How Weather Conditions Affect Health and Well-Being in Singapore Essay

How Weather Conditions Affects Health and Well-Being Singapore is an island lying north of the Equator. Because it lies in the tropics, it receives sufficient sunlight, has broadly speaking uniform temperature and crush, high humidity and abundant rainfall. Singapores climate can be categorized into two main monsoon seasons. During the Northeast Monsoon season, from December to earlier March, northeast winds prevail. It is cloudy with frequent afternoon rain in the early(a) part of this season. From late March to May, winds and showers are present mostly during afternoon and early evening. The Southwest Monsoon season, from June to September, has more isolated showers in the late morning and early afternoon. From October to November, winds and breezes prevail, with showers and thunder in the late afternoon and early evening. The temperature in Singapore ranges from 31 deg C at maximum and 24 at minimum. The pressure fluctuates between 1012 and 1005hPa, average being 1009hPa. Re lative humidity drops from the high 90s in the early morning to 60% in the mid-afternoon. Average humidity is 84%. There is no distinct wet or dry season in Singapore. Maximum rainfall occurs in December and April while drier months include February and July. Sunlight is also present as Singapore lies near the Equator.Some health conditions can be affected by bear conditions such as pregnancy, arthritis, Fibromyalgia Syndrome, fatigue, muscle pain, migraines, sinus headaches, multiple Sclerosis, loss of coordination or balance and Raynauds phenomenon, where blood black market to fingers are disrupted and asthma. During pregnancy, conditions conditions like temperature and humidity changes lead to fluctuation in barometric pressure. This is dangerous as a abrupt... ...one on how tropical weather affects health as little data could be found online. There should also be more research done on how warmer weather can affect health and well-being.Pregnant people may be affected by sud den changes in barometric pressure. As a sudden drop of air pressure outside equals an increase of air pressure inside, the pressure increase can trigger labour. Hence hospitals should stabilize the air pressure to prevent this.Due to the change in time geographical zone of Singapore, natural circadian rhythms are affected and people may feel energetic in the late night and lethargic during the early morning. This is because the time zone is not synchronized with the natural time as Singapores time zone was synchronized with Beijing. This moved time forward and thus affected the natural circadian rhythm. Hence time should be shifted back to its original state.

Seventh Heaven :: Alice Hoffman Magical Realism Essays

Seventh HeavenSeventh Heaven by Alice Hoffman was published by G.P. Putnams Sons in 1991. Many characteristics of sorcerous realism were expressed in an excerpt from this novel. From reading this, I have learnt that magic appears to me as being real. My comprehending of this novel was more because or realism. Alice Hoffmans attitude in this book seemed to be that she set it as an example of wizardly realism and she made the lectors curious about what was going to happen next. Hoffman made this book pleasant so that indorsers would want to read this book. She had a good attitude toward what she was writing about and how the book was presented.The book consisted of many supernatural elements. Some of the charming elements in this book were that the crow in Mrs. Oliveras chimney might find a pack of matches, rub them the wrong way, and poof up goes the house in flames (17). I feel that this is something that would never happen. I think it is impossible for a crow to light a match and catch a house on fire. some other magical element was the fact that no one in the neighborhood could tell the houses apart when they were first built and carve out of the potato farms. (10). I dont think that many people would get confused about what house was theirs and what block that it was on. Everyone should know what is theirs.Realistic elements in this book stood out better than the magical elements because I felt that closely of the book seemed like everyday life. Realistic elements were that the guys in the neighborhood wanted to get together and renovate Mrs. Oliveras house and to swap it to someone instead of it setting there looking run down (18-19). I think that guys in the community would fix a house to cause to help the neighborhood look nice and to keep people from getting hurt.The purpose of the magical elements is to keep the reader tuned into the book. The author tries her best to make the reader experience textualization. She does this by making the book have plenty of realistic elements and throwing in magical element to make the experience different. Magical elements stood out to me the some were when the women in the neighborhood made a big deal about Nora being divorced and how Donna Durgin had never met a divorced person in her accurate life (82).Seventh Heaven Alice Hoffman Magical Realism EssaysSeventh HeavenSeventh Heaven by Alice Hoffman was published by G.P. Putnams Sons in 1991. Many characteristics of magical realism were expressed in an excerpt from this novel. From reading this, I have learnt that magic appears to me as being real. My comprehending of this novel was more because or realism. Alice Hoffmans attitude in this book seemed to be that she set it as an example of magical realism and she made the readers curious about what was going to happen next. Hoffman made this book pleasurable so that readers would want to read this book. She had a good attitude toward what she was writing about and how the book was presented.The book consisted of many magical elements. Some of the magical elements in this book were that the crow in Mrs. Oliveras chimney might find a pack of matches, rub them the wrong way, and poof up goes the house in flames (17). I feel that this is something that would never happen. I think it is impossible for a crow to light a match and catch a house on fire. some other magical element was the fact that no one in the neighborhood could tell the houses apart when they were first built and carve out of the potato farms. (10). I dont think that many people would get confused about what house was theirs and what block that it was on. Everyone should know what is theirs.Realistic elements in this book stood out better than the magical elements because I felt that most of the book seemed like everyday life. Realistic elements were that the guys in the neighborhood wanted to get together and renovate Mrs. Oliveras house and to remove it to someone instead of it setting there looking run down (18-19). I think that guys in the community would fix a house to savour to help the neighborhood look nice and to keep people from getting hurt.The purpose of the magical elements is to keep the reader tuned into the book. The author tries her best to make the reader experience textualization. She does this by making the book have plenty of realistic elements and throwing in magical element to make the experience different. Magical elements stood out to me the most were when the women in the neighborhood made a big deal about Nora being divorced and how Donna Durgin had never met a divorced person in her full life (82).

Connection of European Trials Essays -- Witch Witches Hunting Essays

Connection of European Trials The entrance hunting trials that took place in Early Modern Period occurred across the expanse of Europe, including in its believers groups of people ranging from kings and popes to peasants and servants. Some historians argue that the witch trials of the Early Modern Period where non connected that the trials where not related to each other and are not part of one large witch-hunt. Other historians argue that the witch hunting trials are related to each other by there similar characteristics. The Early-Modern witch trails where both related and separate entities that where born with the help of elite and common peoples belief in Catholic as well as Protestant locations including Geneva, Trier, Scotland, England, Bamberg, and Runchain. One thing that both the Catholics and the Protestants did have in common was the belief that the Bible was the word of God. Partly because of this belief both Protestant and Catholics participated in the prac tice of witch hunting. John Calvin began the Calvinist movement that began in Geneva Switzerland. Calvin lived from AD 1509-1564 during the Early Modern Period. During the Protestant Reformation Calvin wrote about witches stating, And Paul, after he has warned us that our struggle is not with flesh and blood, but with the princes of the air, with the powers of darkness, and spiritual wickedness Eph. 612.1 This passage that Calvin took from the Bible helps to illustrate the feelings of the new Protestant perform toward witches and witchcraft. The passage also shows that the Protestant church, like the Catholics, believe that witches are a threat and that something must be done to dispose of their goddamn and wicked presence. The demonic ... ...ty of Pennsylvania Press, 1972. The Confessions of the Chelmsford Witches of England. In Witchcraft in Europe 1100- 1700, modify by Alan Kors and Edward Peters. Philadelphia University of Pennsylvania Press, 1972. The Persecution in Scotland. In Witchcraft in Europe 1100-1700, edited by Alan Kors and Edward Peters. Philadelphia University of Pennsylvania Press, 1972. The Persecutions at Bamberg. In Witchcraft in Europe 1100-1700, edited by Alan Kors and Edward Peters. Philadelphia University of Pennsylvania Press, 1972. The Persecutions at Trier. In Witchcraft in Europe 1100-1700, edited by Alan Kors and Edward Peters. Philadelphia University of Pennsylvania Press, 1972. The Trial of Suzanne Gaudry. In Witchcraft in Europe 1100-1700, edited by Alan Kors and Edward Peters. Philadelphia University of Pennsylvania Press, 1972.

Saint Augustine Essay -- essays research papers

Saint Augustine was born on 354 CE in Tagaste, Africa.His given name was Aurelius Augustinus. His father wasPatricius, a pagan who was call Christian before hedied, and his mother was Monica, a baptized Christian withan influential role in the life of her son. Augustine isregarded as one of the most smart Christian theologiansand bishops of all time. His works and actions have left amajor imprint on the Church and its doctrine.As a boy, Augustine was not baptized and grew up in theRoman Empire. He studied under the local schoolmastersin Tagaste until he turned fifteen and move to continue hisstudies in Madaurus. From Madaurus, he moved toCarthage for advanced studies in rhetoric and law. It was inCarthage that he took a concubine and later had a sonnamed Adeodatus from her. It was in this period of his lifethat embraced Manichaeism, which is a intuitive feeling that one godis responsible for all good and another responsible for allevil. Augustines belief in Manichaeism prompted Monica,his mother, not to allow his entrance into the familys house.Even with her actions, she continued praying and hopingthat Augustine would find the Lord. After he ended hisstudies in Carthage, he became a teacher and wasconstantly on the move without blue Africa.Augustine stopped teaching and moved to Milan where hegained the position of Public Orator. In Milan, Augustinemet Ambrose, the Bishop of Milan. Augustine grew to loveAmbroses allegorical interpretations of the Bible and thisled to his appreciation and bare-assed reasonableness of theScripture. He also studied and learned to appreciatePlatos works and started linking a lot of his works into themeaning and messages in the Bible. Augustines family,including his mother, joined him in Milan. Her constantprayers for his conversion to Christianity and the strictethical demands of Ambrose made Augustinesappreciation of Platos work grew deeper. It made him nochoice, but to convert to Christianity. On east wind Sunday of387 CE in Milan, Augustine along with his son and hisfriend, Alypius, were baptized by Ambrose. After Augustines mother passed away, he traveledthroughout the Roman Empire. He wrote many of hisbooks on theology along the way. He had no aspirations ofpriesthood, but through a mere chance visit at Hippo inAfrica the bisho... ...howed that the Church is the union of all people intoChrist. He defined free will, Christian sacraments, andoriginal sin. His argument with the Donatists clarifiedChristian doctrines for further generations to come.Pelagianism was the final broil that Augustinehandled. They believed that Gods grace is not needed forsalvation, but only good works. Augustine fought thiscontroversy by explaining that grace was necessary forsalvation because without it, people would be even moresinful. The leader of Pelagianism, Pelagius, never met withAugustine, but as a result of this controversy, "Gods savinggrace" was clearly understood through Augustinesarguments.On Augu st 28, 430 CE in Hippo, Augustine died. He isregarded as one of the greatest and intelligent saints of theChurch. He clarified Church doctrines, establishedmonasteries for new priests, educated many into themeaning of Christianity and made Christianity humanisticwhich is nurtured by Gods love and grace. Brown, Peter. Augustine of Hippo. Berkeley University ofCalifornia Press, 1967.Possidius. The Life of Saint Augustine. VillanovaAugustinian Press, 1988.

Red Faction :: essays research papers

***************************29. Refuse Disposal Plant***************************Open the elevator access by hitting the button to the right. Headforward, and Gryphon entrust run up ahead to the incinerator controls.======================================================================== Hendrix This is Ultors main trash disposal facility. ================================================================================================================================================ Hendrix Thats an Ultor Personnel Suppression combot. I didnt == know they had any on Mars ================================================================================================================================================ Hendrix I cant mark off Orions group on my monitors anymore. ========================================================================While Gryphon takes the first left, go forward up the rage and throughthe door. Climb the ladder up ahead, then climb a second ladder to get to the roof. Youll be shot at by the storm as you climb, but justignore it and get up there. Hell have a tough time hurting you whileyoure up here. Follow the instructions and you shouldnt lose a singlepoint of health, despite the holes he might create with his rocket salads.As you look nigh the outer edge of the massive roof, youll seeseveral groups of ammo crates. Make your way around the outer edge asyou make your way to each one to find ammo, medkits, armor, and a rocket ballista. Make sure to snuggle up to each crate and jump on top of itto make sure you get every last report of ammo and equipment that youcan. Once youre done, equip the rocket launcher and climb back downusing a ladder away from the one you climbed up. The combot should be shaft at you, so begin making your way to the lower level door. Mostimportantly, keep moving and take some shots at the combot with therocket launcher while you can. Keep taking shots at the combot andluring him out the large door on the ground floor until Hendrix cuts in.======================================================================== Hendrix Theres a disposal pit at the last(a) point of this level. ================================================================================================================================================ Hendrix Try to lure the bot out onto the pits covering, then open == the doors from the control room. ========================================================================At this point, youve damaged the combot enough and hes really tickedoff. Now just handle around and run back to the disposal pit. Once thecombot approaches the last corridor leading to the pits covering, runto the right and climb the ladder to the control room (where Gryphon iscowering in fear). The combot will be unloading everything hes got atyou, but ignore it and quickly hit the right button to open the pit,then hit the left button to turn on the ceiling fan to blow the combotdown into it. Ra ther than waiting and watching to make sure it happens,

Red Faction :: essays research papers

***************************29. Refuse Disposal Plant***************************Open the elevator door by hitting the button to the right. Headforward, and gryphon will run up ahead to the incinerator controls.======================================================================== Hendrix This is Ultors main trash disposal facility. ================================================================================================================================================ Hendrix Thats an Ultor Personnel Suppression combot. I didnt == know they had any on Mars ================================================================================================================================================ Hendrix I cant see Orions group on my monitors anymore. ========================================================================While Gryphon takes the first left, go forward up the ramp and throughthe door. Climb the ladder up ahead, therefore climb a second ladder to doctorto the roof. Youll be shot at by the combat as you climb, but just sheer it and get up there. Hell have a tough time hurting you whileyoure up here. Follow the instructions and you shouldnt lose a singlepoint of health, condescension the holes he might create with his rockets.As you look more or less the step forwarder edge of the massive roof, youll seeseveral groups of ammo crates. Make your mien around the outer edge asyou make your way to each one to find ammo, medkits, armor, and a rocket onager. Make sure to snuggle up to each crate and jump on top of itto make sure you get every finis piece of ammo and equipment that youcan. Once youre done, equip the rocket launcher and climb back lotusing a ladder away from the one you climbed up. The combot should beshooting at you, so begin making your way to the lower level door. Mostimportantly, keep moving and take some shots at the combot with therocket launcher while you can. Keep taking shots at the combot andluring him out the l arge door on the ground floor until Hendrix cuts in.======================================================================== Hendrix Theres a disposal pit at the lowest point of this level. ================================================================================================================================================ Hendrix Try to lure the bot out onto the pits covering, past open == the doors from the control room. ========================================================================At this point, youve damaged the combot enough and hes really tickedoff. Now just turn around and run back to the disposal pit. Once thecombot approaches the last corridor leading to the pits covering, runto the right and climb the ladder to the control room (where Gryphon iscowering in fear). The combot will be unloading everything hes got atyou, but ignore it and quickly hit the right button to open the pit,then hit the left button to turn on the ceiling fan to blow the combotdown into it. Rather than waiting and watching to make sure it happens,

Construction resources Essay

Construction resources make use of physics in many of their products. Two that I consent picked are solar collectors and their use of insutation. Insulators countenance the purpose of preventing conduction, convection and radiation to unwanted areas. There are a wide range of techniques and materials use in doing this and I am going to look in depth at this idea. The company uses solar collectors in many different ways, their main purpose is to collect solar energy and foment energy. They shag be used to heat and power many different appliances.photovoltaic cells Photovoltaic cells transfer the energy carried by the electromagnetic waves that make up sunlight directly to an electrical circuit in order to make a current flow. Light excites electrons to move from one bottom to an new(prenominal) through semi conductive silicon materials. Only a fraction of a volt is produced so a large array is unavoidable to produce appreciable currents, usually about 20W to 100W. The Photovolta ic cells produce DC current so if alternate current AC is desired then an inverter is required.AC form is commsolely available from the mains. Solar cells Sunlight may also be used to heat pee directly. This buttocks be done in two ways, water can be circulated through pipes which run through the rachis of the solar cells, producing water at around 60oC suitable for household use. The other method is to focus sunlight from a large are to a small spot using an array of mirrors, this can produce temperatures of up to 4000oC and its quite feasible to produce power stations from this, however it has not been implemented yet.Here is on way in which the hot water can be used, this is quite economical and environmentally friendly. Solar panels are quite versatile and can be put on many hoods of simply in the garden, which makes them popular when it comes to heating swimming pools of hot water for showers. Any material that is a poor conductor of heat and electricity can be used as an insulator. Thermal insulating materials reduce the flow of heat between hot and cold regions.Thermal insulation may have to fulfill one or more of three functions to reduce thermal conduction in the material, in which heat is transferred by electrons to reduce thermal convection currents, which can be set up in air- or liquid-filled spaces and to reduce radiation heat transfer, in which thermal energy is transported by electromagnetic waves. These are three ways in which heat energy can be transferred. Conduction, where heat energy is transferred through solid materials, metals are normally the best conductors. There are a number of factors that reach thermal conductivity.Increasing the area of the cross-section of the solid through which hear flows increases the rate of flow, however increasing the thickness of the wall decreases the flow of heat. Provided that a besotted state has been reached (where the temperature of any point is not changing through time) then the rate of flo w of heat ? Q/? t is given by ?Q/? t A ? T/? x Besides the physical dimensions of the materials, another factor affecting rate of flow is the properties of the material, the Thermal Conductivity of the materials ? through which the heat energy is locomotion is the constant of proportionality in this relationship, so?Q/? t = -? A ? T/? x The units for thermal conductivity are watts per metre per kelvin. The Quantity ?Q/? t is called the temperature gradient. On my visit to Construction resources I noticed that they have used these Physical conclusions in there insulation. One in particular is Homatherm a wall and roof insulator. The slabs are thick which means less flow of heat, and have a thermal conductivity is only 0. 04 W/mK. Conduction in terms of particles The particles in metals and non-metals are arranged differently, which gives them their different characteristics.In a non-metal the particles have forces between them that can be described like springs. When heat is deliver ed to the solid the oscillations of the particles being heat will increase in amplitude. For heat to be conducted the adjacent particles must also receive the extra energy, as particles oscillations increase in frequency the neighboring particles also gradually increase in frequency as heat is transferred to them. This is a very slow process. In a metal the arrangement is different, bimetallic boding occurs between metal atoms where a sea of delocalized electrons hold the positive metal ions together in a lattice.It is these free electrons that are responsible for the game conductivity character of metals. When a metal is heated the metals ions vibrate with an increase frequency. When an electron hits these ions with extra energy, they receive this extra energy and move faster. This electron can then travel to another parts of the lattice, to an ion that hasnt received any of this heat energy and collide with it, transferring its energy to this ion. Now that ion has extra energy and vibrates with a greater frequency. This process is a mass faster as there are many electrons in a metal lattice.U-Values Architects and heating engineers use U-Values to conduct the flow of heat energy through building materials. The U-Values is quoted for a given thickness of a particular material, and is based on actual measurements made using the material. The U-Value is defined as U-value = Rate of energy flow Area X Temperature difference Construction resources uses this knowledge of U-Values when designing building materials to reduce the energy redundant in the form of heat going out of the build and to achieve the ultimate goal of reducing CO2 emissions.But there are taken for granted(predicate) limitations with reducing thermal conductivity, you can only reduce it to a certain amount, 0. 025 in air, so there will always be heat loss. as well as too much insulation in the house will lead to what construction resources refers to as internal pollution. This has lead t o construction resources researching in trying to start out the energy that is used to build and transport he construction materials. Bibliography Microsoft Encarta Heinemann Advanced Science Salters Horners AS Advanced Physics Strengths and Limitations of Photovoltaic cellsSince one photovoltaic cell only produces a small fraction of a volt, large arrays are required to produce appreciable voltages. This requires a lot of space and the cells need to be in a place where no shadows will be cast on them. This limits what they can be used for. likewise the amount of power produced by the photovoltaic cells directly depends on the amount of sunlight, so they cannot be relied on to produce power for something that constantly needs it. The only way this can be overcome is if some power was stored for a rainy day. But this again would take up space.Also an array of solar collector will be expensive and have a long pay back time. However it is environmentally friendly and after the pay ba ck time, savings can be made. Also photovoltaic cells can be made look attractive depending on where they are placed. Strengths and Limitations of solar cells The solar cells have similar limitations to the photovoltaic cells in regards to positioning and cost but a small array is only required to produce enough hot water for something like residential showers or central heating. This is economical and environmentally friendly and solar cells are quite versatile.

How These Bible Verses Affect Me

The Bible is a written collection of scriptures with divine inspiration to be lead and guided by the Holy Spirit. I passel say that these Sacred Scriptures of God are representation of our life. Death and life are In the power of the tongue and they that approve It shall eat the fruit in that respectof. Proverbs 1821 Every word that comes verboten from our mouth is powerful enough to make an impact on our life. In my own experience, I have witnessed the bedevilment and the pain that pinches my heart any time others utter vile words without considering my feelings.Deep inside, dejection and sadness swallow up my entire being alive. Sure, jokes are funny, save there are times that they have completely crossed the line. From then on, I told myself, l must stand up and fight for myself. That is why every time see people bullied by others I was like glaring at my own reflection. I know what they feel. Ergo, I should help and reach out my hand for them. Whilst on the other hand, I realized that I myself too should also be mindful and watchful of the words I say, for my actions, words, and deeds entrust come back to me.They allow reflect on my personality and will describe who I really am. As stated In the Golden overtop Do not do unto other hat you dont want others do unto you. For I am sure that neither death nor life, nor angels nor rulers, nor things present nor things to come, nor powers, nor height nor depth, nor anything else in all creation, will be able to separate us from the love of God in Christ Jesus our Lord. Romans 838-39 As I fortuity my Journey, my life, obstacles keep on blocking my way.As I step farther, hurdles and trials hinder me from going to the right path. Along my way, I may often trip and change surface fall flat onto the ground but I never give up, hence continuing what I ought to be. Life Is a coin with its two sides recreation and pain. Only one can become visible at a time. However, remember, the other side has its turn. P roblems may arouse, as well as sorrow and foiling may ramp on your elite.On the other hand, I believe that even In the darkest nights of my life, there Is always the brighter side that will help me seek for consent and give me the strength to persevere and overcome all of these. Nevertheless, they would not be possible without the presence of the love of God in Jesus Christ, our Lord. His unconditional love gives me strength, wisdom, hope, and the spirit to surpass all these hindrances that comes beneath me. My elite, bounded by in his spirit, is always present in our midst, thus, will never separate me from Him.He is the savior that will guide and save me from being lost. These two verses, analogous to what I experienced and learned, gave a huge impact in my life. I should respect the people rough me and should treat each other equally. If you ask me, one should never hurt others and never say words that will offend them. God is always there for me, guiding, helping and never le aving my side. No one can separate me from God, for being with Him is my destiny and that will be my way towards salvation.

My intercultural communication at school Essay

My ESL classes at Heald College took six months and these were necessary requirements so that I could pursue major courses in Architecture. There were other students like me from other countries and it felt well-grounded to belong to a group that was being taught the American way of life and finale. It felt good because it gave me the feeling that I was not al angiotensin-converting enzyme in my hardships. It did not feel so bad and odd being with a group struggling with nuances of the side of meat language and pronunciation and the idiosyncrasies of American nicety.While I considered myself an out-group when I was with natural-born Americans, I felt like a member of an in-group when I was with the foreign students. Probably, this feeling springs from the point that one naturally feels out of place when all others behave and talk in a different way. There is some comfort gained from knowing that one is just like the others. My struggle to fit in was at times frustrating, as I co uld not be properly understood by the instructor on account of misplaced accents or outright mispronunciations.At first I really felt stupid when the teacher would rephrase my statement and ask if it was what I meant. When I would say it was not what I meant, another classmate would stillt in with a genuine intention to help me out but I would end up all the much confused and misunderstood. Realizing the setbacks of my lack of facility with the language, I resolved more than ever to master the English language. My Intercultural converse at Work Working as a busboy and later, as a waiter in a restaurant gave me the chance to encounter more people and gain more facility with the English language.Finding work was quite an exhilarating experience for me. I felt that I could already advertise my ideas clearly and so I gained more confidence. Of course, there were still some instances when some patrons would fail to understand how I pronounce my words but whenever much(prenominal) a thing happened, I would remember the word and practice saying it correctly when I got home. I had discarded my Dari-English dictionary and I could already get hold of on a conversation with a native speaker of American English without fumbling for the right word or halting.I once attempt to be friendly with other waiters hoping to generate deeper camaraderie spell we were up and just about with our tasks. To my horror, I found that Americans seem to be single-minded and so cerebrate when they go about their work that banter is misconstrued as not being serious about work. From then on, I learned to be very business-like while doing my work, bearing in mind that I was not working in an Afghan setting Conclusion This profound reflection on my sojourn in America has made me realize a lot of things that I once took for granted.First, I realized that I possess the resilience and tenacity that is a earmark of a true Afghan. I used to take for granted the lore and history that cod to the Afghan experience of a succession of foreign invasions, Afghans ar by nature strong in spirit and character. Indeed, it is no easy task for an 18-year-old boy to force himself to become an adult almost overnight due to peculiar personal circumstances. I realized too that exposure to another culture widens ones horizons and makes one more tolerant of other people.Assimilation into another culture truly begins and is facilitated by an earnest desire to learn the language to a level that one speaks it like one born into the language itself. Together with this is a keen sense of awareness of the seemingly insignificant cultural differences between ones culture of origin and the host culture. Cultural sensitivity, if I whitethorn use the term, enables one to steer clear of problematic situations. Indeed, it is not always wise to act and behave according to ones cultural penchant when one is in another cultural setting.My intercultural communication with the Peterson family was at first regulated by cues from the reactions of Mr. Peterson. I came to know what he disliked by detect how he reacted. Yet, the sincere sharing of cultural traditions was the factor that truly made our relationship rise up to a higher level. Mutual respect and soul resulted from the open sharing and this could not have happened by merely observing each others reactions. Truly, if one wants to have a meaningful intercultural communication, it moldiness begin with an honest sharing of culture and traditions.It makes it possible to view the other person from their own cultural perspective. The snatch I stopped my automatic equivalence of Afghan and American values whenever I encountered an unfamiliar cultural practice, I became more accepting of American culture. As Holliday, Hyde and Kullman advise, Put aside simplistic notions about what is real and unreal in your perception of another culture, and appreciate that every society is as complex and culturally varied as your own.(1 0) The moment I began to have this attitude in my attempt to fully integrate myself into American society, I whitethorn have taken the first step towards intercultural competency. Alred and Byram (2002) define intercultural competence as the ability to behave appropriately in intercultural situations. the ability to stabilize ones self-identity while mediating between cultures (340). I may have taken the first step but I still have a long way to go. All people regardless of culture have the innate need to feel a sense of belongingness.I am sure this is the reason why I felt at lull with my fellow Afghans as I did with other foreign students who were struggling to deal with language difficulties. Yet, once a genuine intercultural communication is initiated, it leads one to learn and inquire at the diversity and richness of all cultures around the world.Works Cited Alred, Geof and Mike Byram. Becoming an Intercultural Mediator A Longitudinal Study of Residence Abroad. Journal of mu ltilingual and Multicultural Development 23. 5 (2002).Retrieved from http//www. multilingual-matters. net/jmmd/023/jmmd0230339.htm on April 25, 2007. Holliday, Adrian, Martin Hyde, and John Kullman. Intercultural Communication An Advanced Resource Book. New York Routledge, 2004. Questia. 25 Apr. 2007 . Robson, Barbara, Juliene Lipson with Farid Younos and Mariam Mehdi. Afghans Their History and Culture, Cross-Cultural and Adjustment Challenges, produce by the Center of Applied Linguistics, The Cultural Orientation Resource Center last updated 6/30/02. Retrieved from http//www. cal. org/co/afghan/acult. html on April 25, 2007.

Accrual Swaps

ACCRUAL SWAPS AND RANGE NOTES PATRICK S. HAGAN BLOOMBERG LP 499 PARK AVENUE NEW YORK, NY 10022 emailprotected NET 212-893-4231 Abstract. present we present the standard methodology for pricing accruement business deals, barf abide bys, and c completelyable collection switch overs and pass observes. Key words. upchuck notes, magazine swaps, accrual notes 1. Introduction. 1. 1. Notation. In our notation to sidereal solar daytime is incessantly t = 0, and (1. 1a) D(T ) = nowadayss give the sack factor for maturity T. For any ap elevationment t in the future, let Z(t T ) be the set of $1 to be delivered at a later find T (1. 1b) Z(t T ) = energy verifier bond, maturity T , as seen at t.These discount factors and zipper voucher bonds argon the wholenesss obtained from the fundss swap curve. Clearly D(T ) = Z(0 T ). We in make up distinct notation for discount factors and zero coupon bonds to remind ourselves that discount factors D(T ) atomic number 18 not rand om we can always obtain the ongoing discount factors from the stripper. Zero coupon bonds Z(t T ) are random, at least until time catches up to see to it t. Let (1. 2a) (1. 2b) These are de? ned via (1. 2c) D(T ) = e? T 0 f0 (T ) = right aways instantaneous foregoing treasure for succession T, f (t T ) = instantaneous forrader site for look 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 occlusive j Coupon stick memorial Fixed coupon accrual swaps (aka time swaps) consist of a coupon arm swapped against a funding leg. speculate that the agreed upon pen come out is, say, k month Libor. Let (1. 3) t0 t1 t2 tn? 1 tn 1 Rfix Rmin Rmax L(? ) Fig. 1. 1. Daily coupon identify be the schedule of the coupon leg, and let the nominal ? xed set up be Rf ix . Also let L(? st ) represent the k month Libor rate ? xed for the interval commencement at ? st and closing curtaining at ? termination ( ? st ) = ? t + k months. Then the coupon nonrecreational for accomplishment j is (1. 4a) where (1. 4b) and (1. 4c) ? j = geezerhood ? st in the interval with Rmin ? L(? st ) ? Rmax . Mj ? j = cvg(tj? 1 , tj ) = day count part 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 an other(a) way, each day ? st in the j th purpose conti only ifes the quantity ? ?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 legs schedule is contructed like a standard swap schedule.The theoretical dates (aka nominal dates) are constructed monthly, quarterly, semi-annually, or annually (dep destroying on the contract terms) backwards from the theoretical dismiss date. any(prenominal) odd coupon is a stub (short period) at the front, unless the contract explicitly secerns long ? rst, short goal, or long last. The m odi? ed following business day convention is workoutd to obtain the actual dates tj from the theoretical dates. The coverage (day count disunite) 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, in that respect are many variants of these coupon legs. The only variations that do not make sense for coupon legs are set-in-arrears and 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 rim is 2 exercisingd in place of the ? x ed rate Rf ix . Minimal coupon accrual swaps receive unmatchable rate each day Libor sets within the range and a second, normally 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 App finisix 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 counterpartys extension-worthiness is a signalise concern. To determine the constitute rank of a range note, virtuoso needs to engage an option adjusted stagger (OAS) to re? ect the extra discounting re? ecting the counterpartys 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 denotation rate L(? st ). save, 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 apprise the coupon leg by replicating the payo? in terms of vanilla caps and ? oors. tip over 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 apprise 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 harbor coupon j by replicating each days contribution in terms of vanilla caplets/? oorlets, and then(prenominal) summing over all days ? st in the period. Let Fdig (t ? st , K) be the prize at date t of a digital ? oorlet on the ? oating rate L(? st ) with stunner K. If the Libor rate L(? st ) is at or below the regard 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 notes payo? for date ? st by handout long and short digitals enamored 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 ) preferably of tj . 2. 1. Hedging considerations. Before ? ing the date couple, 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-month 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 frankincense the bullish spread, and its limit, the digitial ? orlet, are directly determined by the market scathes of vanilla ? oors on L(? st ). Digital ? oorlets may develop an unbounded ? -risk as the ? xing date is approached. To overturn this di? culty, most ? rms book, footing, 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 pugnacity of the ? rm. Alternatively, some banks remove 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 desks 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 legs 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 assumptive only parallel shifts in the forward curve The ratio Z(? ix tj )/Z(? f ix ? end ) is the manifestation of the date mismatch. To take 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 end ) = 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 rates 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 co unt 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 devil 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 Rmin ). Setting t to the ? xing date ? f ix shows that this combination matches the contribution from day ? st in eq. 2. 11a. Therefore, this grammatical construction gives the value of the contribution of day ? t for all earlier dates t0 ? t ? ? f ix as come up. 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 grammatical construction 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 monkeylied 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 banks 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 the 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 rates ? 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 ) trustworthy + 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 legs ? oating rate, and ri i collapses to 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 non-white 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 ? oorlets forward rate as seen at date t. Todays ? oorlet value is only if (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 right aways forward Lib or rate is (2. 20c) L0 (? st ) = To obtain at presents 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 todays 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 sterns 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 days contribution is very tedious. Normally, one calculates each days contribution for the current period and two or three months afterward.After that, one usually replaces the sum over dates ? with an constitutional, 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 liquidator, 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 wit h quarterly wield dates starting in year 3 to receive the residuum 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 learnments 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? view Bermudan swaption. This criterion requires using the same the interest rate sticker and normalization method for Bermudan accrual notes as would be used for Bermudan swaptions.Following standard practice, one would calibrate the Bermudan accrual note to the diagonal swaptions struck at the accrual notes e? ective strikes. For example, a 10NC3 accrual sw ap 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 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 swaps funding leg ? i This usually results in strikes Ref f that are not too far from the money. In the precedent 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 todays market 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 paym ent.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 ingrained 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 models prices of standard ? oorlets. This enables us to compose the accrual swap 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 push back. The last tint 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 variability 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 todays variance is zero ? (0) = 0. The ratio (3. 3a) V (t, x) ? V (t, x) ? N (t, x) is usually called the bring down value of the deal. Since N (0, 0) = 1, todays value coincides with todays 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 todays prices.. De? ne Z(t, x T ) to be the value of a zero coupon bond with maturity T , as seen at t, x. Its value can be found by substituting 1 for V (T, X) in the Martingale valuation f ormula. 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,xT 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 todays 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 ra te 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 true rate for the interval and bs(? ) is the Libor rates 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 va lue 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 = 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) 1 Z(t, x ? st ) ? 1 + bs(? st ) ? Z(t, x ? end ) 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 ). S o 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 Blacks formulas 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) = assets 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 abash ? , which is the ? oorlets price volatility, with the commonly quoted 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 model parameters h(t) and ? (t) are known. (In Appendix A we show one popular calibration method). Let the Bermudans 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 for intervals ik , ik + 1, . . . , n. ? The rollback works by induction. anticipate that in the previous rollback steps, we have calculated the reduced value (3. 11a) V + (tex , x) k = value at tex of all rest 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 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 take 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 todays 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 todays values of the standard ? orlets F (0, 0 ? st , K) = D(? end ) 1 + ? (K ? bs)N (? 1 ) ? 1 + ? (L0 ? bs)N (? 2 ) log 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 todays 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 todays market price of the ? oorlet is is the assets log normal volatility according to the LGM model.We did not calibrate the LGM model to these ? oorlets. It is virtually certain that matching todays 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 = 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 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 normality 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 models 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 sharpen up or smear out the digital ? oorlets payo? to match todays value at L0 /K. This results in slightly positive or negative price corrections at divers(a) values of L/K, but these corrections average out 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 make up in a ? ctitious exercise fee to match todays coupon value. Adding a fee gives a positive or negative prepossession 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 todays 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. Unlike 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 swaps 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 issuers credit-worthiness is a key concern. One needs to use an option adjusted spread (OAS) to obtain the extra discounting re? ecting the counterpartys 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 not e 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 notes value relative to the issuers other bonds, then one should use the issuers discount curve for DA (? ) the OAS then measures the notes richness or cheapness compared to the other bonds of issuer A. If one wishes to ? nd the notes value relative to its credit risk, then the OAS calculation should use the issuers risky discount curve or for the issuers credit ratin gs risky discount curve for DA (? ). If one wishes to ? nd the absolute OAS, then one should use the swap markets 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 markets idiosyncratic preference or adversion of the bond. When valuing a callable range note, we are making 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 todays 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 surrogate the coupons discount factors D(tj ) with the bo nd-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 Blacks 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 todays forward rate (4. 4e) Finally, (4. 4f) ? = ? end ? tj . ? end ? ? 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? pass.Clearly the value of the callable range note is the valu e 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 todays value of the non-callable range note in 4. 4a, and VA (0) is todays 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 = 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 = ? and the forward Libor value is (4. 6e) (4. 6f) L? L (tex , x ? st ) k Dst Dend 1 = ? The only remaining issue is calibration. For range notes, we should use constant mean atavism 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 todays 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 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 o ne 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 fractions for interval i using the ? xed leg and ? oating leg day count bases. (For simplicity, we are cheating slightly by applying the ? oating legs 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=1Under the LGM model, todays 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. To 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 swaptions 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 Blacks 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 ?. 2This 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 deals duration as well as its moneyness. Appendix B. Floating rate accrual notes. 21

Frank Lloyd Wright Research Paper

Frank Lloyd W proper(a) and His Impact Frank Lloyd Wright was a very powerful designer and architect who inspired the next century of builders to go beyond their normal standards and break free from the confines of the current building barriers. He utilize aspects of nature to compliment his buildings, and k novel how to perfectly arrange the complex angles and structures to set his projects apart from all others. Frank Lloyd Wright changed the future of architecture with his high attention to interior design, detail, simplicity, and environmental influences.Franks upbringing seems to have impacted his choices and views in deportment. Wrights father was a Baptist minister and a musician, possibly where Wright may have received several(prenominal) of his creativity. His mother was a school teacher, and she introduced Wright to the Froebel Kindergarten Method, which familiarized children to solely geometric forms and certain patterns on grids. Many say that this was a major part t o Wrights modern geometric procedures and focus on design and function (Wright, Frank Lloyd Overview).Frank Lloyd Wright held many jobs that added to his architect career early in his life. The first main influence was his odd-job(prenominal) employment with a Madison, Wisconsin contractor, working as a draftsmans apprentice for Joseph Silsbee. Wright also worked with Dankmar Alder and Louis Sullivan. Wright developed a close relationship with Louis Sullivan, who had a form follows function philosophy, just like Wright did later in life. JRank Encyclopedia says, By the time Wright was in his early 20s, he had worked on some of the almost impressive buildings in Chicago. This quote puts in perspective just how much experience Wright received from his collaborations with Adler and Sullivan (Wright, Frank Lloyd Overview). Wright said himself, describing his work as organic architecture that which proceeds, persists, and creates, according to the nature of man and his circumstance s as they both change. This quote shows what Wright wanted his buildings to convey a view of the natural world through the plant of beautiful architecture.The homes Wright created in his earlier years held strong horizontal lines and shapes and roofs that were low pitched and had large overhangs. The interiors of many of his houses were influenced by Japanese designs they had large open positions, hardly a(prenominal) closed corners, and large windows to allow for the flow of natural light and to not hide the beauty of nature. The homes were heated by coils built into the slabs of concrete that circulated warm water, heating system the house evenly, known as radiant heat.Wright developed a new type of building material, using pre-cast concrete blocks that were reinforced with metal. Several of his houses were built with his new method. Frank Lloyd Wright pushed the standards, creating a new era of architecture that he was paving as he worked (Wright, Frank Lloyd Overview). Alt hough Frank Lloyd Wright receives so much praise nowadays, during his life time and especially his earlier years, he was not as glorified. Only roughly 10% of his over 470 buildings and more than 1200 designs received public commissions (McCarter).Having 4 wives and difference his first family, Wright wasnt seen as the most likable person, but his designs were irresistible (Wright, Frank Lloyd Overview). He has set a standard that has not yet been matched those during his time had not realized his creativity. As Wright continued in his career, people began to notice his uniqueness and powers. Frank Lloyd Wright did not receive the full recognition he merited during his lifetime. Many people can now agree that Frank Lloyd Wright is the single most influential architect and builder to have lived in the recent centuries.When asked about memorable architects, virtually every first-year college student entering American universities could name Frank Lloyd Wright (McCarter). A quote f rom the Taliesin Architects website truly shows the impact Wright had The greatest workman this country has ever produced seems at last to be coming into his own. Americas other great artistsour painters, sculptors, composers dont really rank with the tops of all time. Theyre not Rembrandt or Michelangelo or Beethoven. Wright alone has that standing. (Frank Lloyd Wright).Wright has many extremely recognizable buildings located in many areas around the world, such as the Unity Temple in oak tree Park, Illinois the Midway Gardens in Chicago the Fallingwater in Mill Run, Pennsylvania the Imperial Hotel in Tokyo, Japan the Beth Shalom Synagogue in Elkins Park, Pennsylvania and the Guggenheim Museum in New York (McCarter). Its easy to tell that Wright has been seriously noticed and is overly well known. But, one may ask, what kind of major impacts has Frank Lloyd Wright had on the world? Well, homes with a surviving room, open garage, or a floor plan with many open spaces have all be en majorly influenced by Mr.Wright. Frank himself pioneered all of these designs. His low and sweeping rooflines, many windows, and central fireplaces changed how houses would be built for centuries to come (Frank Lloyd Wright). In 1952, Wright completed a home for a World War II veteran that is only accessible for a person confined to a wheelchair, such as the owner Kenneth Laurent. The house is built on completely one level, with curving walls, added space for turning radii, and lowered light switches and heighted electric plugs giving the perfect example for just how much Wright could bend to certain needs all while maintaining his person-to-person designs.Mr. Laurent described Wright in an interview pacing up and down the room as he listened to the requirements for the proposed home. Mr. Laurent said, Mr. Wright seemed to be designing the home in his mind right then and there. (Menendez). Being able to complete complex projects such as this home shows how talented and unique Frank Lloyd Wright was. Frank Lloyd Wrights advanced designs and send on thinking easily makes him without a doubt one of the greatest architects to have lived in the unify States.He changed the future of architecture inventing new ways to have homes turn over in a more beautiful and efficient way. His complex but simple interior design elements make his homes and buildings a stable to modern American architecture. His respect to the beauty of nature brings the simplistic gloriousness of the outdoors to reside in the structure and designs of his homes. Frank Lloyd Wright made his name known and marked the history of the United States and the world for many, many years to come. Bibliography Frank Lloyd Wright. Building The Windy City.Web. 24 Mar. 2011. http//library. thinkquest. org/J002846/a_wright. htm. McCarter, Robert. The Other Traditional American Architecture Frank Lloyd Wright and Louis I. Kahn. Phi Kappa Phi Forum. 2003. Web. 9 Mar. 2011. http//find. galegroup. com/gtx/ start. do? prodId=AONE. Menendez, Frank. An kind Jewel. Paralyzed Veterans of America. Mar. 2010. Web. 9 Mar. 2011. Wright, Frank Lloyd Overview. Jrank Encyclopedia. Web. 15 Mar. 2011. http//encyclopedia. jrank. org/articles/pages/6404/Wright-Frank-Lloyd.

Strategy and Planning Essay

Strategy and training essay 1. Executive Summary In this research paper we will briefly discuss about the essential factors that involved in strategical planning and decision making and the permute concern and the importance of change management. The essay is solely about the important strategic tools which can be implemented in order to get an excellent strategic plan. In conclusion the back bone of any presidential term which t residuals to become a success lies upon each and e real decision they make and they must betroth into consideration the forces which act on the macro environment and in the micro environment of the organisation. . Introduction Planning and Strategy atomic number 18 the essential factors which act as the back bone of an organisation. A go with with an excellent strategic planning and change management capabilities is sure to get a concrete food market plunk for in this dynamic handicraft environment. Some organizations tend to leave the strategic part of business planning and end up in bankruptcy. Strategy is the management to achieving short and long term goals. Planning is the means of achieving the goals through outline while exploiting the available resources. 3. Main Body (a) Strategy and Planning The term strategy is derived from the Greek word strategos which means the highest ranking army general. So we can derive from these words the importance of strategy and the out come of strategy. Strategy is a tool which helps to create an advantage over the competition in the dynamic business environment. To create a strategic plan for an organisation there are certain constraints which are to be followed in order to achieve the optimal benefits from the decisions we make through strategy and planning.The maximum exploitation of our resources is the key to advantage over the competitors. Marketing is one of the main areas in which a intimately versed strategic plan can make the organisation to attract nodes here the de legacy in which we describe the organizations motives, goals and the dedication to the customers reflect in our adverts. So, cleverly placing the right factors in the right place is a challenge in the strategic planning decision making.In this ever changing business environment the only way of survival of an organisation depends upon the decisions it makes. Thus strategic planning is the key factor to success. In this research we will briefly discuss about the tools of strategy and the outcomes of implementing these tools in an organisation. We will discuss about plodding analysis, PESTEL analysis and Potters Five Force. 3 (b). turn out Analysis SWOT analysis is one of the tools in strategic planning and decision making.This tool helps us to judge the organizations position in the business environment. SWOT stands for Strength, Weakness, Opportunities and Threats. Strength The organizations factors which are most influential with assess to customers, market share, and gull sur vey and so on. The strength of an organisation whitethornbe its good will, an excellent customer relationship management and many other sensitive factors which lies on the organizations side. Weakness The things which are an advantage of the competitor and a disadvantage to our organisation.The weakness factor may depend upon our organizations own business policies and or work ethics which binds us to a corner, the competitors takes advantage of this issue and exploit the opportunity nevertheless fine correct our strategy and decision making will bring in a drastic change. Opportunities The chance in which all the things fall in line for our organisation to exploit and to make a profit of it is the definition for opportunities. Opportunities can occur cod to a change in trend of customers, brand switching an so on.It is wise for an organisation to make use of such opportunities. Threats The main and most important factor is the threats from rivals and competitors. If an organi sation fails to with stand the threats from the competitors then it will surely end up in bankruptcy, the competition may come up with innovative ways of marketing and attracting customers so it is our bounded duty to over come the competitors by creating innovative ways of marketing through strategic planning.Example for SWOT analysis Soft alcoholism giant Coca-Cola Strengths One of the most successful organisations of all time. The brand value of Coca-Cola is worth more than $68 million in 2009. the company manages its products over 200 countries with over 700 products which is a gigantic goal to achieve but maintaining it as the leader of its kind is still another battle. People tend to be faithful to the Coca-Cola brand because of its one main product the cola drink.So a wide variety of different products from the company with the same brand name tends to market the products for itself. (www. interbrand. com) Weaknesses The Coca-ColaCompany products are drinkings so the main weakness for them is that they have this lobby of beverage segment and this is might be their weakness because in the long run they might hit the ground if any other company produces a ruin beverage. And nowadays people tend to make choices more often than in the past so it is a vulnerable area to be looked upon OpportunitiesPeople are refer about their health now more than ever so the introduction of Coke Zero a naked product from the Coca-ColaCompany is catching up a considerable amount of market share because of its new distinct taste without added sugar in some countries but basically its a new drink with zero calories, the company claims. The company is exploiting the current trend of health criteria. Threats The main competitor of Coca-ColaCompany is the PepsiCo Inc, the Pepsi Cola Company introduces new products in response to the Coca-ColaCompany. The Coca-ColaCompany is trying very hard to cope up with the situation. During 998 the PepsiCo Inc managed to out smart the co mpany. So without the proper strategic planning PepsiCo will soon go beyond the Cokes victory 3(c). The bray Analysis The most important tool of strategic planning is the PESTEL analysis. The PESTEL analysis is a tool which helps us to judge our position in the market with respect to the macro environment. PESTLE stands for P semipolitical Forces, E Economical Forces, S - fond Forces, T -Technological Forces, L Legislative Forces and E Environmental Forces. These forces influence the overall performance, market value, finance and customer relationship of the company.We will briefly discuss about the PESTEL forces one by one by implementing into an organisation. Example Integrated Electronics (INTEL) Semiconductors. The Intel Semiconductors are the producers of the famous Intel microchips which literally runs the computer industry but with some rivals such as AMD Inc (Advanced Micro devices). Here we will apply the PESTEL analysis to know the factors that are influencing the Intel semiconductors. Political The implementation of new taxes and the internationalism of the products tends to make the company to adhere to strict rules in each country which the company is ventured on.So Intel is focusing upon its new products onto buyers and not on further research and development because of the political pressure around the organisation. Economic The semiconductor industry is in rise. So, the financial status of the company is rising repayable to a smaller amount of competitors there is lower amount of threats so the economic factor of the company is in concrete position as of now. Social The company is present in an ever changing dynamic business environment so it has to be sensitive to changes that occur in the environment and then adapt to the changes that takes place.Technological The semiconductor industry is ever growing in rapidly increasing speed. So, the technological advancements of the industry must be up to date due to high growth ratio of proce ssors which are use in computers across the globe. The research and development team comes up with new innovative ideas to integrate the preceding(prenominal) processing unit into the new one with advanced technology so the computation power increases more than double the time the predecessor microchip used to compute. LegislativeThe legal factors are important factors of the company because every country that the company venture into must strictly adhere to that particular countrys rules and regulations. Environment being a semiconductor company the company must look after environmental issues which affect the world so the semiconductor devices must be environmentally friendly and cycle of the devices plays a vital role in the environmental factor. The company has a good background of making environmentally friendly chips and recycling units so the environmental issue is under control. (http//ivythesis. typepad. com) 3(d). Porters Five ForcesThe porters five forces is a business strategy tool which is used to analyse the position of a company and assessing its strength with respect to the competitors in this case new entrants and their impact in this ever changing environment. The porters five forces are Existing rivalry among firms Threat of new entrants determinants of supplier power Determinants of buyer power Threat of substitute products Example AIRBUS The porters five applied to the Airbus aviation company who have been the most influential carriers of all time apart from Boeing. These two major aircraft companies were literally in a fight for gaining the considerable mount of market share. Since the introduction of Airbus A380 the Boeing Company has lost messiness of customers. The new entrant in our case the A380 is so much advanced than the legendary Boeing 747 which literally ruled the skies. The Boeing 747 was a fuel efficient and economically well versed passenger and good carrier aircraft but the major flaw in the Boeing decision making is th at it was not concentrating on the future of the aircraft but it produced a wide variant of Boeing 700 series but the cleverly devised strategic plan of Airbus was a nightmare to the Boeing corporation.The A380 has an excellent performance with regardless of its massive size the big jet can people double the time the Boeing 747 used to carry. Since the aviation industry has only fewer concentrations of rivals the market share is huge compared to other industries. So with the new A380, Airbus has a considerable amount of stake in the airline industry. But Boeings answer to A380 is the new 787 dreamliner. Which is more fuel efficient but in the case of size the A380 outsmarts the Boeing. Importance of Change The change management in an organisation is the most important factor which influences the future of an organisation.It describes an strategic approach in modifying or altering an organizations internal factors such as teams, individuals etc. there are different ways to achieve t he change in an organisation by collaborative which is done by providing workshops and seminars explaining about the change to the people working in the organization. Consultative the people are informed about the change and the feedback of the change is taken into consideration. Directive the members of the staff are informed about the details of the change and the sensitivity of the change.Coercive the members are forced to follow the change. Example of Change The encyclopaedism of Iberia by British Airways The British Airways acquired the Iberia for $7 billion and it make the third largest airline in the world. The member of the staff was quickly adapted to the change in different management team. The Iberia staff members were given a change management session on order to cope up with the new acquisition. The acquisition made the stock holders value to increase and they were given a 55 percent of the new airlines share. 4. ConclusionWe have so far discussed about some of the strategical planning tools and the ways in which they can be exploited. And in conclusion we can say that the way in which we approach the organisation with the necessary tools and strategic planning will make a gigantic impact on the organisation in the present and in the future. In this ever changing business environment the challenge for each organisation is to identify the needs of the customers and to create an appropriate strategic plan to approach the problem which will provide the optimum results which can benefit the customers as well as the organisation. . References SWOT Analysis, online, Available http//www. marketingteacher. com/Lessons/lesson_swot. htm PESTLE Analysis, online, Available http//rapidbi. com/pestle/Introduction-to-the-PESTLE-analysis-tool. html Coca-Cola Case Study, Online, Available http//www. interbrand. com/best_global_brands. aspx http//finance. mapsofworld. com/brand/value/coca-cola. html Intel case Study, Online, Available http//ivythesis. typepad . com/term_paper_topics/2009/09/intel-case-study. html Read more http//www. americanessays. com/study-aids/free-essays/management/strategy-and-planning. phpixzz2OPwRLxoO