Visual Literacy in Business Essay Example | Topics and Well Written Essays - 250 words - 4

Visual Literacy in Business - Essay Example The primary purpose of this literature is to win the interest of people to buy this idea or influence them to watch the whole movie (Lopate, 2006). For instance, the trailer for the movie â€Å"That Sugar† is to motivate to try to know the contents in the documentary in detail. That will enable viewer make a reliable decision as far as sugar concerned. The motion pictures are more influential as compared to images and still photos in that; the motion pictures show the exact flow of the movie thus bringing out the theme in the movie apparently. Therefore, motion pictures are more convincing because they bring a real life scenario. However, the motion picture is the ones that can effectively communicate the flow of the movie as well as the themes in the film. Moreover, motion pictures will aid in quick introduction of the main characters to the viewer, thus generating a desire to watch the entire film. Finally, the trainer also enables an individual to know what kind of movie to watch. That is the trailer will communicate to the viewer the whole about of the movie thus avoiding at an early stage or redevelops the desire to watch it. Therefore, the trailer is an advertising tool that producers use to catch the attention and desire of the viewers to the

Reading Aloud Essay Example for Free

Reading Aloud Essay 1. 0 Introduction More than half of our lives today concern about reading ability. Through reading people are being introduce to values and knowledge. People usually read to get a job, as a hobby or to fulfill their dream. Reading always intertwine with literacy and the experiences that one felt via reading often different with each other. Different people have different understanding on text read. Reading materials can be magazines, newspapers, books, research articles, journals and other written form of texts. As long as people can write there will be always a text to read. Moreover, one academic skill that is especially important for school success is reading proficiency (Bernhardt, 2005). For children to succeed academically it is essential that they develop the foundational reading skills that will allow them to obtain knowledge fluidly through text and increased opportunities for learning (Joseph, 2006). Thus, reading proficiency is a fundamental skill which will help students to engage with new input or knowledge. However not all proficient readers have the ability to read fluently. According to Hudson (2009), reading proficiency constitutes of reading for fluency and reading for comprehension: reading for fluency deals with the accurate reading behavior which deals with reading with correct pronunciation. Whereas reading for comprehension deals with reading for understanding a text. Reading fluency is the most important component in reading skills and the numbers of students who can acquire a good reading fluency is not great (Schatschneider, 2006). Most curriculum in schools assumes that all students are literate and they can accumulate knowledge via reading (Rasinski Hoffman, 2003). However not all readers can read a passage fluently even though they can comprehend the passage well (Baker, Smolkowski, Katz, Fien, Seeley, Kame’enui, et al. , 2008). Hence, for years reading fluency was the forgotten stepchild of the reading curriculum because teachers and reading scholars were more interested in moving students as quickly as possible into silent reading, not the level of expressiveness in oral reading (Rasinski et al., 2008). Profoundly according to Elena Lilles et al. (2008) if students struggle with reading fluency, they will consequently struggle with other academic areas. Serious reading fluency problems in school limits success in most academic tasks and promote academic exclusion from intellectually rewarding and challenging educational opportunities (Entwisle, Alexander Olson, 2004). Ellen, Ramp, Anderson Martin, (2007) ask if students capable of comprehending a passage, will they also capable to read the same passage with correct pronunciation? They also explain that if the students can achieve a good level in comprehending a text in English, it does not guarantee that the students can acquire an efficient level in oral reading. As a consequence, students will face problems in using English language orally as medium of communicative language in society. However, students who are struggling to read can be helped through monitoring their oral reading fluency through a suitable reading approach (Schatschneider, 2006). Daly III, Chafouleas Skinner, (2005) has come out with several reading approach. There are 1)reading aloud, where the students need to read aloud a passage given several time before being evaluate by teachers, 2) phrase drill error correction, where the error words are being repeated until the students acquire the phonic sound, 3) performance feedback, where the students need to give feedback on passage that they have read. 4) Listening while reading, where the students need to read the passage aloud in the class and the students who are listening will learn how to pronounce certain words. 5) Folding in flashcard instruction, where the students learn to pronounce a word through flash cards. In conclusion, students should be able to acquire reading proficiency both in fluency and comprehending a reading text. The consequences of lack in reading proficiency could affect their academic performance. As solution reading habits should be implement in school. Students should practice reading in order to prevent them from being a struggle readers. Students who can comprehend a reading text does not mean that they can read the text fluently with correct pronunciation. Reading problem especially in reading fluency can be improved through appropriate intervention which seem to be suitable with the students. Thus, students’ weaknesses in reading a text should be identified in order to enhance their reading proficiency especially in reading fluency 1. 1 Background of the study Malaysia is characterised by a multilingual society where its population is made up of people who come from various ethnic and linguistic backgrounds (Harison, 2010). Thus, Malaysia has different kind of races that use different kind of languages. Malay students will use Malay language to communicate, the Indian students will use Tamil language and the Chinese students will use Mandarin language. Most of the time they will read books that related to their languages because of the need to enhance their reading on their mother tongue (Abdul Rashid, Chew Muhammad Kamarul, 2006). Hence, due to this matter, reading in English might being neglected or being less focused in school. When teacher conducting an oral test to our students, the interference of their mother tongue occurs and that made our students become a struggle reader (Siti Norliana, Roszainora Muthusamy, 2009). Most of the reading activities in Malaysian Secondary Schools focus on understanding a comprehension text which later the knowledge that they comprehend will be used to answer the questions given (KBSM, 2001). Therefore, secondary students reading skills was not optimally being explored. Teachers will ask several students to read aloud and most of the time will be silent reading. Indeed, the students only built their knowledge but they cannot read fluently which most of them having problem related to reading fluency (Siti Norliana, Roszainora Muthusamy, 2009). In order to find a solution for the reading problem related to reading fluency, a precise study on how can we help our Malaysian secondary students become a fluent reader is important to carry out. Reading fluency like has been mention above focused on students’ speed of accurate reading (Hudson et al. , 2005). Thus, this research focused on how we can help struggle readers to enhance their reading fluency. This study used reading aloud approach where the students need to read passage given orally. Reading passages were chosen from the Malaysian secondary English Textbook as the reading materials. CBM was used to measure the accurate reading or the percentages of correct word read during the reading sessions. Accurate reading focuses on 1) words pronounce correctly, 2) words read incorrectly which consist errors of mispronunciations, substitutions, and omissions, 3) three second rule which the words will be counted as an error when the teacher help the students to pronounce it after they hesitate in pronouncing the words for three seconds. Exactly as the guidelines provided by Daly III, Chafouleas Skinner, (2005, p. 78). Thus, the result of this research sought to reveal on how far the reading will aloud approach can improve Malaysian secondary students reading fluency. 1. 2 Problem statement Models of World Englishes has been coined by Professor Braj B. Kachru on 1985 (Phillipson, 2008). This model explains how English widely spread and used worldwide. According to Kachru, B. B. , Kachru, Y. , Nelson, C. L. , (2006), there are three circle models which can classify English as native language (ENL), English as a second language (ESL), and English as foreign language (EFL). The three circles model are: The current sociolinguistic profile of English may be viewed in terms of three concentric circles . . . The inner Circle refers to traditional cultural and linguistic bases of English. The Outer Circle represents the institutionalised non- native varieties (ESL) in regions that have passed through extended periods of colonization . . . The Expanding Circle includes the regions where the performance varieties of the languages are used essentially in EFL contexts. (Kachru, B. B. , Kachru, Y. , Nelson, C. L. , 2006). Concisely, this model explains about three circles which roughly classify three different English learners worldwide. 1) The Inner Circle houses countries, like the United States, United Kingdom, Australia and so on, where English is traditionally the primary or native language (English as Native Language). 2) The Outer Circle comprises countries where English has a long history of institutionalized functions, usually owing to a colonial past, and is used intra-nationally among fellow citizens who are usually bilingual (English as Second Language). Finally, 3) The Expanding Circle consists of countries in which English has no special status, but may be taught as a foreign language (English as Foreign Language) (Rajadurai, 2010). Applied to Malaysia, our country has traditionally been assigned Outer Circle status due to the British colonization. (David Govindasamy 2003). Rajah stated that the independence of Malaya in 1957, however, saw a continuing change in attitude towards the English language, in favor of the Malay language. English continued to be a dominant language. (as cited in Lee Su Kim, Lee King Siong, Wong Azizah, 2010). Competence in English had become a crucial partition in Malaysian society after the independence, carving out for itself a role in the economical areas, in education and placing the society status or standard among Malaysian. (Lee Su Kim, Lee King Siong, Wong Azizah, 2010). In the Malaysian school context, where English is officially stated and taught as a second language, learning English as a second language (ESL) in class always poses many language and cultural obstacles (Melor et al,.2012). Thus, there is a widespread concern among educators about students who not having the ability to read or students who is struggling to read in English (Zulhilmi, 2005). In News Straits Times newspaper dated on 12th February 2006 reported a survey conducted by the National Union of the Teaching Profession (NUTP) on students’ ability to read in English among secondary schools students in Malaysia. From the 70 secondary schools population of 73,858 students were analyzed, and from the analysis there are 34,890 students who could not read in English. What is more shocking is that some of these students are in Form five and having had eleven years of schooling and learning English as a second language yet they failed to achieve the basic skills of reading in English. Thus, students’ reading ability in second language need to be developed so that they can become fluent readers. (Noorliza, 2006). In order for students to succeed in reading fluently the teacher need to focus on enhancing their reading fluency (Baker, 2008). Baker also claimed that acquiring fluency in reading can also be considered important because it is also a part of developing reading ability. One of the ways to help students in enhancing their reading fluency is through reading aloud approach. (Hale et al. , 2007). Thus, this research will show the insight of struggling readers enhancing their reading fluency through reading aloud approach. 1. 3 Rationale of the study Several models of reading development suggest that reading fluency is a one of the most important components of effective reading ( Kuhn Stahl, 2003). Normally, students who are struggling to read will take longer time to develop their reading fluency. Due to this subject matter using reading aloud strategy seem to be an effective and convenient way to help our struggling readers. (Compton, Fuchs, D. , Fuchs, L. S. , Bryant, 2006) Rasinski Padak (2008) claims that reading aloud approach should be an instructional routine in all classrooms, including those for student who experience difficulty in reading. Another advantage of reading aloud, especially for those who find reading difficult, is that it familiarizes students with the style and form of written language. It also provides students with a model of what fluent reading should sound like. (Goering Baker, 2010). Furthermore, Nurazila et al. , (2011) claim that the use of reading aloud approach is less being attempt to be studied by our Malaysian researchers. Indeed, they also stated that this reading approach strengths and weaknesses still need to be investigate deeper. So, the rationales of this study open up the chance to enhance students reading fluency using reading aloud approach. 1. 4 The Purpose and Objective of the study. The purpose of this study is to investigate the reading fluency development of our secondary school students using reading aloud approach. Henceforth, the objectives of this study are: 1. To identify whether reading aloud approach can improve students reading fluency or not 2. To identify whether reading aloud approach has a positive influence on the subjects’ percentages of correct words read during the treatment was given. 3. To identify students common errors in their reading session. 1. 5 Research questions Thus, research questions of this study will be: 1. To what extent does reading aloud approach improve students’ reading fluency? 2. To what extent does reading aloud approach influences the subjects’ percentages of correct words read in each reading sessions during the treatment was given? 3. To what extent does students make errors in their reading sessions? 1. 6 Significance of the study 1. 6. 1. Society level At the society level, this study can help in developing number of peoples who can use English as a second language fluently which can cater jobs requirement as English is being used as second language in Malaysia. This study will also help the new generation equip with proper level English proficiency that can help them face the globalization era which English is widely being used as universal language. 1. 6. 2. School level. This study will help school to improve their students reading abilities in order to gain better result in major examination in English subject. It is also will give an idea on how to save struggle readers in the schools. 1. 6. 3. Teacher At the pedagogical level, this study can assist the teachers in solving problems in reading fluency among the students. Students who are struggling readers could be helped through the approach used in this research which later could help teachers to easily conduct the lesson of the day without facing any problem in teaching and learning session. 1. 6. 4. Students. This research might help the student in enhancing their reading proficiency. Students not only competent in reading to gain comprehension and knowledge but also can convey the knowledge fluently through oral reading using accurate pronunciation, intonation and reading rate. 1. 7 Limitation of the study 1. 7. 1. Time limitation In order to carry out this research, a consistent time management should be taken care of. This study has to be conducted 4 times a week, for 5 weeks. So, a proper supervision of time is needed. However, in school the periodic timetable given and the school’s programs will interrupt the suggested period of the research. 1. 8 Definition of terms 1. 9. 1 Reading fluency Reading fluency is often defined as â€Å"the ability to read rapidly with ease and accuracy and to read with appropriate expression and phrasing. (Grabe, 2008). In this study, reading fluency refers to accurately read the words in one minute without making errors in the reading sessions. 1. 9. 2. 1 Accurate reading. According to Fletcher, Francis, Morris Lyon (2005) accurate reading is the ability to recognize word, how to sound a words which involve the process of pronouncing words correctly with the correct pronunciation. In this research, accurate reading refers to reading a passage accurately without making errors in reading. The errors consist of mispronunciation, substitutions and omission, and three second rules which the words will be counted as an error when the teacher help the students to pronounce it after they hesitate in pronouncing the words for three seconds. 1. 9. 2 Reading aloud approach Rasinski (2003) defined reading aloud as a process of sounding the words in written forms loudly with appropriate facial expression, rhythm and use the correct punctuation marks in the texts read. In this study reading aloud means students need to read passage given orally in front of teacher. Teacher will assist the students through monitoring the students’ reading. Students will be given chances to read the passages repeatedly before the students’ progress in reading were taken. 1. 9. 3 Curriculum Based Measurement Curriculum-based measurement (CBM) is a standardize format for assessing reading fluency in one minute time. (Daly III, Chafouleas Skinner, 2005). In this study, CBM refers to students reading progress in one minute. CBM will show the level of students’ reading fluency gain through reading treatment given for the students. 1. 9. 4 Struggling readers Struggling readers refers to students who fail to recognize words, fail to comprehend various types of text, have little motivation to read and spend less time in reading. (Chard, Vaughn Tyler, 2002). In this research struggling readers refers to students who fail to pronounce a word with correct intonation using appropriate reading speed and students who can comprehend the texts but having difficulties in sounding the words in a text.

Dayan During His Military Career History Essay

Dayan During His Military Career History Essay 1. Moshe Dayan was a well-known Israeli military leader and politician. He was born in May 1915 in Degania near the Sea of Galilee in Palestine which was a part of the Ottoman Empire. Dayan was the youngest son of Shmuel and Dvorah. With the beginning of his life Dayan joined Haganah the Jewish military organisation against Arab attacks when he was 14. He joined the Palestine Supernumerary Police in 1938 and became sergeant then he was imprisoned by British in Acre  prison in 1939 with another forty two of his subordinates due to maintaining quantity of illegal rifles. They were released in 1941 after Chaim Weizmanns (first President of Israel) investigation in London then he joined British Army as an officer. During World War II in Syria-Lebanon Campaign Dayan was wounded and he lost his left eye due to a rifle shot fired by a sniper from quite a few hundred yards away, due to the nature of wound he could not use artificial eye. Thereafter he dressed in a black eye patch. 2. Key appointments of Dayan during his military career were, Haganah  General Staff working on Arab affairs. The first Commander of the  89th Armoured Battalion. Military Commander of Jewish controlled areas in  Jerusalem. October 1949 he was promoted to the rank of Major General and appointed as the Commander Southern Command. In 1952 he was appointed to the Operational Commander of the Northern Command. Head of Operations General Branch. Appointed as Chief of Staff in December 1953. 3. Key appointments of Dayan during his political career were: Minister of Agriculture. Minister of Defense.   Foreign Minister. 4. During the period of Minister of Defence Dayan conducted several major operations. They were Six Day War in 1967 and Yom Kippur War in 1973. 5. Then as the Foreign Minister he was the key person to implement the  Camp David Accords, a peace agreement with Egypt. 6.   In 1981 Dayan formed a new political party called  Telem. During the 1981 election Telem party won two seats but countrys greatest military and political leader, Israels legacy or legendary hero closed his eye shortly due to a serious heart attack In Tel Aviv. AIM 7. The aim of this presentation is to study and analysis leadership qualities of Moshe Dayan the legendary hero of Israel. SEQUENCE 8. This presentation unfolds under following sequence. Military career. Political career. Leadership qualities. Comparison with his Counterparts Conclusion. MILITARY CAREER 9. When he was only 14 years old, Dayan joined the  Haganah, an underground organization that defended Jewish settlements from Arab attacks. In 1936, Sergeant Dayan served with several regiments when the British in charge of Palestine authorized an attachment of the Haganah. Dayan gained command of one of the Mobile Guards of the Jewish Settlement Police in 1937. By 1938, he had risen to be an instructor, training Sergeant for the Auxiliary Force. During the riots in 1936 to 1939,  he served with the special police force in the Jezreel Valley and Galilee. 10. When the British banned the  Haganah in 1939, Dayan was arrested and imprisoned for two years. Upon his release in 1941, Dayan joined the British army, where he served with the forces that liberated Lebanon and Syria from Vichy France during World War II. Dayan was wounded in battle in Lebanon and lost his left eye. He began to wear the black eye patch that later became his identity. He remained active in the  Haganah until 1948. 11. War of independence  began when he commanded the defense of Jewish settlements in the Jordan Valley as a major in 1948. Later he commanded the battalion that attacked the city of Lydda and helped to halt Egyptian forces on the southern front. In August 1948, he promoted to the rank of Lieutenant Colonel and he was appointed commander of the Jerusalem  front. In 1949, he participated in ceasefire  talks with Jordanian officials in Rhodes. By the conclusion of the conflict in 1949, Dayan wore the rank of Major General and became in charge of the Southern Command at Beersheba. Dayans military proficiency allowed him to rise to the appointment of chief of operations at General Headquarters in 1952. 12. During the post war years, Dayan pioneered to organize a professional Israeli Defence Force (IDF) in 1953 and he became the Chief of Staff of the IDF. In 1956, during the Sinai campaign Dayan defeated the Egyptians in eight short days. In Israel and around the world, the Black Eye Patched General became the symbol of Jewish military proficiency. 13. Dayans skills in training and his aggressiveness and flexibility on the battlefield made the IDF one of the worlds most efficient and effective fighting forces of all time. In 1958, Moshe Dayan retired from the Army. WITH BRITISH ARMY 14. During his tenure with British army, he served with the forces that liberated Lebanon and Syria from Vichy France during World War II. He practiced the experience he gains from the past especially the guerrilla tactics. Later he cooperated with British Intelligence to set up a broadcasting network for clandestine operations behind enemy lines. That demonstrated his capability on the intelligent aspect which he gained confidence on his command in future. BATTLE OF LYDDA 15. In 1948 when he commanded the 89th armoured battalion that attacked the city of Lydda and helped to halt Egyptian forces on the southern front when he was Lieutenant Colonel. Afternoon of 11 July, Israels moved into Lydda. The raid was carried out on Dayans initiative without coordinating it with his commander. Using a column of jeeps led by a Marmon Harrington armored vehicle with a cannon taken from the Arab Legion the day before he launched the attack in daylight,  driving through the town from east to west machine gunning anything that moved, then along the Lydda-Ramle road firing at militia posts until they reached the train station in Ramle. Troops faced heavy fire from the Arab Legion in the police stations in Lydda and on the Lydda-Ramle road. 16. The raid lasted 47 minutes, leaving 100 to 150 Arabs dead. Six died and 21 were wounded on the Israeli side. The high casualty rate was caused by confusion over which Dayans troops were. The IDF was led by an armored car seized from the Arab Legion. Residents may have believed the Arab Legion had arrived, only to encounter Dayans forces shooting at everything as they ran from their homes. Dayan shows his leadership qualities of courage and initiative during this campaign. CEASEFIRE TALKS WITH JORDAN 17. In 1949, he participated in ceasefire  talks with Jordanian officials at Rhodes. Dayan served on a commission held in Rhodes which had assembled to try to work out a settlement between the Jewish and the Arabs. Between 1949 and 1950, he held secret talks with King Abdullah of Jordan. The King was one of the most influential Arabs in the region and his input and support was vital if the area was to become peaceful as opposed to a hotbed of Malcontents. However, at these meetings, Dayan proved to be a tough negotiator and refused to compromise. As a result, nothing came out of these meetings that would lead to stability in the Middle East. AS THE CHIEF OF STAFF 18. Dayan became the Chief of Staff of the IDF, and the entire Israeli military began to take on his personality. Dayan carried out a major reorganization of the Israeli army, which included: a. Raising the Intelligence and Training Branches of the Israeli Army. b. Surrendering the activities of stores and procurement to the civilian Ministry of Defense. c. Revamping the mobilization scheme and ensuring earmarking for adequate equipment. d. Starting a military academy for officers of the rank of major and above. e. Emphasized strike forces (Air Force, Armour) and on training of Commando battalions. f. Developed a youth wing for military training. 19. This is where he highlighted his great qualities of leadership of sound knowledge, planning capability and organizing ability. SINAI CAMPAIGN 20. Israeli units parachuted into the eastern approaches of the Mitla Pass near the Suez Canal on 29 October 1956. It was a political objective rather than tactical or strategic objective. The action provided the pretext for a French and British ultimatum to Israel and Egypt, calling on both sides to cease hostilities and withdraw from the Canal area. For diversionary reasons, Israeli forces also advanced on southern and central axes. 21. The following day, October 30, Britain and France issued the planned ultimatum, but to no effect, as heavy fighting between Egyptian and Israeli units persisted. In a swift, sweeping operation of 100 hours, under the leadership of then Chief of the General Staff, Moshe Dayan, the entire Sinai peninsula fell into Israeli hands, at a cost of 231 soldiers killed. In this stage he practiced his initiative much more comprehensive manner as a real leader who took decision past where opponent never had a chance to reflect. 22. In Israel and around the world, the Black Eye Patched General became the symbol of Jewish military proficiency. As a custom, Dayan disliked on anything not directly related to combat readiness. He emphasized weapons marksmanship, advantages of use of terrain, and an overall aggressiveness. POLITICAL CAREER 23. The world of politics and government was not strange to Moshe Dayan because as chief of staff he carried parliamentary responsibilities for conduct of military affairs in large capacity. At the end of his term as chief of staff he shed off uniform and joined at the Hebrew university of Jerusalem as a student in the political science faculty for period of two year. It was helped him to make better foundation to approach political field in perfect way. With that foundation he joined Israel`s labor party and elected and joined as a prestige member of Knesset (parliament) for Mapai area on 3 November 1959. AS AGRICULTURAL MINISTER 24. Dayan was appointed as minister of agriculture in the government of David Ben Gurion from 1959 unit 1964. This subject was not new field to him because he born and brought up in a farming atmosphere field. The orchard, the cowshed, the season of planting and harvesting were deeply infused in his blood more than tanks, guns and fighting. With his inherent experience he was able to identify problems which were faced in the agriculture field. He found that farmers faced financial difficulties and technical problems due to low prices for products, high production cost and financial difficulties to buy new tool and machineries. He analysed and identified agriculture systems of other counties. By analyzing, he was able to establish a planning authority, production and marketing council for each branch of agriculture. He made regional offices throughout the country where local farmers could receive agriculture guidance and services. AS MINISTER OF DEFENCE 25. Dayans reputation as an effective leader grew when he was appointed Minister of Defense under Levi Eshkol just in time for the Six-Day War in 1967 against Egypt, Jordan and Syria. During Yom Kippur war his actions was critically condemned by people of Israel due to huge frailer of Israel military force. The nations lack of preparation was blamed on Defense Minister Dayan and an outraged public demanded his resignation. This was caused him to give resignation to Meir in 1974 and he left his appointment. SIX DAY WAR IN 1967 26. The Six-Day War was initiated by  General Moshe Dayan as the Israelis Defence Minister. Although General Dayan did not take part in most of the planning before the Six-Day War of June 1967, his appointment contributed to the Israeli success. When the Syrians were shelling Israeli villages in Upper Galilee, Dayan was the one who made the decision to launch a full-scale attack against the Syrians. Rather than wait to be attacked, the Israelis launched a hugely successful military campaign against its perceived enemies. The air forces of Egypt, Jordan, Syria and Iraq were all destroyed in fifth June. By seventh of June, many Egyptian tanks had been destroyed in the Sinai Desert and Israeli forces reached the Suez Canal. On the same day, the whole of the west bank of the Jordan River had been cleared of Jordanian forces. The Golan Heights were captured from Syria and Israeli forces moved 30 miles into Syria itself. YOM KIPPUR WAR 1973 27. Egyptian President Anwar Sadat launched a surprise attack against Israel. On Yom Kippur, October 6, 1973, Egyptian armies crossed the Suez Canal, moved anti-aircraft missiles into the canal area, and waged war on Israel. Israeli losses were high and Israel had too short a supply of equipment to conduct a prolonged war. 28. On October 22, a cease-fire was declared, but the Israeli publics confidence had been severely shaken. Israel had been unprepared for the surprise attack and unable to repulse it quickly. The president of the Supreme Court set up a commission to investigate the performance of generals during the war. The commission recommended the resignation of the Chief of Staff, but reserved judgment on Dayan. The press and the public, however, condemned him. After attending a military funeral at which bereaved parents had called him a murderer of their sons, Dayan submitted his resignation to Meir in 1974. AS FOREIGN MINISTER 29. Year 1977, newly elected Prime Minister Menachem Begin gave him a second chance by offering him the post of Minister of Foreign Affairs. Although Dayan was from the opposition Labor Party, he accepted the appointment because he believed, I could significantly influence Israel`s moves towards achieving a peace arrangement with our neighboring Arab States and with the Palestinian inhabitants of Judea, Samaria and Gaza Strip. 30. In May 1977, Dayan began negotiating with the Egyptians. As lead negotiator, Dayan began with the premise of receiving an Arab acceptance of Israeli rule over Judea, Samaria and Gaza, in exchange for a return of Sinai to Egypt. He negotiated for 18 months, and held secret meetings with officials in India, Iran, England and Morocco. 31. With help from U.S. president and mediator Jimmy Carter, Dayan met with the Egyptians first at Leeds Castle and later at Camp David. Eventually, a peace agreement, the Camp David Accords, was drawn up and signed at 11 p.m. on Sunday September 17, 1978 32. In 1979, Dayan resigned as Foreign Minister. Dayan and Begin disagreed about the building of settlements in the territories and Dayan was frustrated by the fact that he was not leading the autonomy talks with the Palestinians. Dayan also felt that he was increasingly being bypassed on foreign policy issues. In 1981, he formed the Telem party, which advocated unilateral disengagement from the territories occupied in 1967. The party received only two mandates in the subsequent elections. LEADERSHIP QUALITIES INITIATIVE 33. Six day war against Egypt, Jordan and Syria is shows Moshe Dayans initiative significantly. When Syrians were shelling Israeli villages Dayan took the initiative to launch a full scale attack against Syrians. He was able to make it successful within very short time, giving deterrence to the Arab countries. It had highlighted the Moshe Dayans initiative and decision taking ability as an effective military leader. KNOWLEDGE 34. Moshe Dayan was a commander who had a fantastic knowledge about own and enemy. He had studied science at the Hebrew University in Jerusalem. He possessed perfect knowledge on his job all the time. Almost immediately the independence of Israel, the new state was attacked by a coalition of neighboring Arab states. Dayan put into practice his knowledge and what he had learnt fighting in World War II. 36. As Minister of Agriculture, he toured with the same zeal that he had as Defence Minister, resorting more to seeing to the implementation of his instruction rather than being confined to an office. Though the Prime Minister and the Cabinet were not too keen on using the expertise of Dayan, they were however forced by the mass to emplace him as the Defence Minister, due to his extensive knowledge on the subject. 37. Dayan was an asset to the Israeli Higher Command as he could discuss operational matters with them at their level and offer practical options. He stressed on the development of the intellectual standards of the officer corps of the IDF and took steps to provide them with a University Education on government expenses. COURAGE 38. His sense of proximity to death explains leading aspects of his character. Further his courage on battle field has been proven as the Chief of Staff. Within five years, from 1948 to 1953 he climbed up to Chief of Staff from the battalion commander. He believed that the appointment means causing the general staff to become revolutionary. When he took up Israel Army in 1951 it was fed up after the failure of Tel Aviv against Syrian Army. He shook up it and changed in to an aggressive army with his commencement of Chief of Staff. 39. Moshe Dayan saw no need for American guarantees of Israels security and strongly opposed Americas conditions, that Israel forswears territorial expansion and military retaliation. In an informal talk with the ambassadors to Washington, London, and Paris, Dayan described military retaliation as a life drug to the Israel Army. First, it obliged the Arab governments to take drastic measures to protect their borders. Second, and this was the essence, it enabled the Israeli government to maintain a high degree of tension in the country and Army. ENTHUSIASM 40. As a young man he was a guard in the village fields, later joined the Haganah. Dayan was arrested in 1939, together with 42 of his friends, for participating in an illegal Haganah commanders course, and was sentenced to ten years imprisonment. Released in 1941, he joined a British Army unit lost an eye in a battle with Vichy forces in Syria. With all those incidents his enthusiasm took him to long way as an exemplary military leader. SELF-CONFIDENCE 41. He suffered heavy criticism for not being prepared for the Arab attack during the Yom Kippur War in October 1973, Dayan became a controversial figure in Israel Although elected to the Ninth Knesset as a Lobour party member, he served as Foreign Minister in the beginning of the government until 1980-1981 elections he formed a new party, Telem, and represented it in the Tenth Knesset. Many Israelis regarded Dayan as their countrys greatest military and political leader. ABILITY TO COMMUNICATE 42. During the crisis preceding the Six Day War Dayan was appointed as Minister of Defense. After successfully conducting the War, Dayan administered the territories occupied by the Israel Army. He conducted a policy of liberal military government, opening the borders to trade and travel between the occupied territories and Arab countries. OTHER SKILLS 43. Dayan was the most fascinating and born leader who enjoyed more power during his leadership experience in both military and political. Although no one question about his overuse of power since he introduce totally new mechanism in military campaign as well in political campaign which helped to develop and ensure the security within the Middle East. Besides it has been helped by his capability of well handling of language which able to negotiate his modernizing ideas with the audience. 44. 1958 he was the Commander-in-Chief of the Israel Army. He successfully commanded the Israel forces throughout the Sinai Campaign of 1956. And also the entire Israeli military began to take on his personality. Dayan carried out a major reorganization of the Israeli army; this is where he highlighted his great qualities of leadership of sound knowledge, planning capability and as an organizer. Dayan ended his Army service in 1958 and in the fall of 1959 was elected to the Knesset as a member of the Mapai party, and became Minister of Agriculture. COMPARISON WITH HIS COUNTERPARTS EVENTS MOSHE DAYAN GAMAL ABDEL NASSER ANWAR SADAT HAFEZ AL-ASSAD Early life Enthusiasm and gain much experience which lead to become a strong leader Auare knowledge not the experience Aqure knowledge Gain profeciency and decentcy which help to become a peaseful leader Military carrier Courage and enthusiasm Gain courage in revolution 1952 Gain courage in revolution 1952 Proficiency in Air force carrier As the Chief of Staff Knowledge, modernizer and originator, Pride-Command Minister of Defence Initiative, planner Cooperation with Gamal, Knowledge As political leader Knowledge Knowledge and courage during Suez crisis 1956, modernizer in politics Initiative, peace negotiator Cunning, Knowledge Six day War Initiative, planner Failure in Initiative and assessing Failure in assessing Yom Kippur war Self-confidence, loss of Initiative Initiative, Enthusiasm 45. Moshe Dayan was an Israeli military warrior and politician who became a supporter for peace too. He was skilled in not only battle but also diplomacy. He played a key role in four wars and also helped to negotiate the historic Israeli-Egyptian peace treaty. Gamal Abdel Nasser was the president of Egypt in the same era as opponent of Dayan. He took the power over Egypt by revolution and became president. He was the only one leader in the region to go against western countries over the Suez crisis in 1956. Anwar Sadat came to power in Egypt with the death of Gamal in1970 who supported Gamal to come in to power. Hafez al-Assad was the president in Syria in that era and he was respect the peace negotiations rather than utilizes force to solve the Meddle-east crisis. KNOWLEDGE 46. Moshe Dayan as a commander had a sound knowledge about own and enemy. He possessed perfect knowledge of his job too. He gains that knowledge from his carrier from the childhood, when he joined with Haganah and from rest of carrier up to became officer in the Army. Then he exercised that knowledge during his period of Chief of staff, where he renovated the IDF and also as an Agriculture Minister where he introduced a new mechanism which help farmers to reach supervises closely for the advices. 47. When we consider the other counterparts, they were also had the same experience in their young life exempt Hafez, where they too able to acquire much knowledge. Gamal and Sadat both were worked in the Egypt Army together and had many experience their career. Latter they were utilized their possessed knowledge to become state leaders. Thereafter Gamal made many changes to economy in Egypt which country had lead towards development. MODERNIZER AND ORIGINATOR 48. Dayan was the most fascinating and origin leader who enjoyed more power during his leadership experience in both military and political career compared to other three leaders. He was always to introduce creative assets in any professional where he command or served without any reluctant. Although, Gamal Abdel Nasser was practice the quality of modernization during his period of presidency to develop the economic aspect in Egypt. COURAGE AND ENTHUSIASM 49. Dayan was the most courageous leader in that era in the region of middle-east. He proved that during his military carrier while he was conducting the operation Lidda and during Sinai campaign. And also as a Defence Minister during Six-day war. The following quotes which Dayan expresses clearly demonstrated his courage over the region: Let us not be afraid to see the hatred that consumes the lives of hundreds of thousands of Arabs who sit around us and wait for the moment when their hands will be able to reach our blood. 50. Gamal and Sadat too had quality of courage where they demonstrated during their revolution to become to power in Egypt. Although that, Hafez Al-Assad not showed much this quality during his carrier because he always respected the proficiency which lead to take peace rather than war in his life. SELF-CONFIDENCE AND INITIATIVE 51. The most powerful leadership quality possessed by Moshe Dayan in his career. While he was performing in military, he always practice this quality even his higher authority disparate. It was significantly demonstrated in Six-day war against Egypt, Syria and Jordan defeating other leaders initiative and assessing capabilities. Gamal and Sadat too possessed with the initiative which they collected from military carrier. Then they took it to practice during their revolution against the government existed in 1952. Compared to all above three leaders, Hafez had less experience on this aspect. PRIDE IN COMMAND 52. This is the leadership quality which Dayan was able to attract the most of the people in the region towards him. And also entire Army also followed him as role model. Comparing to Moshe Dayan other three leader never had this quality in their carrier. The following quote also emphasis his pride over command which he practiced throughout his life. Our American friends offer us money, arms, and advice. We take the money, we take the arms, and we decline the advice.   COMPARISION WITHIN THE COUNTRY AS A POLITICAL LEADER 53. Not only as a military leader but also as a political leader he succeeded. While he was performing as a Minister of Agriculture, introduce a new mechanism to enhance the field of agriculture in the Israel: establish a planning authority, production and marketing council for each branch of agriculture. He made regional offices throughout the country where local farmers could receive agriculture guidance and services. This was where he saws his sound leadership qualities in out of military scenario where he proved that leaders are always created by the military. 54. Comparatively to former agricultural ministers such as Kadish luz (1955-1959), Peretez Naftali (1952-1955), Levi Eshkol (1951-1952), who served in Israel, Moshe Dayan made brilliant magnificent contribution to enhance agriculture development during his period as agriculture minister. His experience, brilliant leader ship qualities and vast knowledge about the field paved way for systematical improvement in various field of agriculture. 55. Once he was given another chance by Menachem Begin to undertake as Foreign Minister, he commence his work believing that he could significantly influence Israels moves towards achieving a peace arrangement with their neighboring Arab States and with the Palestinian inhabitants of Judea and Samaria and the Gaza Strip. During this period he was able to get all other counter pert to the peace table and he could make others to think twice prior to take a decision against Israel. And it paved way for disparities in between his Arab opponents. 56. That differentiated the leadership qualities of Moshe Dayan from other contemporary leaders within the country and even from the region was significant. CONCLUSION 57. Moshe Dayan who was born to the world on May 20, 1915 where is not having a piece of land even nationality for his people, he was able to build a country called Israel, piece by piece. He gave a county to people who did not have country. He gave a nationality to people who fought for their identity. Moshe Dayan became one of Israels most famous men and found fame as a military leader associated with victories that were seemingly impossible within the  Middle East  conflicts. Dayan developed the aura of a military superman. 58. Throughout Moshe Dayans life as an Israel military and political leader number of leadership qualities can be identified and proved himself to nation long way from creating country for Jews even didnt had piece of land on their own. His courage, determination, knowledge, self confidence, enthusiasm, will power, integrity, loyalty, approachability build a country within another country, gave recognition to the nation Jews state Israel. Moshe Dayan was a good diplomat who believed peace, a hugely successful military leader who developed a legendary status. But he never forgot his ambition, once he stated his view on USA: Our American friends offer us money, arms, and advice. We take the money, we take the arms, and we decline the advice.   59. Dayans never forgot his nation, his country when liberating land from Arabs. He never forgot other nations in the world with keeping national strategy. Dayans career is probably unequalled in Israels short history. He successfully crossed over to politics and held a number of highly influential government posts before he left politics. Senior military figures had tried to do the same move from the military to politics but many have failed. 60. Moshe Dayan was a leader who is a leader of the sense of the word. He possessed several qualities through his whole life as one of the greatest leader in the history who became a legend in his own life time. He loved his enemy too. He always gave his warm hand for peace but with an eagle eye. On 16 October 1981 this great leader General Moshe Dayan left the world to Shamayim (heaven) in Tel Aviv.

Fear of Flying and Classical Conditioning Theory Essays -- Classical C

How Lauren may have learned of her Fear of Flying? How Lauren learned she had a fear in flying? Using the Classical Conditioning theory the possibilities could be endless. Classical conditioning in simple terms is the method in which one determines why and the cause of a condition as well as what has brought it about. There are many stimulus both conditioned and unconditioned that can cause fear or other problems, but the major reason for causes regarding the fear of flying has been mentioned in several articles regarding anxiety disorders. Fear of flying is created by the unconscious mind as a protective mechanism. When using the neutral stimulus explanation, Lauren may not have had a relevant response of interest. Lauren may have learned something or heard someone from her past that caused the continuous fear. Due to the facts in this case, there’s little information to provide us regarding Lauren. First we know she’s afraid to fly, but we have no further information regarding the condition that caused the fear or the circumstances to what led to this fear. The first step in Pavlov’s theory is trying to discover how Lauren’s fear came about, but without more information one can only speculate or guess how Lauren’s condition developed. Pavlov’s theory states several actions and read actions that could have caused Lauren’s Condition. The conditioning of the plane could be neutral stimulus, and the activities on the plane is the unconditioned stimulus. During condit...

The Destruction of Willy Loman in Arthur Millers Death of a Salesman :: Death Salesman essays

The Destruction of Willy Loman in Arthur Miller's Death of a Salesman  Ã‚  Ã‚  Ã‚  Ã‚   Willy Loman is a travelling salesman who has worked for the Wagner firm for 34 years. He is now 61 years old and his job has been taken off salary and put on commission. He has a family and he boasts to them that he is "vital in New England," but in fact he isn’t vital anywhere. Willy has many strong beliefs that he strives to achieve. He wants to own his own business and he wants to be "bigger than Uncle Charley" and especially he wants to be a great success and he tries to emulate Dave Singleman. He wishes to die the "Death of a Salesman" and have many buyers and salesmen mourn for him. He also tries to be a good father, and husband. However Willy’s aims in life have been useless as he hasn’t really achieved anything. He got fired by Howard, his sons are both failures and they abandoned him in a restaurant toilet. His relationship with his wife is plagued by his guilt for committing adultery. He has to borrow $50 a week from Charley. He can’t even keep his mind on one thing for a long time. He can’t drive a car. Willy gets so fed up with all of these things that he want’s to commit suicide and eventually, he does. This topic suggests that Willy’s deterioration occurs because the principals he believes in. To a large extent this is true. After 34 years of Willy’s life, he loses his job. To a normal person under normal circumstances, being retrenched is a time when you feel useless. But for Willy, since everything else is going wrong at the same time, he feels like a useless old man. Willy thought that just because he named his boss, that he would have a secure future with the company but as Charley said "them things don’t mean anything? You named him Howard, but you can’t sell that." Even though Willy wasn’t even getting paid a salary, Howard didn’t want him to even represent the company in case Willy "cracked up" again. Although Willy is mostly destroyed by his own ideals there are other things that destroy him as well, like Howard, Happy and Biff. Willy is emotionally destroyed when Howard fires him. Then, both of his sons disown and abandon him in Frank’s Chop House.

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