge the digital ? oorlet as if it were the spread in eq. 2. 7a with ? set to 5bps or 10bps, depending on the aggressiveness of the ? rm. Alternatively, some banks choose to super-replicate or sub-replicate the digital, by booking and hedging it as (2. 7b) or (2. 7c) Fsub (t; ? st , K, ? ) = 1 {F (t; ? st , K) ? F (t; ? st , K ? ?)} Fsup (t; ? st , K, ? ) = 1 {F (t; ? st , K + ? ) ? F (t; ? st , K)} depending on which side they own. One should price accrual swaps in accordance with a desk’s policy for using central- or super- and sub-replicating payo? s for other digital caplets and ? oorlets. 2. 2. Handling the date mismatch. We re-write the coupon leg’s contribution from day ? st as ? ?j Rf ix Z(? f ix ; tj ) ? V (? f ix ; ? st ) = Z(? f ix ; ? end ) Mj Z(? f ix ; ? end ) ? 0 4 (2. 8) if Rmin ? L(? st ) ? Rmax otherwise . f(t,T) L(? ) tj-1 ? tj ? end T Fig. 2. 1. Date mismatch is corrected assuming only parallel shifts in the forward curve The ratio Z(? ix ; tj )/Z(? f ix ; ? end ) is the manifestation of the date mismatch. To handle this mismatch, we approximate the ratio by assuming that the yield curve makes only parallel shifts over the relevent interval. See ?gure 2. 1. So suppose we are at date t0 . Then we assume that (2. 9a) Z(? f ix ; tj ) Z(t0 ; tj ) ? [L(? st )? Lf (t0 ,? st )](tj en d ) = e Z(? f ix ; ? end ) Z(t0 ; ? end ) Z(t0 ; tj ) = {1 + [L(? st ) ? Lf (t0 , ? st )](? end ? tj ) +  ·  ·  · } . Z(t0 ; ? end ) Z(t0 ; ? st ) ? Z(t0 ; ? end ) + bs(? st ), ? Z(t0 ; ? end ) Here (2. 9b) Lf (t0 , ? st ) ? is the forward rate for the k-month period starting at ? t , as seen at the current date t0 , bs(? st ) is the ? oating rate’s basis spread, and (2. 9c) ? = cvg(? st , ? end ), is the day count fraction for ? st to ? end . Since L(? st ) = Lf (? f ix , ? st ) represents the ? oating rate which is actually ? xed on the ? xing date ? ex , 2. 9a just assumes that any change in the yield curve between tj and ? end is the same as the change Lf (? f ix , ? st ) ? Lf (t0 , ? st ) in the reference rate between the observation date t0 , and the ? xing date ? f ix . See ? gure 2. 1. We actually use a slightly di? erent approximation, (2. 10a) where (2. 10b) ? = ? end ? tj . ? end ? ? st Z(? ix ; tj ) Z(t0 ; tj ) 1 + L(? st ) ? Z(? f ix ; ? end ) Z(t0 ; ? end ) 1 + Lf (t0 , ? st ) We prefer this approximation because it is the only linear approximation which accounts for the day count basis correctly, is exact for both ? st = tj? 1 and ? st = tj , and is centerred around the current forward value for the range note. 5 Rfix Rmin L0 Rmax L(? ) Fig. 2. 2. E? ective contribution from a single day ? , after accounting for the date mis-match. With this approximation, the payo? from day ? st is ? 1 + L(? st ) (2. 11a) V (? f ix ; ? ) = A(t0 , ? st )Z(? f ix ; ? end ) 0 as seen at date t0 . Here the e? ctive notional is (2. 11b) A(t0 , ? st ) = if Rmin ? L(? st ) ? Rmax otherwise 1 ? j Rf ix Z(t0 ; tj ) . Mj Z(t0 ; ? end ) 1 + Lf (t0 , ? st ) We can replicate this digital-linear-digital payo? by using a combination of two digital ? oorlets and two standard ? oorlets. Consider the combination (2. 12) V (t; ? st ) ? A(t0 , ? st ) {(1 + Rmax )Fdig (t, ? st ; Rmax ) ? (1 + ? Rmin )Fdig (t, ? st ; Rmin ) F (t, ? st ; Rmax ) + ? F (t, ? st ; Rmi n ). Setting t to the ? xing date ? f ix shows that this combination matches the contribution from day ? st in eq. 2. 11a. Therefore, this formula gives the value of the contribution of day ? t for all earlier dates t0 ? t ? ? f ix as well. Alternatively, one can replicate the payo? as close as one wishes by going long and short ? oorlet spreads centerred around Rmax and Rmin . Consider the portfolio (2. 13a) A(t0 , ? st )  © ? V (t; ? st , ? ) = a1 (? st )F (t; ? st , Rmax + 1 ? ) ? a2 (? st )F (t; ? st , Rmax ? 1 ? ) 2 2 ? 1 ? a3 (? st )F (t; ? st , Rmin + 2 ? )+ a4 (? st )F (t; ? st , Rmin ? 1 ? ) 2 a1 (? st ) = 1 + (Rmax ? 1 ? ), 2 a3 (? st ) = 1 + (Rmin ? 1 ? ), 2 ? ? a2 (? st ) = 1 + (Rmax + 1 ? ), 2 a4 (? st ) = 1 + (Rmin + 1 ? ). 2 with (2. 13b) (2. 13c) Setting t to ? ix yields (2. 14) ? V = A(t0 , ? st )Z(? f ix ; ? end ) 0 if L(? st ) < Rmin ? 1 ? 2 1 + L(? st ) if Rmin + 1 ? < L(? st ) < Rmax ? 1 ? , 2 2 ? 0 if Rmax + 1 ? < L(? st ) 2 6 with linear ramps between Rmin ? 1 ? < L(? st ) < Rmin + 1 ? and Rmax ? 1 ? < L(? st ) < Rmax + 1 ?. As 2 2 2 2 above, most banks would choose to use the ? oorlet spreads (with ? being 5bps or 10bps) instead of using the more troublesome digitals. For a bank insisting on using exact digital options, one can take ? to be 0. 5bps to replicate the digital accurately.. We now just need to sum over all days ? t in period j and all periods j in the coupon leg, (2. 15) Vcpn (t) = n X This formula replicates the value of the range note in terms of vanilla ? oorlets. These ? oorlet prices should be obtained directly from the marketplace using market quotes for the implied volatilities at the relevent strikes. Of course the centerred spreads could be replaced by super-replicating or sub-replicating ? oorlet spreads, bringing the pricing in line with the bank’s policies. Finally, we need to value the funding leg of the accrual swap. For most accrual swaps, the funding leg ? ? pays ? oating plus a margin. Let th e funding leg dates be t0 , t1 , . . , tn . Then the funding leg payments are (2. 16) f ? ? cvg(ti? 1 , ti )[Ri lt + mi ]  ¤ A(t0 , ? st )  ©? 1 + (Rmax ? 1 ? ) F (t; ? st , Rmax + 1 ? ) 2 2 j=1 ? st =tj? 1 +1 ?  ¤ ? 1 + (Rmax + 1 ? ) F (t; ? st , Rmax ? 1 ? ) 2 2 ?  ¤ ? 1 + (Rmin ? 1 ? ) F (t; ? st , Rmin + 1 ? ) 2 2 ?  ¤ ? + 1 + (Rmin + 1 ? ) F (t; ? st , Rmin ? 1 ? ) . 2 2 tj X ? paid at ti , i = 1, 2, †¦ , n, ? f ? ? where Ri lt is the ? oating rate’s ? xing for the period ti? 1 < t < ti , and the mi is the margin. The value of the funding leg is just n ? X i=1 (2. 17a) Vf und (t) = ? ? ? cvg(ti? 1 , ti )(ri + mi )Z(t; ti ), ? ? where, by de? ition, ri is the forward value of the ? oating rate for period ti? 1 < t < ti : (2. 17b) ri = ? ? Z(t; ti? 1 ) ? Z(t; ti ) true + bs0 . + bs0 = ri i i ? ? ? cvg(ti? 1 , ti )Z(t; ti ) true is the true (cash) rate. This sum Here bs0 is the basis spread for the funding leg’s ? oating rate, and ri i collapses t o n ? X i=1 (2. 18a) Vf und (t) = Z(t; t0 ) ? Z(t; tn ) + ? ? ? ? cvg(ti? 1 , ti )(bs0 +mi )Z(t; ti ). i If we include only the funding leg payments for i = i0 to n, the value is ? (2. 18b) ? Vf und (t) = Z(t; ti0 ? 1 ) ? Z(t; tn ) + ? n ? X ? ? ? cvg(ti? 1 , ti )(bs0 +mi )Z(t; ti ). i i=i0 2. 2. 1. Pricing notes. Caplet/? oorlet prices are normally quoted in terms of Black vols. Suppose that on date t, a ? oorlet with ? xing date tf ix , start date ? st , end date ? end , and strike K has an implied vol of ? imp (K) ? ? imp (? st , K). Then its market price is (2. 19a) F (t, ? st , K) = ? Z(t; ? end ) {KN (d1 ) ? L(t, ? )N (d2 )} , 7 where (2. 19b) Here (2. 19c) d1,2 = log K/L(t, ? st )  ± 1 ? 2 (K)(tf ix ? t) 2 imp , v ? imp (K) tf ix ? t Z(t; ? st ) ? Z(t; ? end ) + bs(? st ) ? Z(t; ? end ) L(t, ? st ) = is ? oorlet’s forward rate as seen at date t. Today’s ? oorlet value is simply (2. 20a) where (2. 20b) d1,2 = log K/L0 (? st )  ± 1 ? (K)tf ix 2 imp , v ? imp (K) tf ix D(? st ) ? D(? end ) + bs(? st ). ?D(? end ) ? j Rf ix D(tj ) 1 . Mj D(? end ) 1 + L0 (? st ) F (0, ? st , K) = ? D(? end ) {KN (d1 ) ? L0 (? )N (d2 )} , and where today’s forward Libor rate is (2. 20c) L0 (? st ) = To obtain today’s price of the accrual swap, note that the e? ective notional for period j is (2. 21) A(0, ? st ) = as seem today. See 2. 11b. Putting this together with 2. 13a shows that today’s price is Vcpn (0) ? Vf und (0), where (2. 22a) Vcpn (0) = n X ? j Rf ix D(tj ) j=1 Mj  ¤ ?  ¤ ? 1 + (Rmax ? 1 ? ) B1 (? st ) ? 1 + (Rmax + 1 ? ) B2 (? st ) 2 2 ? [1 + L0 (? st )] ? t =tj? 1 +1  ¤ ?  ¤ ? 1 + (Rmin ? 1 ? ) B3 (? st ) ? 1 + (Rmin + 1 ? ) B4 (? st ) 2 2 ? , ? [1 + L0 (? st )] tj X n ? X i=1 (2. 22b) Vf und (0) = D(t0 ) ? D(tn ) + ? ? ? ? cvg(ti? 1 , ti )(bs0 +mi )D(ti ). i Here B? are Black’s formula at strikes around the boundaries: (2. 22c) (2. 22d) with (2. 22e) K1,2 = Rmax  ± 1 ? , 2 K3,4 = Rmin  ± 1 ?. 2 B? (? st ) = K? N (d? ) ? L0 (? st )N (d? ) 1 2 d? = 1,2 log K? /L0 (? st )  ± 1 ? 2 (K? )tf ix 2 imp v ? imp (K? ) tf ix Calculating the sum of each day’s contribution is very tedious. Normally, one calculates each day’s contribution for the current period and two or three months afterward. After that, one usually replaces the sum over dates ? with an integral, and samples the contribution from dates ? one week apart for the next year, and one month apart for subsequent years. 8 3. Callable accrual swaps. A callable accrual swap is an accrual swap in which the party paying the coupon leg has the right to cancel on any coupon date after a lock-out period expires. For example, a 10NC3 with 5 business days notice can be called on any coupon date, starting on the third anniversary, provided the appropriate notice is given 5 days before the coupon date. We will value the accrual swap from the viewpoint of the receiver, who would price the callable accrual swap as the full accrual swap (coupon leg minus funding leg) minus the Bermudan option to enter into the receiver accrual swap. So a 10NC3 cancellable quarterly accrual swap would be priced as the 10 year bullet quarterly receiver accrual swap minus the Bermudan option – with quarterly exercise dates starting in year 3 – to receive the remainder of the coupon leg and pay the remainder of the funding leg. Accordingly, here we price Bermudan options into receiver accrual swaps. Bermudan options on payer accrual swaps can be priced similarly. There are two key requirements in pricing Bermudan accrual swaps. First, as Rmin decreases and Rmax increases, the value of the Bermudan accrual swap should reduce to the value of an ordinary Bermudan swaption with strike Rf ix . Besides the obvious theoretical appeal, meeting this requirement allows one to hedge the callability of the accrual swap by selling an o? setting Bermudan swaption. This criterion requires using the same the interest rate model and calibration method for Bermudan accrual notes as would be used for Bermudan swaptions. Following standard practice, one would calibrate the Bermudan accrual note to the â€Å"diagonal swaptions† struck at the accrual note’s â€Å"e? ective strikes. † For example, a 10NC3 accrual swap which is callable quarterly starting in year 3 would be calibrated to the 3 into 7, the 3. 25 into 6. 75, †¦ , the i 8. 75 into 1. 25, and the 9 into 1 swaptions. The strike Ref f for each of these â€Å"reference swaptions† would be chosen so that for swaption i, (3. 1) value of the ? xed leg value of all accrual swap coupons j ? i = value of the ? oating leg value of the accrual swap’s funding leg ? i This usually results in strikes Ref f that are not too far from the money. In the preceding section we showed that each coupon of the accrual swap can be written as a combination of vanilla ? oorlets, and therefore the market value of each coupon is known exactly. The second requirement is that the valuation procedure should reproduce today’s m arket value of each coupon exactly. In fact, if there is a 25% chance of exercising into the accrual swap on or before the j th exercise date, the pricing methodology should yield 25% of the vega risk of the ? oorlets that make up the j th coupon payment. E? ectively this means that the pricing methodology needs to use the correct market volatilities for ? oorlets struck at Rmin and Rmax . This is a fairly sti? requirement, since we now need to match swaptions struck at i Ref f and ? oorlets struck at Rmin and Rmax . This is why callable range notes are considered heavily skew depedent products. 3. 1. Hull-White model. Meeting these requirements would seem to require using a model that is sophisticated enough to match the ? oorlet smiles exactly, as well as the diagonal swaption volatilities. Such a model would be complex, calibration would be di? ult, and most likely the procedure would yield unstable hedges. An alternative approach is to use a much simpler model to match the diagonal swaption prices, and then use â€Å"internal adjusters† to match the ? oorlet volatilities. Here we follow this approach, using the 1 factor linear Gauss Markov (LGM) model with internal adjusters to price Bermudan options on accrual swaps. Speci ? cally, we ? nd explicit formulas for the LGM model’s prices of standard ? oorlets. This enables us to compose the accrual swap â€Å"payo? s† (the value recieved at each node in the tree if the Bermudan is exercised) as a linear combination of the vanilla ? orlets. With the payo? s known, the Bermudan can be evaluated via a standard rollback. The last step is to note that the LGM model misprices the ? oorlets that make up the accrual swap coupons, and use internal adjusters to correct this mis-pricing. Internal adjusters can be used with other models, but the mathematics is more complex. 3. 1. 1. LGM. The 1 factor LGM model is exactly the Hull-White model expressed as an HJM model. The 1 factor LGM model has a single state variable x that determines the entire yield curve at any time t. 9 This model can be summarized in three equations. The ? st is the Martingale valuation formula: At any date t and state x, the value of any deal is given by the formula, Z V (t, x) V (T, X) (3. 2a) = p(t, x; T, X) dX for any T > t. N (t, x) N (T, X) Here p(t, x; T, X) is the probability that the state variable is in state X at date T , given that it is in state x at date t. For the LGM model, the transition density is Gaussian 2 1 e? (X? x) /2[? (T ) (t)] , p(t, x; T, X) = p 2? [? (T ) ? ?(t)] (3. 2b) with a variance of ? (T ) ? ?(t). The numeraire is (3. 2c) N (t, x) = 1 h(t)x+ 1 h2 (t)? (t) 2 , e D(t) for reasons that will soon become apparent. Without loss of generality, one sets x = 0 at t = 0, and today’s variance is zero: ? (0) = 0. The ratio (3. 3a) V (t, x) ? V (t, x) ? N (t, x) is usually called the reduced value of the deal. Since N (0, 0) = 1, today’s value coincides with today’s reduced value: (3. 3b) V (0, 0) ? V (0, 0) = V (0, 0) ? . N (0, 0) So we only have to work with reduced values to get today’s prices.. De? ne Z(t, x; T ) to be the value of a zero coupon bond with maturity T , as seen at t, x. It’s value can be found by substituting 1 for V (T, X) in the Martingale valuation formula. This yields (3. 4a) 1 2 Z(t, x; T ) ? Z(t, x; T ) ? = D(T )e? (T )x? 2 h (T )? (t) . N (t, x) Since the forward rates are de? ned through (3. 4b) Z(t, x; T ) ? e? T t f (t,x;T 0 )dT 0 , ? taking ? ?T log Z shows that the forward rates are (3. 4c) f (t, x; T ) = f0 (T ) + h0 (T )x + h0 (T )h(T )? (t). This last equation captures the LGM model in a nutshell. The curves h(T ) and ? (t) are model parameters that need to be set by calibration or by a priori reasoning. The above formula shows that at any date t, the forward rate curve is given by today’s forward rate curve f0 (T ) plus x times a second curve h0 (T ), where x is a Gaussian random variable with mean zero and variance ? (t). Thus h0 (T ) determines possible shapes of the forward curve and ? (t) determines the width of the distribution of forward curves. The last term h0 (T )h(T )? (t) is a much smaller convexity correction. 10 3. 1. 2. Vanilla prices under LGM. Let L(t, x; ? st ) be the forward value of the k month Libor rate for the period ? st to ? end , as seen at t, x. Regardless of model, the forward value of the Libor rate is given by (3. 5a) where (3. 5b) ? = cvg(? st , ? end ) L(t, x; ? st ) = Z(t, x; ? st ) ? Z(t, x; ? end ) + bs(? st ) = Ltrue (t, x; ? st ) + bs(? st ), ? Z(t, x; ? end ) is the day count fraction of the interval. Here Ltrue is the forward â€Å"true rate† for the interval and bs(? ) is the Libor rate’s basis spread for the period starting at ? . Let F (t, x; ? st , K) be the value at t, x of a ? oorlet with strike K on the Libor rate L(t, x; ? st ). On the ? xing date ? f ix the payo? is (3. 6) ?  ¤+ F (? f ix , xf ix ; ? st , K) = ? K ? L(? f ix , xf ix ; ? st ) Z(? f ix , xf ix ; ? end ), where xf ix is the state variable on the ? xing date. Substituting for L(? ex , xex ; ? st ), the payo? becomes (3. 7a)  · ? + F (? f ix , xf ix ; ? st , K) Z(? f ix , xf ix ; ? st ) Z(? f ix , xf ix ; ? end ) . = 1 + ? (K ? bs(? st )) ? N (? ix , xf ix ) N (? f ix , xf ix ) Z(? f ix , xf ix ; ? end ) Knowing the value of the ? oorlet on the ? xing date, we can use the Martingale valuation formula to ? nd the value on any earlier date t: Z 2 1 F (t, x; ? st , K) F (? f ix , xf ix ; ? st , K) e? (xf ix ? x) /2[? f ix ] =q dxf ix , (3. 7b) N (t, x) N (? f ix , xf ix ) 2? [? f ix ? ?] where ? f ix = ? (? f ix ) and ? = ? (t). Substituting the zero coupon bond formula 3. 4a and the payo? 3. 7a into the integral yields (3. 8a) where log (3. 8b) ? 1,2 =  µ 1 + ? (K ? bs) 1 + ? (L ? bs)  ¤ ?  ± 1 (hend ? hst )2 ? f ix ? ?(t) 2 q , (hend ? hst ) ? f ix ? (t)  ¶ F (t, x; ? st , K) = Z(t, x; ? end ) {[1 + ? (K ? bs)]N (? 1 ) ? [1 + ? (L ? bs)]N (? 2 )} , and where L ? L(t, x; ? st ) = (3. 8c)  µ  ¶ 1 Z(t, x; ? st ) ? 1 + bs(? st ) ? Z(t, x; ? end )  ¶  µ 1 Dst (hend ? hst )x? 1 (h2 ? h2 )? end st 2 = e ? 1 + bs(? st ) ? Dend 11 is the forward Libor rate for the period ? st to ? end , as seen at t, x. Here hst = h(? st ) and hend = h(? end ). For future reference, it is convenient to split o? the zero coupon bond value Z(t, x; ? end ). So de? ne the forwarded ? oorlet value by (3. 9) Ff (t, x; ? st , K) = F (t, x; ? st , K) Z(t, x; ? end ) = [1 + ? (K ? bs)]N (? 1 ) ? [1 + ? L(t, x; ? st ) ? bs)]N (? 2 ). Equations 3. 8a and 3. 9 are just Black’s formul as for the value of a European put option on a log normal asset, provided we identify (3. 10a) (3. 10b) (3. 10c) (3. 10d) 1 + ? (L ? bs) = asset’s forward value, 1 + ? (K ? bs) = strike, ? end = settlement date, and p ? f ix ? ? (hend ? hst ) v = ? = asset volatility, tf ix ? t where tf ix ? t is the time-to-exercise. One should not confuse ? , which is the ? oorlet’s â€Å"price volatility,† with the commonly quoted â€Å"rate volatility. † 3. 1. 3. Rollback. Obtaining the value of the Bermudan is straightforward, given the explicit formulas for the ? orlets, . Suppose that the LGM model has been calibrated, so the â€Å"model parameters† h(t) and ? (t) are known. (In Appendix A we show one popular calibration method). Let the Bermudan’s noti? cation dates be tex , tex+1 , . . . , tex . Suppose that if we exercise on date tex , we receive all coupon payments for the K k0 k0 k intervals k + 1, . . . , n and recieve all funding leg payments f or intervals ik , ik + 1, . . . , n. ? The rollback works by induction. Assume that in the previous rollback steps, we have calculated the reduced value (3. 11a) V + (tex , x) k = value at tex of all remaining exercises tex , tex . . . , tex k k+1 k+2 K N (tex , x) k at each x. We show how to take one more step backwards, ? nding the value which includes the exercise tex k at the preceding exercise date: (3. 11b) V + (tex , x) k? 1 = value at tex of all remaining exercises tex , tex , tex . . . . , tex . k? 1 k k+1 k+2 K N (tex , x) k? 1 Let Pk (x)/N (tex , x) be the (reduced) value of the payo? obtained if the Bermudan is exercised at tex . k k As seen at the exercise date tex the e? ective notional for date ? st is k (3. 12a) where we recall that (3. 12b) ? = ? end (? st ) ? tj , ? end (? st ) ? ? st ? = cvg(? st , ? end (? st )). 12 A(tex , x, ? t ) = k ?j Rf ix Z(tex , x; tj ) 1 k , Mj Z(tex , x; ? end ) 1 + Lf (tex , x; ? st ) k k Reconstructing the reduced value of the payo? (see equation 2. 15) yields (3. 13a) Pk (x) = N (tex , x) k n X ? j Rf ix Z(tex , x; tj ) k Mj N (tex , x) ? k tj X j=k+1 st =tj? 1 +1 ? 1 + (Rmax ? 1 ? ) 2 Ff (tex , x; ? st , Rmax + 1 ? ) k 2 1 + Lf (tex , x; ? st ) k ? ? 1 + (Rmax + 1 ? ) 2 Ff (tex , x; ? st , Rmax ? 1 ? ) k 2 1 + Lf (tex , x; ? st ) k 1 + (Rmin ? 1 ? ) 2 Ff (tex , x; ? st , Rmin + 1 ? ) k 2 1 + Lf (tex , x; ? st ) k 1 + (Rmin + 1 ? ) 2 + Ff (tex , x; ? st , Rmin ? 1 ? ) k 2 1 + Lf (tex , x; ? st ) k ? n ? X ? ? Z(tex , x, tik ? 1 ) ? Z(tex , x, tn ) Z(tex , x, ti ) k k k ? ? cvg(ti? 1 , ti )(bsi +mi ) ? ex , x) ex , x) . N (tk N (tk i=i +1 k ? This payo? includes only zero coupon bonds and ? oorlets, so we can calculate this reduced payo? explicitly using the previously derived formula 3. 9. The reduced valued including the kth exercise is clearly ? ? Pk (x) V + (tex , x) V (tex , x) k k = max , at each x. (3. 13b) N (tex , x) N (tex , x) N (tex , x) k k k Using the Martingale valuation formula we can â€Å"roll di? erences, trees, convolution, or direct integration to Z V + (tex , x) 1 k? 1 (3. 3c) =p N (tex , x) 2? [? k ? ? k? 1 ] k? 1 back† to the preceding exercise date by using ? nite compute the integral V (tex , X) ? (X? x)2 /2[? k k? 1 ] k dX e N (tex , X) k at each x. Here ? k = ? (tex ) and ? k? 1 = ? (tex ). k k? 1 At this point we have moved from tex to the preceding exercise date tex . We now repeat the procedure: k k? 1 at each x we t ake the max of V + (tex , x)/N (tex , x) and the payo? Pk? 1 (x)/N (tex , x) for tex , and then k? 1 k? 1 k? 1 k? 1 use the valuation formula to roll-back to the preceding exercise date tex , etc. Eventually we work our way k? 2 througn the ? rst exercise V (tex , x). Then today’s value is found by a ? nal integration: k0 Z V (tex , X) ? X 2 /2? V (0, 0) 1 k0 k0 dX. (3. 14) V (0, 0) = =p e N (0, 0) N (tex , X) 2 k0 k0 3. 2. Using internal adjusters. The above pricing methodology satis? es the ? rst criterion: Provided we use LGM (Hull-White) to price our Bermudan swaptions, and provided we use the same calibration method for accrual swaps as for Bermudan swaptions, the above procedure will yield prices that reduce to the Bermudan prices as Rmin goes to zero and Rmax becomes large. However the LGM model yields the following formulas for today’s values of the standard ? orlets: F (0, 0; ? st , K) = D(? end ) {[1 + ? (K ? bs)]N (? 1 ) ? [1 + ? (L0 ? bs)]N (? 2 )} log  µ  ¶ 1 + ? (K ? bs)  ± 1 ? 2 tf ix 2 mod 1 + ? (L0 ? bs) . v ? mod tf ix 13 (3. 15a) where (3. 15b) ?1,2 = Here (3. 15c) L0 = Dst ? Dend + bs(? st ) ? Dend is today’s forward value for the Libor rate, and (3. 15d) q ? mod = (hend ? hst ) ? f ix /tf ix 3. 2. 1. Obtaining the market vol. Floorlets are quoted in terms of the ordinary (rate) vol. Suppose the rate vol is quoted as ? imp (K). Then today’s market price of the ? oorlet is is the asset’s log normal volatility according to the LGM model. We did not calibrate the LGM model to these ? oorlets. It is virtually certain that matching today’s market prices for the ? oorlets will require using q an implied (price) volatility ? mkt which di? ers from ? mod = (hend ? hst ) ? f ix /tf ix . (3. 16a) where (3. 16b) Fmkt (? st , K) = ? D(? end ) {KN (d1 ) ? L0 N (d2 )} d1,2 = log K/L0  ± 1 ? 2 (K)tf ix 2 imp v ? imp (K) tf ix The price vol ? mkt is the volatility that equates the LGM ? oorlet value to this market value. It is de? ned implicitly by (3. 17a) with log (3. 17b) ? 1,2 =  µ  ¶ 1 + ? (K ? bs)  ± 1 ? 2 tf ix 2 mkt 1 + ? (L0 ? bs) v ? kt tf ix [1 + ? (K ? bs)]N (? 1 ) ? [1 + ? (L0 ? bs)]N (? 2 ) = ? KN (d1 ) ? ?L0 N (d2 ), (3. 17c) d1,2 = log K/L0  ± 1 ? 2 (K)tf ix 2 imp v ? imp (K) tf ix Equivalent vol techniques can be used to ? nd the price vol ? mkt (K) which corresponds to the market-quoted implied rate vol ? imp (K) : (3. 18) ? imp (K) = 1 + 5760 ? 4 t2 ix +  ·  ·  · 1+ imp f ? mkt (K) 1 2 1 4 2 24 ? mkt tf ix + 5760 ? mkt tf ix  µ log L0 /K  ¶ 1 + ? (L0 ? bs) 1 + ? (K ? bs) 1+ 1 2 24 ? imp tf ix log If this approximation is not su? ciently accurate, we can use a single Newton step to attain any reasonable accuracy. 14 igital floorlet value ? mod ? mkt L0/K Fig. 3. 1. Unadjusted and adjusted digital payo? L/K 3. 2. 2. Adjusting the price vol. The price vol ? mkt obtained from the market price will not match the q LGM model’s price vol ? mod = (hend ? hst ) ? f ix /tf ix . This is easily remedied using an internal adjuster. All one does is multiply the model volatility with the factor needed to bring it into line with the actual market volatility, and use this factor when calculating the payo? s. Speci? cally, in calculating each payo? Pk (x)/N (tex , x) in the rollback (see eq. 3. 13a), one makes the replacement k (3. 9) (3. 20) (hend ? hst ) q q ? mkt ? f ix ? ?(tex ) =? (hend ? hst ) ? f ix ? ?(t) k ? mod q p = 1 ? ?(tex )/? (tf ix )? mkt tf ix . k With the internal adjusters, the pricing methodology now satis? es the second criteria: it agrees with all the vanilla prices that make up the range note coupons. Essentially, all the adjuster does is to slightly â€Å"sharpen up† or â€Å"smear out† the digital ? oorlet’s payo? to match today’s value at L0 /K. This results in slightly positive or negative price corrections at various values of L/K, but these corrections average out to zero when averaged over all L/K. Making this volatility adjustment is vastly superior to the other commonly used adjustment method, which is to add in a ? ctitious â€Å"exercise fee† to match today’s coupon value. Adding a fee gives a positive or negative bias to the payo? for all L/K, even far from the money, where the payo? was certain to have been correct. Meeting the second criterion forced us to go outside the model. It is possible that there is a subtle arbitrage to our pricing methodology. (There may or may not be an arbitrage free model in which extra factors – positively or negatively correlated with x – enable us to obtain exactly these ? orlet prices while leaving our Gaussian rollback una? ected). However, not matching today’s price of the underlying accrual swap would be a direct and immediate arbitrage. 15 4. Range notes and callable range notes. In an accrual swap, the coupon leg is exchanged for a funding leg, which is normally a standard Libor leg plus a margin. U nlike a bond, there is no principle at risk. The only credit risk is for the di? erence in value between the coupon leg and the ? oating leg payments; even this di? erence is usually collateralized through various inter-dealer arrangements. Since swaps are indivisible, liquidity is not an issue: they can be unwound by transferring a payment of the accrual swap’s mark-to-market value. For these reasons, there is no detectable OAS in pricing accrual swaps. A range note is an actual bond which pays the coupon leg on top of the principle repayments; there is no funding leg. For these deals, the issuer’s credit-worthiness is a key concern. One needs to use an option adjusted spread (OAS) to obtain the extra discounting re? ecting the counterparty’s credit spread and liquidity. Here we analyze bullet range notes, both uncallable and callable. The coupons Cj of these notes are set by the number of days an index (usually Libor) sets in a speci? ed range, just like accrual swaps: ? tj X ? j Rf ix 1 if Rmin ? L(? st ) ? Rmax (4. 1a) Cj = , 0 otherwise Mj ? =t +1 st j? 1 where L(? st ) is k month Libor for the interval ? st to ? end (? st ), and where ? j and Mj are the day count fraction and the total number of days in the j th coupon interval tj? 1 to tj . In addition, these range notes repay the principle on the ? nal pay date, so the (bullet) range note payments are: (4. 1b) (4. 1c) Cj 1 + Cn paid on tj , paid on tn . j = 1, 2, . . . n ? 1, For callable range notes, let the noti? ation on dates be tex for k = k0 , k0 + 1, . . . , K ? 1, K with K < n. k Assume that if the range note is called on tex , then the strike price Kk is paid on coupon date tk and the k payments Cj are cancelled for j = k + 1, . . . , n. 4. 1. Modeling option adjusted spreads. Suppose a range note is issued by issuer A. ZA (t, x; T ) to be the value of a dollar paid by the note on date T , as seen at t, x. We assume that (4. 2) ZA (t, x; T ) = Z(t, x; T ) ? (T ) , ? (t) De? ne where Z(t, x; T ) is the value according to the Libor curve, and (4. 3) ? (? ) = DA (? ) . e D(? ) Here ? is the OAS of the range note. The choice of the discount curve DA (? ) depends on what we wish the OAS to measure. If one wishes to ? nd the range note’s value relative to the issuer’s other bonds, then one should use the issuer’s discount curve for DA (? ); the OAS then measures the note’s richness or cheapness compared to the other bonds of issuer A. If one wishes to ? nd the note’s value relative to its credit risk, then the OAS calculation should use the issuer’s â€Å"risky discount curve† or for the issuer’s credit rating’s risky discount curve for DA (? ). If one wishes to ? nd the absolute OAS, then one should use the swap market’s discount curve D(? , so that ? (? ) is just e . When valuing a non-callable range note, we are just determining which OAS ? is needed to match the current price. I. e. , the OAS needed to match the market’s idiosyncratic preference or adversion of the bond. When valuing a callable range note, we are ma king a much more powerful assumption. By assuming that the same ? can be used in evaluating the calls, we are assuming that (1) the issuer would re-issue the bonds if it could do so more cheaply, and (2) on each exercise date in the future, the issuer could issue debt at the same OAS that prevails on today’s bond. 16 4. 2. Non-callable range notes. Range note coupons are ? xed by Libor settings and other issuerindependent criteria. Thus the value of a range note is obtained by leaving the coupon calculations alone, and replacing the coupon’s discount factors D(tj ) with the bond-appropriate DA (tj )e tj : (4. 4a) VA (0) = n X j=1 ?j Rf ix DA (tj )e tj Mj  ¤ ?  ¤ ? 1 + (Rmax ? 1 ? ) B1 (? st ) ? 1 + (Rmax + 1 ? ) B2 (? st ) 2 2 ? [1 + L0 (? st )] ? st =tj? 1 +1  ¤ ?  ¤ ? 1 + (Rmin ? 1 ? ) B3 (? st ) ? 1 + (Rmin + 1 ? ) B4 (? st ) 2 2 ? ? [1 + L0 (? st )] +DA (tn )e tn . tj X Here the last term DA (tn )e n is the value of the notional repaid at maturity. As before, the B? are Black’s formulas, (4. 4b) B? (? st ) = Kj N (d? ) ? L0 (? st )N (d? ) 1 2 (4. 4c) d? = 1,2 log K? /L0 (? st )  ± 1 ? 2 (K? )tf ix 2 imp v ? imp (K? ) tf ix (4. 4d) K1,2 = Rmax  ± 1 ? , 2 K3,4 = Rmin  ± 1 ? , 2 and L0 (? ) is today’s forward rate: (4. 4e) Finally, (4. 4f) ? = ? end ? tj . ? en d ? ? st L0 (? st ) = D(? st ) ? D(? end ) ? D(? end ) 4. 3. Callable range notes. We price the callable range notes via the same Hull-White model as used to price the cancelable accrual swap. We just need to adjust the coupon discounting in the payo? function. Clearly the value of the callable range note is the value of the non-callable range note minus the value of the call: (4. 5) callable bullet Berm VA (0) = VA (0) ? VA (0). bullet Berm (0) is the today’s value of the non-callable range note in 4. 4a, and VA (0) is today’s value of Here VA the Bermudan option. This Bermudan option is valued using exactly the same rollback procedure as before, 17 except that now the payo? is (4. 6a) (4. 6b) Pk (x) = N (tex , x) k ? tj X st =tj? 1 +1 j=k+1 n X ? j Rf ix ZA (tex , x; tj ) k Mj N (tex , x) ? k 1 + (Rmax ? 1 ? ) 2 Ff (tex , x; ? st , Rmax + 1 ? ) k 2 1 + Lf (tex , x; ? st ) k ? ? + (Rmax + 1 ? ) 2 Ff (tex , x; ? st , Rmax ? 1 ? ) k 2 1 + Lf (tex , x; ? st ) k 1 + (Rmin ? 1 ? ) 2 Ff (tex , x; ? st , Rmin + 1 ? ) k 2 1 + Lf (tex , x; ? st ) k 1 + (Rmin + 1 ? ) 2 + Ff (tex , x; ? st , Rmin ? 1 ? ) k 2 1 + Lf (tex , x; ? st ) k ZA (tex , x, tn ) ZA (tex , x, tk ) k k + ? Kk ex , x) N (tk N (tex , x) k Here the bond speci? c reduced zero coupon bond value is (4. 6c) ex ex 1 2 ZA (tex , x, T ) D(tex ) k k = DA (T )e (T ? tk ) e? h(T )x? 2 h (T )? k , ex , x) N (tk DA (tex ) k ? the (adjusted) forwarded ? oorlet value is Ff (tex , x; ? st , K) = [1 + ? (K ? bs)]N (? 1 ) ? [1 + ? (L(tex , x; ? t ) ? bs)]N (? 2 ) k k log (4. 6d) ? 1,2 =  µ  ¶ 1 + ? (K ? bs)  ± 1 [1 ? ?(tex )/? (tf ix )]? 2 tf ix k mkt 2 1 + ? (L ? bs) p , v 1 ? ?(tex )/? (tf ix )? mkt tf ix k  ¶ Z(tex , x; ? st ) k ? 1 + bs(? st ) Z(tex , x; ? end ) k  ¶ (hend ? hst )x? 1 (h2 ? h2 )? ex end st k ? 1 + bs(? 2 e st ) 1 = ?  µ and the forward Libor value is (4. 6e) (4. 6f) L? L (tex , x; ? st ) k  µ Dst Dend 1 = ? The only remaining issue is calibration. For range notes, we should use constant mean reversion and calibrate along the diagonal, exactly as we did for the cancelable accrual swaps. We only need to specify the strikes of the reference swaptions. A good method is to transfer the basis spreads and margin to the coupon leg, and then match the ratio of the coupon leg to the ? oating leg. For exercise on date tk , this ratio yields (4. 7a) n X ?k = ? j Rf ix DA (tj )e (tj ? tk ) Mj Kk DA (tk ) j=k+1 (?  ¤ ?  ¤ 1 + (Rmax ? 1 ? ) B1 (? st ) ? 1 + (Rmax + 1 ? ) B1 (? st ) 2 2 ? [1 + L0 (? st )] ? st =tj? 1 +1 )  ¤ ?  ¤ ? 1 + (Rmin ? 1 ? ) B3 (? st ) ? 1 + (Rmin + 1 ? ) B3 (? st ) 2 2 ? 1 + Lf (tex , x; ? st ) k tj X + DA (tn )e (tn ? tk ) Kk DA (tk ) 18 As before, the Bj are dimensionless Black formulas, (4. 7b) B? (? st ) = K? N (d? ) ? L0 (? st )N (d? ) 1 2 d? = 1,2 log K? /L0 (? st )  ± 1 ? 2 (K? )tf ix 2 imp v ? imp (K? ) tf ix K3,4 = Rmin  ± 1 ? , 2 (4. 7c) (4. 7d) K1,2 = Rmax  ± 1 ? , 2 and L0 (? st ) is today’s forward rate: Appendix A. Calibrating the LGM model. The are several methods of calibrating the LGM model for pricing a Bermudan swaption. The most popular method is to choose a constant mean reversion ? , and then calibrate on the diagonal European swaptions making up the Bermudan. In the LGM model, a â€Å"constant mean reversion ? † means that the model function h(t) is given by (A. 1) h(t) = 1 ? e t . ? Usually the value of ? s selected from a table of values that are known to yield the correct market prices of liquid Bermudans; It is known empirically that the needed mean reversion parameters are very, very stable, changing little from year to year. ? 1M 3M 6M 1Y 3Y 5Y 7Y 10Y 1Y -1. 00% -0. 75% -0. 50% 0. 00% 0. 25% 0. 50% 1. 00% 1. 50% 2Y -0. 50% -0. 25% 0. 00% 0. 25% 0. 50% 1. 00% 1. 25% 1. 50% 3Y -0. 25% 0. 00% 0. 25% 0. 50% 1. 00% 1. 25% 1. 50% 1. 75% 4Y -0. 25% 0. 00% 0. 25% 0. 50% 1. 00% 1. 25% 1. 50% 1. 75% 5Y -0. 25% 0. 00% 0. 25% 0. 50% 1. 00% 1. 25% 1. 50% 1. 75% 7Y -0. 25% 0. 00% 0. 25% 0. 50% 1. 00% 1. 25% 1. 50% 1. 75% 10Y -0. 25% 0. 0% 0. 25% 0. 50% 1. 00% 1. 25% 1. 50% 1. 75% Table A. 1 Mean reverssion ? for Bermudan swaptions. Rows are time-to-? rst exercise; columns are tenor of the longest underlying swap obtained upon exercise. With h(t) known, we only need determine ? (t) by calibrating to European swaptions. Consider a European swaption with noti? cation date tex . Suppose that if one exercises the option, one recieves a ? xed leg worth (A. 2a) Vf ix (t, x) = n X i=1 Rf ix cvg(ti? 1 , ti , dcbf ix )Z(t, x; ti ), and pays a ? oating leg worth (A. 2b) Vf lt (t, x) = Z(t, x; t0 ) ? Z(t, x; tn ) + n X i=1 cvg(ti? 1 , ti , dcbf lt ) bsi Z(t, x; ti ). 9 Here cvg(ti? 1 , ti , dcbf ix ) and cvg(ti? 1 , ti , dcbf lt ) are the day count fraction s for interval i using the ? xed leg and ? oating leg day count bases. (For simplicity, we are cheating slightly by applying the ? oating leg’s basis spread at the frequency of the ? xed leg. Mea culpa). Adjusting the basis spread for the di? erence in the day count bases (A. 3) bsnew = i cvg(ti? 1 , ti , dcbf lt ) bsi cvg(ti? 1 , ti , dcbf ix ) allows us to write the value of the swap as (A. 4) Vswap (t, x) = Vf ix (t, x) ? Vf lt (t, x) n X = (Rf ix ? bsi ) cvg(ti? 1 , ti , dcbf ix )Z(t, x; ti ) + Z(t, x; tn ) ? Z(t, x; t0 ) i=1 Under the LGM model, today’s value of the swaption is (A. 5) 1 Vswptn (0, 0) = p 2 (tex ) Z e? xex /2? (tex ) 2 [Vswap (tex , xex )]+ dxex N (tex , xex ) Substituting the explicit formulas for the zero coupon bonds and working out the integral yields (A. 6a) n X (Rf ix ? bsi ) cvg(ti? 1 , ti , dcbf ix )D(ti )N Vswptn (0, 0) = where y is determined implicitly via (A. 6b) y + [h(ti ) ? h(t0 )] ? ex p ? ex i=1 A A ! ! y + [h(tn ) ? h(t0 )] ? ex y p ? D(t0 )N p , +D(tn )N ? ex ? ex A ! n X 2 1 (Rf ix ? bsi ) cvg(ti? 1 , ti , dcbf ix )e? [h(ti )? h(t0 )]y? 2 [h(ti )? h(t0 )] ? ex i=1 +D(tn )e? [h(tn )? h(t0 )]y? [h(tn )? h(t0 )] 1 2 ? ex = D(t0 ). The values of h(t) are known for all t, so the only unknown parameter in this price is ? (tex ). One can show that the value of the swaption is an increasing function of ? (tex ), so there is exactly one ? (tex ) which matches the LGM value of the swaption to its market price. This solution is easily found via a global Newton iteration. T o price a Bermudan swaption, one typcially calibrates on the component Europeans. For, say, a 10NC3 Bermudan swaption struck at 8. 2% and callable quarterly, one would calibrate to the 3 into 7 swaption struck at 8. 2%, the 3. 25 into 6. 5 swaption struck at 8. 2%, †¦ , then 8. 75 into 1. 25 swaption struck at 8. 25%, and ? nally the 9 into 1 swaption struck at 8. 2%. Calibrating each swaption gives the value of ? (t) on the swaption’s exercise date. One generally uses piecewise linear interpolation to obtain ? (t) at dates between the exercise dates. The remaining problem is to pick the strike of the reference swaptions. A good method is to transfer the basis spreads and margin to the coupon leg, and then match the ratio of the coupon leg to the funding leg to the equivalent ratio for a swaption. For the exercise on date tk , this ratio is de? ed to be 20 n X ? j D(tj ) (A. 7a) ? k = Mj D(tk ) ? j=k+1 D(tn ) X D(ti ) + cvg(ti? 1 , ti )(bs0 +mi ) ? i D(tk ) i=1 D(tk ) n  ¤ ?  ¤ 1 + (Rmax ? 1 ? ) B1 (? st ) ? 1 + (Rmax + 1 ? ) B2 (? st ) 2 2 ? [1 + L0 (? st )] st =tj? 1 +1  ¤ ?  ¤ ? 1 + (Rmin ? 1 ? ) B3 (? st ) ? 1 + (Rmin + 1 ? ) B4 (? st ) 2 2 ? ? [1 + L0 (? st )] tj X ? where B? are Black’s formula at strikes around the boundaries: (A. 7b) B? (? st ) = ? D(? end ) {K? N (d? ) ? L0 (? st )N (d? )} 1 2 d? = 1,2 log K? /L0 (? st )  ± 1 ? 2 (K? )tf ix 2 imp v ? imp (K? ) tf ix (A. 7c) with (A. 7d) K1,2 = Rmax  ± 1 ? , 2 K3,4 = Rmin  ± 1 ?. 2 This is to be matched to the swaption whose swap starts on tk and ends on tn , with the strike Rf ix chosen so that the equivalent ratio matches the ? k de? ned above: (A. 7e) ? k = n X i=k+1 (Rf ix ? bsi ) cvg(ti? 1 , ti , dcbf ix ) D(ti ) D(tn ) + D(tk ) D(tk ) The above methodology works well for deals that are similar to bullet swaptions. For some exotics, such as amortizing deals or zero coupon callables, one may wish to choose both the tenor of the and the strike of the reference swaptions. This allows one to match the exotic deal’s duration as well as its moneyness. Appendix B. Floating rate accrual notes. 21