How Society Viewed Love and Marriage in Pride and...

How Society Viewed Love and Marriage in Pride and Prejudice by Jane Austen Jane Austen was born in 1775 and spent most of her life in the countryside in a village called Steventon, Hampshire. She was the daughter of a clergyman, Reverend George Austen and her mother was called Cassandra Austen. She had a brief education starting at the age of seven and ending at eleven, when she settled at home. Like women in Austen’s society, she had little education due to the beliefs at the time; the only education she would have received would likely have been to up her social status, through marriage. She wrote â€Å"Pride and Prejudice† to portray society’s views of love and marriage to the reader and to shoe that marriages take place for†¦show more content†¦Ã¢â‚¬Å"†¦ Mr Collins to be sure neither sensible nor agreeable; his society was irksome and his attachment to her must be imaginary. But still he would be a husband†¦Ã¢â‚¬  This reflects the social beliefs of, at least, the middle classes at the time. Evidently all that Charlotte wants out of life is a comfortable home and enough wealth to sustain this. â€Å" I’m not a romantic girl you know. I never was, I only ask for a comfortable home †¦Ã¢â‚¬ . The use of the word â€Å"only† emphasises the fact that Charlotte wants this and nothing else. The consequences of her not marring well would be to severely limit her options i.e. she would have to become a governess or an old maid for a wealthy couple, but this would not support her once she had reached an age at which, she could no longer work. When Charlotte Married Mr Collins she seemed happy in their relationship, even though its not based on love :- she has all that she wants out of marriage. She does however ignore her husbands silliness and does not have any bad words to say about him. â€Å" When Mr Collins said anything of which his wife might reasonably, which certainly was not unseldom†¦ once of twice she could discern a faint blush; but in general Charlotte wisely did not hear†. From this we gather that the orthodoxy of society has been embedded in her behaviour. She possibly perceives that she is happy because society deems that now she is married she should beShow MoreRelatedJane Austen s Pride And Prejudice1697 Words   |  7 Pagesan Oppressive Society Jane Austen once said, â€Å"it is a truth universally acknowledged, that a single man in possession of a good fortune, must be in want of a wife† (Austen 1). In other words, women of the nineteenth century were deemed dependent on men. They were to join an advantageous marriage to remain respectable and achieve a higher social class. Jane Austen’s Pride and Prejudice concerns the social norms of the eighteenth and early nineteenth centuries—a patriarchal society ruled by men whoRead More Explore Jane Austen’s attitude to marriage in Pride and Prejudice1671 Words   |  7 PagesExplore Jane Austen’s attitude to marriage in Pride and Prejudice Looking at the social, historical and cultural context In the 19th century when Austen wrote ‘Pride and Prejudice’, the way in which marriage was viewed was very different. It would have been expected of a young woman to find a ‘suitable’ partner for marriage before they were thirty, as after this they could be seen as an embarrassment to their family. By suitable, it does not mean in the way in which marriage is viewed todayRead MoreGender Roles . Regency England Is The Time Period In England1544 Words   |  7 Pagesarchitecture, clothes, furniture, literature, and politics, establishing a new style of society. This time period is specifically distinguished as between the â€Å"Georgian Era† and the â€Å"Victorian Era.† Ultimately England’s society as an aeon acclaimed for its fine arts came new economic, social, and political changes. Specifically, females were given new roles in their society, in which some women did not approbate. With Pride and Prejudice’s opening line: It is a truth universally acknowledged that a singleRead MoreJane Austen s Pride And Prejudice1048 Words   |  5 Pages In the novel Pride and Prejudice, love at first sight is mocked throughout the characters in this nineteenth century society. Jane Austen portrays irony in certain characters’ romances. Some characters were expected to be together due to â€Å"love at first sight,† yet Austen creates conflict which disrupts these relationships. Jane Austen mocks conventional romantic novels by developing hardships and obstacles among characters’ romances. Austen shatters the expectations of â€Å"love at first sight† andRead MorePride and Prejudice is a British novel written by Jane Austen. This book is one of the most800 Words   |  4 PagesPride and Prejudice is a British novel written by Jane Austen. This book is one of the most cherished love stories in English Literature. Pride and Prejudice was written in the early 1800’s to replicate the relationships between men and women in Austen’s time. She portrayed Elizabeth, the second eldest of the Bennet dau ghters as fearless, independent, and more concerned about marrying for love than marrying for social status and stability. Elizabeth is able to still able to have the expectationsRead MoreJane Austen s Life And Prejudice1430 Words   |  6 PagesJane Austen was born on December 16, 1775 in Steventon, Hampshire, England. She was the seventh child and second daughter of Cassandra and George Austen. Jane Austen s life was one of the most transformative eras in British history. The American Revolution, The French Revolution, family and societal views caused Jane Austen s life to be influenced in several ways. Jane Austen was a conservative female who spent most of her time writing novels that reflected her views on love, war, reputation, andRead MoreLiving in Social Classes in Jane Austins Novels816 Words   |  3 PagesJane Austin is a great author from the 1800s. I really enjoyed the few books of hers, that I did get around to reading .I enjoyed the movie adaptions even more, Jane f ocused on things she thought about and used humor to point out the lives of the middle class, the wealthy, nobility, and families in different financial standings, battling to keep up or with their social status. Jane puts most of her focus and writing into the importance of marriage. She lived with her family her entire lifeRead MoreNorthanger Abbey as a Precursor to Pride and Prejudice Essay1614 Words   |  7 PagesJane Austen’s Northanger Abbey is frequently described as a novel about reading—reading novels and reading people—while Pride and Prejudice is said to be a story about love, about two people overcoming their own pride and prejudices to realize their feelings for each other. If Pride and Prejudice is indeed about how two stubborn youth have misjudged each other, then why is it that this novel is so infrequently viewed to be connected to Austen’s original novel about misjudgment and reading one’s fellowsRead MorePride And Prejudice Essay1808 Words   |  8 PagesIntroduction Pride refers to a deep and consistent feeling of satisfaction of one’s possession, class and achievements. Pride as a theme has been developed in the novel pride and prejudice by Jane Austen. Characters can claim better treatment and status in the society and also relationships based on their family possessions as well as exploits. Mr. Bennett daughters are also proud because they are beautiful which gives them the audacity to boast to men who want their hand in marriage (Gao, HaiyanRead MorePride and Prejudice by Jane Austen1324 Words   |  5 PagesShould one follow society’s rules for marriage, love, class, and gender expectations or their heart regardless of the judgment of others? Jane Austen explores these themes in her novel Pride and Prejudice, which takes place in the early nineteenth century. In this famous novel, Elizabeth Bennet, who is the protagonist, is intelligent, witty, and the most sensible of the five Bennet sisters, who all face challenges with s ocial rules and expectations. Conflicts and parental pressure arise through

The Learning And Teaching Strategies - 876 Words

Many great learning and teaching strategies are presented in the text, How we learn†¦ of which a few learning tactics, namely discrimination, perceptual learning, chunking and interleaving will be briefly discussed as to how they have relevance in my context. To begin we will examine discrimination, the brain’s ability to â€Å"detect minute differences in sights, sounds and textures. [Such is considered]†¦one of the first steps we take in making sense of the world† (Carey, 2012, p.180). This learning technique occurs often during HT training as students detect differences in breathing patterns of those being treated, l see slight tremors or body twitches that accompanies release of stagnate energy and perceived smoothness of textures in the human energy field (Mentgen Bulbrook, 2011). Next, perceptual learning occurs as the brain takes notice of subtle differences in techniques, such as hand positions that seem the same but may not have the same therapeutic effect. Another learning strategy that can be useful in HT training is chunking which is the simple act of grouping bits and pieces of information together to make a whole (or at minimal a segment) of a concept. For example words stringed together create sentences and sentences which are connected formulate paragraph. In the case of HT chunking is necessary as a series of hand movements and/or places equate to a specific healing technique. The final relevant strategy from the text is that if interleaving, â€Å"a cognitive scienceShow MoreRelatedTeaching / Learning Strategies For Learning1621 Words   |  7 Pagesdifferent strategies for learning, the student feels confident to explore, and mistakes most likely are to be minimized and engagement increased. Lessons and activities that enhance and support learning are teaching /learning strategies that when cooperatively done contributes to a lifelong learning/knowledge. Students skills activities allow me to provide feedback that can be shared and/or discussed by the whole class increasing learning. Opportunities lead to sharing that leads to learning. By literallyRead MoreTeaching Strategies For Teaching And Learning Essay2254 Words   |  10 PagesThe teacher s role in teaching and learning process are of considerable importance, no matter what teaching practices are taken for effective acquisition and learning. Effective teaching involves a deep knowledge of a specific student, a teacher teach and where his student is on his learning journey. The main purpose of this report is to address the teacher s effective classroom teaching practices and how these practices influence the child s development and learning. The report also identifiedRead MorePedagogy, Teaching and Learning Strategies1012 Words   |  5 PagesPedagogy: Teaching and Learning Strategies During my PGCE placements I felt I have used a wide variety of assessment, teaching and learning strategies. I feel that this is absolutely necessary in terms of providing the best possible service to the pupils. To be repetitive, predictable and non-contemporary with such strategies is to compromise the effectiveness of your teaching and limit the chance of achieving your Learning Objectives. It is crucial at the outset to understand that the pupils underRead MoreDiscussing the Teaching and Learning Strategies1571 Words   |  7 PagesTeaching and learning strategies used in an actual session and to be delivered during professional practice. The strategies to be used will depend on several different components, e.g. the ability, knowledge and background of learners, the subject, differing learning styles etc. John Dewey (1859-1952) believed that formal schooling was falling short of its potential. He emphasised facilitating learning through promoting various activities rather than by using a traditional teacher-focused methodRead MoreEffective Teaching And Learning Strategies Essay1428 Words   |  6 Pages Marzano’s Effective Teaching and Learning Strategies Effective schools make a big difference in student achievement. Effective leadership makes a positive difference, too. Effective teachers, however, directly impact student learning and achievement. It’s been shown that teachers who have a large repertoire of effective instructional strategies teach differently (Tyson).They’re more intentional in their objectives, strategies, and intended outcomes. And, have better results. Robert Marzano, an educationalRead MoreTeaching Learning Styles And Strategies Essay1855 Words   |  8 Pagesapproach. In this process of learning the language learners adopt their own methods and strategies depending on their styles of learning. They select the more appropriate strategies to fit their learning styles. This paper focuses on the role of learning styles and strategies in a language classroom. It is the responsibility of the teacher to be aware of the learning styles of the learners so as to make the learning teaching process an effective one. Moreover the learning style of one learner is differentRead MoreTeaching Strategies For Learning Style Preferences853 Words   |  4 PagesBeck matches three teaching strategies to learning style preferences. During which he attempts to link the 4MAT system, Dunn’s LSI, and the Renzulli Smith’s LSI to teaching strategies associated to display individual learning preferences linked to the brain’s hemispheres. The 4MAT system and Dunn’s LSI is teacher driven teaching strategies while the Renzulli Smith’s LSI allows student input into their teaching strategies. In linking the 4MAT system to teaching strategies, Beck further exploresRead MoreTeaching And Learning Strategies For Teaching English Language Learners4159 Words   |  17 PagesAbstract A systematic review of studies that utilized an effective technology tool and/or technology-program in primary classrooms for teaching English Language Learners was conducted. The search produced 476 potential studies, of which the most recent 6 studies that met inclusion criteria were selected. The results of these 6 studies were Introduction English is one of the most important languages spoken around the world; so much so, that several countries whose native language is not EnglishRead MoreTeaching Strategies For Students With A Learning Environment Suitable For Learning Essay2029 Words   |  9 Pagesways on how to engage and motivate students to do their work. As an educator, I have to be constantly changing my teaching strategies to meet the diverse needs of the students and establish a learning environment suitable for learning. In order for teachers to become leaders, they have to keep educating themselves professionally and implement new teaching methods to enhance the learning of the students. In addition, effective teachers have to implement activities that are meaningful to the studentsRead MoreTeaching Strategies Used to Promote Active Learning Essay1159 Words   |  5 PagesActive Learning According to the book Promoting Active Learning by Chet Meyers and Thomas Jones and referenced by Kathleen McKinney, active learning means (1) that learning is by nature an active endeavor and (2) that different people learn in different ways. The first segment of the definition by nature an active endeavor is important because it allows the student to develop her critical thinking skills as well as practice her writing techniques. The student can discuss class material in

Accrual Swaps Free Essays

ACCRUAL SWAPS AND RANGE NOTES PATRICK S. HAGAN BLOOMBERG LP 499 PARK AVENUE NEW YORK, NY 10022 PHAGAN1@BLOOMBERG. NET 212-893-4231 Abstract. We will write a custom essay sample on Accrual Swaps or any similar topic only for you Order Now Here we present the standard methodology for pricing accrual swaps, range notes, and callable accrual swaps and range notes. Key words. range notes, time swaps, accrual notes 1. Introduction. 1. 1. Notation. In our notation today is always t = 0, and (1. 1a) D(T ) = today’s discount factor for maturity T. For any date t in the future, let Z(t; T ) be the value of $1 to be delivered at a later date T : (1. 1b) Z(t; T ) = zero coupon bond, maturity T , as seen at t. These discount factors and zero coupon bonds are the ones obtained from the currency’s swap curve. Clearly D(T ) = Z(0; T ). We use distinct notation for discount factors and zero coupon bonds to remind ourselves that discount factors D(T ) are not random; we can always obtain the current discount factors from the stripper. Zero coupon bonds Z(t; T ) are random, at least until time catches up to date t. Let (1. 2a) (1. 2b) These are de? ned via (1. 2c) D(T ) = e? T 0 f0 (T ) = today’s instantaneous forward rate for date T, f (t; T ) = instantaneous forward rate for date T , as seen at t. f0 (T 0 )dT 0 Z(t; T ) = e? T t f (t,T 0 )dT 0 . 1. 2. Accrual swaps (? xed). ?j t0 t1 t2 †¦ tj-1 tj †¦ tn-1 tn period j Coupon leg schedule Fixed coupon accrual swaps (aka time swaps) consist of a coupon leg swapped against a funding leg. Suppose that the agreed upon reference rate is, say, k month Libor. Let (1. 3) t0 t1 t2  ·  ·  · tn? 1 tn 1 Rfix Rmin Rmax L(? ) Fig. 1. 1. Daily coupon rate be the schedule of the coupon leg, and let the nominal ? xed rate be Rf ix . Also let L(? st ) represent the k month Libor rate ? xed for the interval starting at ? st and ending at ? end (? st ) = ? t + k months. Then the coupon paid for period j is (1. 4a) where (1. 4b) and (1. 4c) ? j = #days ? st in the interval with Rmin ? L(? st ) ? Rmax . Mj ? j = cvg(tj? 1 , tj ) = day count fraction for tj? 1 to tj , Cj = ? j Rf ix ? j paid at tj , Here Mj is the total number of days in interval j, and Rmin ? L(? st ) ? Rmax is the agreed-upon accrual range. Said another way, each day ? st in the j th period contibutes the amount ? ?j Rf ix 1 if Rmin ? L(? st ) ? Rmax (1. 5) 0 otherwise Mj to the coupon paid on date tj . For a standard deal, the leg’s schedule is contructed like a standard swap schedule. The theoretical dates (aka nominal dates) are constructed monthly, quarterly, semi-annually, or annually (depending on the contract terms) backwards from the â€Å"theoretical end date. † Any odd coupon is a stub (short period) at the front, unless the contract explicitly states long ? rst, short last, or long last. The modi? ed following business day convention is used to obtain the actual dates tj from the theoretical dates. The coverage (day count fraction) is adjusted, that is, the day count fraction for period j is calculated from the actual dates tj? 1 and tj , not the theoretical dates. Also, L(? t ) is the ? xing that pertains to periods starting on date ? st , regardless of whether ? st is a good business day or not. I. e. , the rate L(? st ) set for a Friday start also pertains for the following Saturday and Sunday. Like all ? xed legs, there are many variants of these coupon legs. The only variations that do not make sense for coupon legs are â€Å"set-in-arrearsâ €  and â€Å"compounded. † There are three variants that occur relatively frequently: Floating rate accrual swaps. Minimal coupon accrual swaps. Floating rate accrual swaps are like ordinary accrual swaps except that at the start of each period, a ? ating rate is set, and this rate plus a margin is 2 used in place of the ? xed rate Rf ix . Minimal coupon accrual swaps receive one rate each day Libor sets within the range and a second, usually lower rate, when Libor sets outside the range ? j Mj ? Rf ix Rf loor if Rmin ? L(? st ) ? Rmax . otherwise (A standard accrual swap has Rf loor = 0. These deals are analyzed in Appendix B. Range notes. In the above deals, the funding leg is a standard ?oating leg plus a margin. A range note is a bond which pays the coupon leg on top of the principle repayments; there is no ? oating leg. For these deals, the counterparty’s credit-worthiness is a key concern. To determine the correct value of a range note, one needs to use an option adjusted spread (OAS) to re? ect the extra discounting re? ecting the counterparty’s credit spread, bond liquidity, etc. See section 3. Other indices. CMS and CMT accrual swaps. Accrual swaps are most commonly written using 1m, 3m, 6m, or 12m Libor for the reference rate L(? st ). However, some accrual swaps use swap or treasury rates, such as the 10y swap rate or the 10y treasury rate, for the reference rate L(? st ). These CMS or CMT accrual swaps are not analyzed here (yet). There is also no reason why the coupon cannot set on other widely published indices, such as 3m BMA rates, the FF index, or the OIN rates. These too will not be analyzed here. 2. Valuation. We value the coupon leg by replicating the payo? in terms of vanilla caps and ? oors. Consider the j th period of a coupon leg, and suppose the underlying indice is k-month Libor. Let L(? st ) be the k-month Libor rate which is ? xed for the period starting on date ? st and ending on ? end (? st ) = ? st +k months. The Libor rate will be ? xed on a date ? f ix , which is on or a few days before ? st , depending on currency. On this date, the value of the contibution from day ? st is clearly ? ? ? j Rf ix V (? f ix ; ? st ) = payo? = Z(? f ix ; tj ) Mj ? 0 if Rmin ? L(? st ) ? Rmax otherwise (2. 1) , where ? f ix the ? xing date for ? st . We value coupon j by replicating each day’s contribution in terms of vanilla caplets/? oorlets, and then summing over all days ? st in the period. Let Fdig (t; ? st , K) be the value at date t of a digital ? oorlet on the ? oating rate L(? st ) with strike K. If the Libor rate L(? st ) is at or below the strike K, the digital ? oorlet pays 1 unit of currency on the end date ? end (? st ) of the k-month interval. Otherwise the digital pays nothing. So on the ? xing date ? f ix the payo? is known to be ? 1 if L(? st ) ? K , (2. 2) Fdig (? f ix ; ? st , K) = Z(? f ix ; ? end ) 0 otherwise We can replicate the range note’s payo? for date ? st by going long and short digitals struck at Rmax and Rmin . This yields, (2. 3) (2. 4) ? j Rf ix [Fdig (? f ix ; ? st , Rmax ) ? Fdig (? f ix ; ? st , Rmin )] Mj ? ?j Rf ix 1 = Z(? f ix ; ? end ) 0 Mj 3 if Rmin ? L(? st ) ? Rmax . otherwise This is the same payo? as the range note, except that the digitals pay o? on ? end (? st ) instead of tj . 2. 1. Hedging considerations. Before ? ing the date mismatch, we note that digital ? oorlets are considered vanilla instruments. This is because they can be replicated to arbitrary accuracy by a bullish spread of ? oorlets. Let F (t, ? st , K) be the value on date t of a standard ? oorlet with strike K on the ? oating + rate L(? st ). This ? oorlet pays ? [K ? L(? st )] on the end date ? end (? st ) of the k-m onth interval. So on the ? xing date, the payo? is known to be (2. 5a) F (? f ix ; ? st , K) = ? [K ? L(? st )] Z(? f ix ; ? end ). + Here, ? is the day count fraction of the period ? st to ? end , (2. 5b) ? = cvg(? st , ? end ). 1 ? oors struck at K + 1 ? nd short the same number struck 2 The bullish spread is constructed by going long at K ? 1 ?. This yields the payo? 2 (2. 6) which goes to the digital payo? as ? 0. Clearly the value of the digital ? oorlet is the limit as ? 0 of (2. 7a) Fcen (t; ? st , K, ? ) = ? 1  © F (t; ? st , K + 1 ? ) ? F (t; ? st , K ? 1 ? ) . 2 2 ? 1  © F (? f ix ; ? st , K + 1 ? ) ? F (? f ix ; ? st , K ? 1 ? ) 2 2 ? ? ? ? 1 ? 1 = Z(? f ix ; ? end ) K + 1 ? ? L(? st ) 2 ? ? ? 0 if K ? 1 ? L(? st ) K + 1 ? , 2 2 if K + 1 ? L(? st ) 2 if L(? st ) K ? 1 ? 2 Thus the bullish spread, and its limit, the digitial ? orlet, are directly determined by the market prices of vanilla ? oors on L(? st ). Digital ? oorlets may develop an unbounded ? -risk a s the ? xing date is approached. To avoid this di? culty, most ? rms book, price, and hedge digital options as bullish ? oorlet spreads. I. e. , they book and hedge the digital ? oorlet as if it were the spread in eq. 2. 7a with ? set to 5bps or 10bps, depending on the aggressiveness of the ? rm. Alternatively, some banks choose to super-replicate or sub-replicate the digital, by booking and hedging it as (2. 7b) or (2. 7c) Fsub (t; ? st , K, ? ) = 1 {F (t; ? st , K) ? F (t; ? st , K ? ?)} Fsup (t; ? st , K, ? ) = 1 {F (t; ? st , K + ? ) ? F (t; ? st , K)} depending on which side they own. One should price accrual swaps in accordance with a desk’s policy for using central- or super- and sub-replicating payo? s for other digital caplets and ? oorlets. 2. 2. Handling the date mismatch. We re-write the coupon leg’s contribution from day ? st as ? ?j Rf ix Z(? f ix ; tj ) ? V (? f ix ; ? st ) = Z(? f ix ; ? end ) Mj Z(? f ix ; ? end ) ? 0 4 (2. 8) if Rmin ? L(? st ) ? Rmax otherwise . f(t,T) L(? ) tj-1 ? tj ? end T Fig. 2. 1. Date mismatch is corrected assuming only parallel shifts in the forward curve The ratio Z(? ix ; tj )/Z(? f ix ; ? end ) is the manifestation of the date mismatch. To handle this mismatch, we approximate the ratio by assuming that the yield curve makes only parallel shifts over the relevent interval. See ?gure 2. 1. So suppose we are at date t0 . Then we assume that (2. 9a) Z(? f ix ; tj ) Z(t0 ; tj ) ? [L(? st )? Lf (t0 ,? st )](tj en d ) = e Z(? f ix ; ? end ) Z(t0 ; ? end ) Z(t0 ; tj ) = {1 + [L(? st ) ? Lf (t0 , ? st )](? end ? tj ) +  ·  ·  · } . Z(t0 ; ? end ) Z(t0 ; ? st ) ? Z(t0 ; ? end ) + bs(? st ), ? Z(t0 ; ? end ) Here (2. 9b) Lf (t0 , ? st ) ? is the forward rate for the k-month period starting at ? t , as seen at the current date t0 , bs(? st ) is the ? oating rate’s basis spread, and (2. 9c) ? = cvg(? st , ? end ), is the day count fraction for ? st to ? end . Since L(? st ) = Lf (? f ix , ? st ) represents the ? oating rate which is actually ? xed on the ? xing date ? ex , 2. 9a just assumes that any change in the yield curve between tj and ? end is the same as the change Lf (? f ix , ? st ) ? Lf (t0 , ? st ) in the reference rate between the observation date t0 , and the ? xing date ? f ix . See ? gure 2. 1. We actually use a slightly di? erent approximation, (2. 10a) where (2. 10b) ? = ? end ? tj . ? end ? ? st Z(? ix ; tj ) Z(t0 ; tj ) 1 + L(? st ) ? Z(? f ix ; ? end ) Z(t0 ; ? end ) 1 + Lf (t0 , ? st ) We prefer this approximation because it is the only linear approximation which accounts for the day count basis correctly, is exact for both ? st = tj? 1 and ? st = tj , and is centerred around the current forward value for the range note. 5 Rfix Rmin L0 Rmax L(? ) Fig. 2. 2. E? ective contribution from a single day ? , after accounting for the date mis-match. With this approximation, the payo? from day ? st is ? 1 + L(? st ) (2. 11a) V (? f ix ; ? ) = A(t0 , ? st )Z(? f ix ; ? end ) 0 as seen at date t0 . Here the e? ctive notional is (2. 11b) A(t0 , ? st ) = if Rmin ? L(? st ) ? Rmax otherwise 1 ? j Rf ix Z(t0 ; tj ) . Mj Z(t0 ; ? end ) 1 + Lf (t0 , ? st ) We can replicate this digital-linear-digital payo? by using a combination of two digital ? oorlets and two standard ? oorlets. Consider the combination (2. 12) V (t; ? st ) ? A(t0 , ? st ) {(1 + Rmax )Fdig (t, ? st ; Rmax ) ? (1 + ? Rmin )Fdig (t, ? st ; Rmin ) F (t, ? st ; Rmax ) + ? F (t, ? st ; Rmi n ). Setting t to the ? xing date ? f ix shows that this combination matches the contribution from day ? st in eq. 2. 11a. Therefore, this formula gives the value of the contribution of day ? t for all earlier dates t0 ? t ? ? f ix as well. Alternatively, one can replicate the payo? as close as one wishes by going long and short ? oorlet spreads centerred around Rmax and Rmin . Consider the portfolio (2. 13a) A(t0 , ? st )  © ? V (t; ? st , ? ) = a1 (? st )F (t; ? st , Rmax + 1 ? ) ? a2 (? st )F (t; ? st , Rmax ? 1 ? ) 2 2 ? 1 ? a3 (? st )F (t; ? st , Rmin + 2 ? )+ a4 (? st )F (t; ? st , Rmin ? 1 ? ) 2 a1 (? st ) = 1 + (Rmax ? 1 ? ), 2 a3 (? st ) = 1 + (Rmin ? 1 ? ), 2 ? ? a2 (? st ) = 1 + (Rmax + 1 ? ), 2 a4 (? st ) = 1 + (Rmin + 1 ? ). 2 with (2. 13b) (2. 13c) Setting t to ? ix yields (2. 14) ? V = A(t0 , ? st )Z(? f ix ; ? end ) 0 if L(? st ) Rmin ? 1 ? 2 1 + L(? st ) if Rmin + 1 ? L(? st ) Rmax ? 1 ? , 2 2 ? 0 if Rmax + 1 ? L(? st ) 2 6 with linear ramps between Rmin ? 1 ? L(? st ) Rmin + 1 ? and Rmax ? 1 ? L(? st ) Rmax + 1 ?. As 2 2 2 2 above, most banks would choose to use the ? oorlet spreads (with ? being 5bps or 10bps) instead of using the more troublesome digitals. For a bank insisting on using exact digital options, one can take ? to be 0. 5bps to replicate the digital accurately.. We now just need to sum over all days ? t in period j and all periods j in the coupon leg, (2. 15) Vcpn (t) = n X This formula replicates the value of the range note in terms of vanilla ? oorlets. These ? oorlet prices should be obtained directly from the marketplace using market quotes for the implied volatilities at the relevent strikes. Of course the centerred spreads could be replaced by super-replicating or sub-replicating ? oorlet spreads, bringing the pricing in line with the bank’s policies. Finally, we need to value the funding leg of the accrual swap. For most accrual swaps, the funding leg ? ? pays ? oating plus a margin. Let the fundin g leg dates be t0 , t1 , . . , tn . Then the funding leg payments are (2. 16) f ? ? cvg(ti? 1 , ti )[Ri lt + mi ]  ¤ A(t0 , ? st )  ©? 1 + (Rmax ? 1 ? ) F (t; ? st , Rmax + 1 ? ) 2 2 j=1 ? st =tj? 1 +1 ?  ¤ ? 1 + (Rmax + 1 ? ) F (t; ? st , Rmax ? 1 ? ) 2 2 ?  ¤ ? 1 + (Rmin ? 1 ? ) F (t; ? st , Rmin + 1 ? ) 2 2 ?  ¤ ? + 1 + (Rmin + 1 ? ) F (t; ? st , Rmin ? 1 ? ) . 2 2 tj X ? paid at ti , i = 1, 2, †¦ , n, ? f ? ? where Ri lt is the ? oating rate’s ? xing for the period ti? 1 t ti , and the mi is the margin. The value of the funding leg is just n ? X i=1 (2. 17a) Vf und (t) = ? ? ? cvg(ti? 1 , ti )(ri + mi )Z(t; ti ), ? ? where, by de? ition, ri is the forward value of the ? oating rate for period ti? 1 t ti : (2. 17b) ri = ? ? Z(t; ti? 1 ) ? Z(t; ti ) true + bs0 . + bs0 = ri i i ? ? ? cvg(ti? 1 , ti )Z(t; ti ) true is the true (cash) rate. This sum Here bs0 is the basis spread for the funding leg’s ? oating rate, and ri i collapses 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 Black vols. Suppose that on date t, a ? oorlet with ? xing date tf ix , start date ? st , end date ? end , and strike K has an implied vol of ? imp (K) ? ? imp (? st , K). Then its market price is (2. 19a) F (t, ? st , K) = ? Z(t; ? end ) {KN (d1 ) ? L(t, ? )N (d2 )} , 7 where (2. 19b) Here (2. 19c) d1,2 = log K/L(t, ? st )  ± 1 ? 2 (K)(tf ix ? t) 2 imp , v ? imp (K) tf ix ? t Z(t; ? st ) ? Z(t; ? end ) + bs(? st ) ? Z(t; ? end ) L(t, ? st ) = is ? oorlet’s forward rate as seen at date t. Today’s ? oorlet value is simply (2. 20a) where (2. 20b) d1,2 = log K/L0 (? st )  ± 1 ? (K)tf ix 2 imp , v ? imp (K) tf ix D(? st ) ? D(? end ) + bs(? st ). ?D(? end ) ? j Rf ix D(tj ) 1 . Mj D(? end ) 1 + L0 (? st ) F (0, ? st , K) = ? D(? end ) {KN (d1 ) ? L0 (? )N (d2 )} , and where today’s forward Libor rate is (2. 20c) L0 (? st ) = To obtain today’s price of the accrual swap, note that the e? ective notional for period j is (2. 21) A(0, ? st ) = as seem today. See 2. 11b. Putting this together with 2. 13a shows that today’s price is Vcpn (0) ? Vf und (0), where (2. 22a) Vcpn (0) = n X ? j Rf ix D(tj ) j=1 Mj  ¤ ?  ¤ ? 1 + (Rmax ? 1 ? ) B1 (? st ) ? 1 + (Rmax + 1 ? ) B2 (? st ) 2 2 ? [1 + L0 (? st )] ? t =tj? 1 +1  ¤ ?  ¤ ? 1 + (Rmin ? 1 ? ) B3 (? st ) ? 1 + (Rmin + 1 ? ) B4 (? st ) 2 2 ? , ? [1 + L0 (? st )] tj X n ? X i=1 (2. 22b) Vf und (0) = D(t0 ) ? D(tn ) + ? ? ? ? cvg(ti? 1 , ti )(bs0 +mi )D(ti ). i Here B? are Black’s formula at strikes around the boundaries: (2. 22c) (2. 22d) with (2. 22e) K1,2 = Rmax  ± 1 ? , 2 K3,4 = Rmin  ± 1 ?. 2 B? (? st ) = K? N (d? ) ? L0 (? st )N (d? ) 1 2 d? = 1,2 log K? /L0 (? st )  ± 1 ? 2 (K? )tf ix 2 imp v ? imp (K? ) tf ix Calculating the sum of each day’s contribution is very tedious. Normally, one calculates each day’s contribution for the current period and two or three months afterward. After that, one usually replaces the sum over dates ? with an integral, and samples the contribution from dates ? one week apart for the next year, and one month apart for subsequent years. 8 3. Callable accrual swaps. A callable accrual swap is an accrual swap in which the party paying the coupon leg has the right to cancel on any coupon date after a lock-out period expires. For example, a 10NC3 with 5 business days notice can be called on any coupon date, starting on the third anniversary, provided the appropriate notice is given 5 days before the coupon date. We will value the accrual swap from the viewpoint of the receiver, who would price the callable accrual swap as the full accrual swap (coupon leg minus funding leg) minus the Bermudan option to enter into the receiver accrual swap. So a 10NC3 cancellable quarterly accrual swap would be priced as the 10 year bullet quarterly receiver accrual swap minus the Bermudan option – with quarterly exercise dates starting in year 3 – to receive the remainder of the coupon leg and pay the remainder of the funding leg. Accordingly, here we price Bermudan options into receiver accrual swaps. Bermudan options on payer accrual swaps can be priced similarly. There are two key requirements in pricing Bermudan accrual swaps. First, as Rmin decreases and Rmax increases, the value of the Bermudan accrual swap should reduce to the value of an ordinary Bermudan swaption with strike Rf ix . Besides the obvious theoretical appeal, meeting this requirement allows one to hedge the callability of the accrual swap by selling an o? setting Bermudan swaption. This criterion requires using the same the interest rate model and calibration method for Bermudan accrual notes as would be used for Bermudan swaptions. Following standard practice, one would calibrate the Bermudan accrual note to the â€Å"diagonal swaptions† struck at the accrual note’s â€Å"e? ective strikes. † For example, a 10NC3 accrual swap which is callable quarterly starting in year 3 would be calibrated to the 3 into 7, the 3. 25 into 6. 75, †¦ , the i 8. 75 into 1. 25, and the 9 into 1 swaptions. The strike Ref f for each of these â€Å"reference swaptions† would be chosen so that for swaption i, (3. 1) value of the ? xed leg value of all accrual swap coupons j ? i = value of the ? oating leg value of the accrual swap’s funding leg ? i This usually results in strikes Ref f that are not too far from the money. In the preceding section we showed that each coupon of the accrual swap can be written as a combination of vanilla ? oorlets, and therefore the market value of each coupon is known exactly. The second requirement is that the valuation procedure should reproduce today’s m arket value of each coupon exactly. In fact, if there is a 25% chance of exercising into the accrual swap on or before the j th exercise date, the pricing methodology should yield 25% of the vega risk of the ? oorlets that make up the j th coupon payment. E? ectively this means that the pricing methodology needs to use the correct market volatilities for ? oorlets struck at Rmin and Rmax . This is a fairly sti? requirement, since we now need to match swaptions struck at i Ref f and ? oorlets struck at Rmin and Rmax . This is why callable range notes are considered heavily skew depedent products. 3. 1. Hull-White model. Meeting these requirements would seem to require using a model that is sophisticated enough to match the ? oorlet smiles exactly, as well as the diagonal swaption volatilities. Such a model would be complex, calibration would be di? ult, and most likely the procedure would yield unstable hedges. An alternative approach is to use a much simpler model to match the diagonal swaption prices, and then use â€Å"internal adjusters† to match the ? oorlet volatilities. Here we follow this approach, using the 1 factor linear Gauss Markov (LGM) model with internal adjusters to price Bermudan options on accrual swaps. Speci ? cally, we ? nd explicit formulas for the LGM model’s prices of standard ? oorlets. This enables us to compose the accrual swap â€Å"payo? s† (the value recieved at each node in the tree if the Bermudan is exercised) as a linear combination of the vanilla ? orlets. With the payo? s known, the Bermudan can be evaluated via a standard rollback. The last step is to note that the LGM model misprices the ? oorlets that make up the accrual swap coupons, and use internal adjusters to correct this mis-pricing. Internal adjusters can be used with other models, but the mathematics is more complex. 3. 1. 1. LGM. The 1 factor LGM model is exactly the Hull-White model expressed as an HJM model. The 1 factor LGM model has a single state variable x that determines the entire yield curve at any time t. 9 This model can be summarized in three equations. The ? st is the Martingale valuation formula: At any date t and state x, the value of any deal is given by the formula, Z V (t, x) V (T, X) (3. 2a) = p(t, x; T, X) dX for any T t. N (t, x) N (T, X) Here p(t, x; T, X) is the probability that the state variable is in state X at date T , given that it is in state x at date t. For the LGM model, the transition density is Gaussian 2 1 e? (X? x) /2[? (T ) (t)] , p(t, x; T, X) = p 2? [? (T ) ? ?(t)] (3. 2b) with a variance of ? (T ) ? ?(t). The numeraire is (3. 2c) N (t, x) = 1 h(t)x+ 1 h2 (t)? (t) 2 , e D(t) for reasons that will soon become apparent. Without loss of generality, one sets x = 0 at t = 0, and today’s variance is zero: ? (0) = 0. The ratio (3. 3a) V (t, x) ? V (t, x) ? N (t, x) is usually called the reduced value of the deal. Since N (0, 0) = 1, today’s value coincides with today’s reduced value: (3. 3b) V (0, 0) ? V (0, 0) = V (0, 0) ? . N (0, 0) So we only have to work with reduced values to get today’s prices.. De? ne Z(t, x; T ) to be the value of a zero coupon bond with maturity T , as seen at t, x. It’s value can be found by substituting 1 for V (T, X) in the Martingale valuation formula. This yields (3. 4a) 1 2 Z(t, x; T ) ? Z(t, x; T ) ? = D(T )e? (T )x? 2 h (T )? (t) . N (t, x) Since the forward rates are de? ned through (3. 4b) Z(t, x; T ) ? e? T t f (t,x;T 0 )dT 0 , ? taking ? ?T log Z shows that the forward rates are (3. 4c) f (t, x; T ) = f0 (T ) + h0 (T )x + h0 (T )h(T )? (t). This last equation captures the LGM model in a nutshell. The curves h(T ) and ? (t) are model parameters that need to be set by calibration or by a priori reasoning. The above formula shows that at any date t, the forward rate curve is given by today’s forward rate curve f0 (T ) plus x times a second curve h0 (T ), where x is a Gaussian random variable with mean zero and variance ? (t). Thus h0 (T ) determines possible shapes of the forward curve and ? (t) determines the width of the distribution of forward curves. The last term h0 (T )h(T )? (t) is a much smaller convexity correction. 10 3. 1. 2. Vanilla prices under LGM. Let L(t, x; ? st ) be the forward value of the k month Libor rate for the period ? st to ? end , as seen at t, x. Regardless of model, the forward value of the Libor rate is given by (3. 5a) where (3. 5b) ? = cvg(? st , ? end ) L(t, x; ? st ) = Z(t, x; ? st ) ? Z(t, x; ? end ) + bs(? st ) = Ltrue (t, x; ? st ) + bs(? st ), ? Z(t, x; ? end ) is the day count fraction of the interval. Here Ltrue is the forward â€Å"true rate† for the interval and bs(? ) is the Libor rate’s basis spread for the period starting at ? . Let F (t, x; ? st , K) be the value at t, x of a ? oorlet with strike K on the Libor rate L(t, x; ? st ). On the ? xing date ? f ix the payo? is (3. 6) ?  ¤+ F (? f ix , xf ix ; ? st , K) = ? K ? L(? f ix , xf ix ; ? st ) Z(? f ix , xf ix ; ? end ), where xf ix is the state variable on the ? xing date. Substituting for L(? ex , xex ; ? st ), the payo? becomes (3. 7a)  · ? + F (? f ix , xf ix ; ? st , K) Z(? f ix , xf ix ; ? st ) Z(? f ix , xf ix ; ? end ) . = 1 + ? (K ? bs(? st )) ? N (? ix , xf ix ) N (? f ix , xf ix ) Z(? f ix , xf ix ; ? end ) Knowing the value of the ? oorlet on the ? xing date, we can use the Martingale valuation formula to ? nd the value on any earlier date t: Z 2 1 F (t, x; ? st , K) F (? f ix , xf ix ; ? st , K) e? (xf ix ? x) /2[? f ix ] =q dxf ix , (3. 7b) N (t, x) N (? f ix , xf ix ) 2? [? f ix ? ?] where ? f ix = ? (? f ix ) and ? = ? (t). Substituting the zero coupon bond formula 3. 4a and the payo? 3. 7a into the integral yields (3. 8a) where log (3. 8b) ? 1,2 =  µ 1 + ? (K ? bs) 1 + ? (L ? bs)  ¤ ?  ± 1 (hend ? hst )2 ? f ix ? ?(t) 2 q , (hend ? hst ) ? f ix ? (t)  ¶ F (t, x; ? st , K) = Z(t, x; ? end ) {[1 + ? (K ? bs)]N (? 1 ) ? [1 + ? (L ? bs)]N (? 2 )} , and where L ? L(t, x; ? st ) = (3. 8c)  µ  ¶ 1 Z(t, x; ? st ) ? 1 + bs(? st ) ? Z(t, x; ? end )  ¶  µ 1 Dst (hend ? hst )x? 1 (h2 ? h2 )? end st 2 = e ? 1 + bs(? st ) ? Dend 11 is the forward Libor rate for the period ? st to ? end , as seen at t, x. Here hst = h(? st ) and hend = h(? end ). For future reference, it is convenient to split o? the zero coupon bond value Z(t, x; ? end ). So de? ne the forwarded ? oorlet value by (3. 9) Ff (t, x; ? st , K) = F (t, x; ? st , K) Z(t, x; ? end ) = [1 + ? (K ? bs)]N (? 1 ) ? [1 + ? L(t, x; ? st ) ? bs)]N (? 2 ). Equations 3. 8a and 3. 9 are just Black’s formul as for the value of a European put option on a log normal asset, provided we identify (3. 10a) (3. 10b) (3. 10c) (3. 10d) 1 + ? (L ? bs) = asset’s forward value, 1 + ? (K ? bs) = strike, ? end = settlement date, and p ? f ix ? ? (hend ? hst ) v = ? = asset volatility, tf ix ? t where tf ix ? t is the time-to-exercise. One should not confuse ? , which is the ? oorlet’s â€Å"price volatility,† with the commonly quoted â€Å"rate volatility. † 3. 1. 3. Rollback. Obtaining the value of the Bermudan is straightforward, given the explicit formulas for the ? orlets, . Suppose that the LGM model has been calibrated, so the â€Å"model parameters† h(t) and ? (t) are known. (In Appendix A we show one popular calibration method). Let the Bermudan’s noti? cation dates be tex , tex+1 , . . . , tex . Suppose that if we exercise on date tex , we receive all coupon payments for the K k0 k0 k intervals k + 1, . . . , n and recieve all funding leg payments f or intervals ik , ik + 1, . . . , n. ? The rollback works by induction. Assume that in the previous rollback steps, we have calculated the reduced value (3. 11a) V + (tex , x) k = value at tex of all remaining exercises tex , tex . . . , tex k k+1 k+2 K N (tex , x) k at each x. We show how to take one more step backwards, ? nding the value which includes the exercise tex k at the preceding exercise date: (3. 11b) V + (tex , x) k? 1 = value at tex of all remaining exercises tex , tex , tex . . . . , tex . k? 1 k k+1 k+2 K N (tex , x) k? 1 Let Pk (x)/N (tex , x) be the (reduced) value of the payo? obtained if the Bermudan is exercised at tex . k k As seen at the exercise date tex the e? ective notional for date ? st is k (3. 12a) where we recall that (3. 12b) ? = ? end (? st ) ? tj , ? end (? st ) ? ? st ? = cvg(? st , ? end (? st )). 12 A(tex , x, ? t ) = k ?j Rf ix Z(tex , x; tj ) 1 k , Mj Z(tex , x; ? end ) 1 + Lf (tex , x; ? st ) k k Reconstructing the reduced value of the payo? (see equation 2. 15) yields (3. 13a) Pk (x) = N (tex , x) k n X ? j Rf ix Z(tex , x; tj ) k Mj N (tex , x) ? k tj X j=k+1 st =tj? 1 +1 ? 1 + (Rmax ? 1 ? ) 2 Ff (tex , x; ? st , Rmax + 1 ? ) k 2 1 + Lf (tex , x; ? st ) k ? ? 1 + (Rmax + 1 ? ) 2 Ff (tex , x; ? st , Rmax ? 1 ? ) k 2 1 + Lf (tex , x; ? st ) k 1 + (Rmin ? 1 ? ) 2 Ff (tex , x; ? st , Rmin + 1 ? ) k 2 1 + Lf (tex , x; ? st ) k 1 + (Rmin + 1 ? ) 2 + Ff (tex , x; ? st , Rmin ? 1 ? ) k 2 1 + Lf (tex , x; ? st ) k ? n ? X ? ? Z(tex , x, tik ? 1 ) ? Z(tex , x, tn ) Z(tex , x, ti ) k k k ? ? cvg(ti? 1 , ti )(bsi +mi ) ? ex , x) ex , x) . N (tk N (tk i=i +1 k ? This payo? includes only zero coupon bonds and ? oorlets, so we can calculate this reduced payo? explicitly using the previously derived formula 3. 9. The reduced valued including the kth exercise is clearly ? ? Pk (x) V + (tex , x) V (tex , x) k k = max , at each x. (3. 13b) N (tex , x) N (tex , x) N (tex , x) k k k Using the Martingale valuation formula we can â€Å"roll di? erences, trees, convolution, or direct integration to Z V + (tex , x) 1 k? 1 (3. 3c) =p N (tex , x) 2? [? k ? ? k? 1 ] k? 1 back† to the preceding exercise date by using ? nite compute the integral V (tex , X) ? (X? x)2 /2[? k k? 1 ] k dX e N (tex , X) k at each x. Here ? k = ? (tex ) and ? k? 1 = ? (tex ). k k? 1 At this point we have moved from tex to the preceding exercise date tex . We now repeat the procedure: k k? 1 at each x we t ake the max of V + (tex , x)/N (tex , x) and the payo? Pk? 1 (x)/N (tex , x) for tex , and then k? 1 k? 1 k? 1 k? 1 use the valuation formula to roll-back to the preceding exercise date tex , etc. Eventually we work our way k? 2 througn the ? rst exercise V (tex , x). Then today’s value is found by a ? nal integration: k0 Z V (tex , X) ? X 2 /2? V (0, 0) 1 k0 k0 dX. (3. 14) V (0, 0) = =p e N (0, 0) N (tex , X) 2 k0 k0 3. 2. Using internal adjusters. The above pricing methodology satis? es the ? rst criterion: Provided we use LGM (Hull-White) to price our Bermudan swaptions, and provided we use the same calibration method for accrual swaps as for Bermudan swaptions, the above procedure will yield prices that reduce to the Bermudan prices as Rmin goes to zero and Rmax becomes large. However the LGM model yields the following formulas for today’s values of the standard ? orlets: F (0, 0; ? st , K) = D(? end ) {[1 + ? (K ? bs)]N (? 1 ) ? [1 + ? (L0 ? bs)]N (? 2 )} log  µ  ¶ 1 + ? (K ? bs)  ± 1 ? 2 tf ix 2 mod 1 + ? (L0 ? bs) . v ? mod tf ix 13 (3. 15a) where (3. 15b) ?1,2 = Here (3. 15c) L0 = Dst ? Dend + bs(? st ) ? Dend is today’s forward value for the Libor rate, and (3. 15d) q ? mod = (hend ? hst ) ? f ix /tf ix 3. 2. 1. Obtaining the market vol. Floorlets are quoted in terms of the ordinary (rate) vol. Suppose the rate vol is quoted as ? imp (K). Then today’s market price of the ? oorlet is is the asset’s log normal volatility according to the LGM model. We did not calibrate the LGM model to these ? oorlets. It is virtually certain that matching today’s market prices for the ? oorlets will require using q an implied (price) volatility ? mkt which di? ers from ? mod = (hend ? hst ) ? f ix /tf ix . (3. 16a) where (3. 16b) Fmkt (? st , K) = ? D(? end ) {KN (d1 ) ? L0 N (d2 )} d1,2 = log K/L0  ± 1 ? 2 (K)tf ix 2 imp v ? imp (K) tf ix The price vol ? mkt is the volatility that equates the LGM ? oorlet value to this market value. It is de? ned implicitly by (3. 17a) with log (3. 17b) ? 1,2 =  µ  ¶ 1 + ? (K ? bs)  ± 1 ? 2 tf ix 2 mkt 1 + ? (L0 ? bs) v ? kt tf ix [1 + ? (K ? bs)]N (? 1 ) ? [1 + ? (L0 ? bs)]N (? 2 ) = ? KN (d1 ) ? ?L0 N (d2 ), (3. 17c) d1,2 = log K/L0  ± 1 ? 2 (K)tf ix 2 imp v ? imp (K) tf ix Equivalent vol techniques can be used to ? nd the price vol ? mkt (K) which corresponds to the market-quoted implied rate vol ? imp (K) : (3. 18) ? imp (K) = 1 + 5760 ? 4 t2 ix +  ·  ·  · 1+ imp f ? mkt (K) 1 2 1 4 2 24 ? mkt tf ix + 5760 ? mkt tf ix  µ log L0 /K  ¶ 1 + ? (L0 ? bs) 1 + ? (K ? bs) 1+ 1 2 24 ? imp tf ix log If this approximation is not su? ciently accurate, we can use a single Newton step to attain any reasonable accuracy. 14 igital floorlet value ? mod ? mkt L0/K Fig. 3. 1. Unadjusted and adjusted digital payo? L/K 3. 2. 2. Adjusting the price vol. The price vol ? mkt obtained from the market price will not match the q LGM model’s price vol ? mod = (hend ? hst ) ? f ix /tf ix . This is easily remedied using an internal adjuster. All one does is multiply the model volatility with the factor needed to bring it into line with the actual market volatility, and use this factor when calculating the payo? s. Speci? cally, in calculating each payo? Pk (x)/N (tex , x) in the rollback (see eq. 3. 13a), one makes the replacement k (3. 9) (3. 20) (hend ? hst ) q q ? mkt ? f ix ? ?(tex ) =? (hend ? hst ) ? f ix ? ?(t) k ? mod q p = 1 ? ?(tex )/? (tf ix )? mkt tf ix . k With the internal adjusters, the pricing methodology now satis? es the second criteria: it agrees with all the vanilla prices that make up the range note coupons. Essentially, all the adjuster does is to slightly â€Å"sharpen up† or â€Å"smear out† the digital ? oorlet’s payo? to match today’s value at L0 /K. This results in slightly positive or negative price corrections at various values of L/K, but these corrections average out to zero when averaged over all L/K. Making this volatility adjustment is vastly superior to the other commonly used adjustment method, which is to add in a ? ctitious â€Å"exercise fee† to match today’s coupon value. Adding a fee gives a positive or negative bias to the payo? for all L/K, even far from the money, where the payo? was certain to have been correct. Meeting the second criterion forced us to go outside the model. It is possible that there is a subtle arbitrage to our pricing methodology. (There may or may not be an arbitrage free model in which extra factors – positively or negatively correlated with x – enable us to obtain exactly these ? orlet prices while leaving our Gaussian rollback una? ected). However, not matching today’s price of the underlying accrual swap would be a direct and immediate arbitrage. 15 4. Range notes and callable range notes. In an accrual swap, the coupon leg is exchanged for a funding leg, which is normally a standard Libor leg plus a margin. U nlike a bond, there is no principle at risk. The only credit risk is for the di? erence in value between the coupon leg and the ? oating leg payments; even this di? erence is usually collateralized through various inter-dealer arrangements. Since swaps are indivisible, liquidity is not an issue: they can be unwound by transferring a payment of the accrual swap’s mark-to-market value. For these reasons, there is no detectable OAS in pricing accrual swaps. A range note is an actual bond which pays the coupon leg on top of the principle repayments; there is no funding leg. For these deals, the issuer’s credit-worthiness is a key concern. One needs to use an option adjusted spread (OAS) to obtain the extra discounting re? ecting the counterparty’s credit spread and liquidity. Here we analyze bullet range notes, both uncallable and callable. The coupons Cj of these notes are set by the number of days an index (usually Libor) sets in a speci? ed range, just like accrual swaps: ? tj X ? j Rf ix 1 if Rmin ? L(? st ) ? Rmax (4. 1a) Cj = , 0 otherwise Mj ? =t +1 st j? 1 where L(? st ) is k month Libor for the interval ? st to ? end (? st ), and where ? j and Mj are the day count fraction and the total number of days in the j th coupon interval tj? 1 to tj . In addition, these range notes repay the principle on the ? nal pay date, so the (bullet) range note payments are: (4. 1b) (4. 1c) Cj 1 + Cn paid on tj , paid on tn . j = 1, 2, . . . n ? 1, For callable range notes, let the noti? ation on dates be tex for k = k0 , k0 + 1, . . . , K ? 1, K with K n. k Assume that if the range note is called on tex , then the strike price Kk is paid on coupon date tk and the k payments Cj are cancelled for j = k + 1, . . . , n. 4. 1. Modeling option adjusted spreads. Suppose a range note is issued by issuer A. ZA (t, x; T ) to be the value of a dollar paid by the note on date T , as seen at t, x. We assume that (4. 2) ZA (t, x; T ) = Z(t, x; T ) ? (T ) , ? (t) De? ne where Z(t, x; T ) is the value according to the Libor curve, and (4. 3) ? (? ) = DA (? ) . e D(? ) Here ? is the OAS of the range note. The choice of the discount curve DA (? ) depends on what we wish the OAS to measure. If one wishes to ? nd the range note’s value relative to the issuer’s other bonds, then one should use the issuer’s discount curve for DA (? ); the OAS then measures the note’s richness or cheapness compared to the other bonds of issuer A. If one wishes to ? nd the note’s value relative to its credit risk, then the OAS calculation should use the issuer’s â€Å"risky discount curve† or for the issuer’s credit rating’s risky discount curve for DA (? ). If one wishes to ? nd the absolute OAS, then one should use the swap market’s discount curve D(? , so that ? (? ) is just e . When valuing a non-callable range note, we are just determining which OAS ? is needed to match the current price. I. e. , the OAS needed to match the market’s idiosyncratic preference or adversion of the bond. When valuing a callable range note, we are ma king a much more powerful assumption. By assuming that the same ? can be used in evaluating the calls, we are assuming that (1) the issuer would re-issue the bonds if it could do so more cheaply, and (2) on each exercise date in the future, the issuer could issue debt at the same OAS that prevails on today’s bond. 16 4. 2. Non-callable range notes. Range note coupons are ? xed by Libor settings and other issuerindependent criteria. Thus the value of a range note is obtained by leaving the coupon calculations alone, and replacing the coupon’s discount factors D(tj ) with the bond-appropriate DA (tj )e tj : (4. 4a) VA (0) = n X j=1 ?j Rf ix DA (tj )e tj Mj  ¤ ?  ¤ ? 1 + (Rmax ? 1 ? ) B1 (? st ) ? 1 + (Rmax + 1 ? ) B2 (? st ) 2 2 ? [1 + L0 (? st )] ? st =tj? 1 +1  ¤ ?  ¤ ? 1 + (Rmin ? 1 ? ) B3 (? st ) ? 1 + (Rmin + 1 ? ) B4 (? st ) 2 2 ? ? [1 + L0 (? st )] +DA (tn )e tn . tj X Here the last term DA (tn )e n is the value of the notional repaid at maturity. As before, the B? are Black’s formulas, (4. 4b) B? (? st ) = Kj N (d? ) ? L0 (? st )N (d? ) 1 2 (4. 4c) d? = 1,2 log K? /L0 (? st )  ± 1 ? 2 (K? )tf ix 2 imp v ? imp (K? ) tf ix (4. 4d) K1,2 = Rmax  ± 1 ? , 2 K3,4 = Rmin  ± 1 ? , 2 and L0 (? ) is today’s forward rate: (4. 4e) Finally, (4. 4f) ? = ? end ? tj . ? en d ? ? st L0 (? st ) = D(? st ) ? D(? end ) ? D(? end ) 4. 3. Callable range notes. We price the callable range notes via the same Hull-White model as used to price the cancelable accrual swap. We just need to adjust the coupon discounting in the payo? function. Clearly the value of the callable range note is the value of the non-callable range note minus the value of the call: (4. 5) callable bullet Berm VA (0) = VA (0) ? VA (0). bullet Berm (0) is the today’s value of the non-callable range note in 4. 4a, and VA (0) is today’s value of Here VA the Bermudan option. This Bermudan option is valued using exactly the same rollback procedure as before, 17 except that now the payo? is (4. 6a) (4. 6b) Pk (x) = N (tex , x) k ? tj X st =tj? 1 +1 j=k+1 n X ? j Rf ix ZA (tex , x; tj ) k Mj N (tex , x) ? k 1 + (Rmax ? 1 ? ) 2 Ff (tex , x; ? st , Rmax + 1 ? ) k 2 1 + Lf (tex , x; ? st ) k ? ? + (Rmax + 1 ? ) 2 Ff (tex , x; ? st , Rmax ? 1 ? ) k 2 1 + Lf (tex , x; ? st ) k 1 + (Rmin ? 1 ? ) 2 Ff (tex , x; ? st , Rmin + 1 ? ) k 2 1 + Lf (tex , x; ? st ) k 1 + (Rmin + 1 ? ) 2 + Ff (tex , x; ? st , Rmin ? 1 ? ) k 2 1 + Lf (tex , x; ? st ) k ZA (tex , x, tn ) ZA (tex , x, tk ) k k + ? Kk ex , x) N (tk N (tex , x) k Here the bond speci? c reduced zero coupon bond value is (4. 6c) ex ex 1 2 ZA (tex , x, T ) D(tex ) k k = DA (T )e (T ? tk ) e? h(T )x? 2 h (T )? k , ex , x) N (tk DA (tex ) k ? the (adjusted) forwarded ? oorlet value is Ff (tex , x; ? st , K) = [1 + ? (K ? bs)]N (? 1 ) ? [1 + ? (L(tex , x; ? t ) ? bs)]N (? 2 ) k k log (4. 6d) ? 1,2 =  µ  ¶ 1 + ? (K ? bs)  ± 1 [1 ? ?(tex )/? (tf ix )]? 2 tf ix k mkt 2 1 + ? (L ? bs) p , v 1 ? ?(tex )/? (tf ix )? mkt tf ix k  ¶ Z(tex , x; ? st ) k ? 1 + bs(? st ) Z(tex , x; ? end ) k  ¶ (hend ? hst )x? 1 (h2 ? h2 )? ex end st k ? 1 + bs(? 2 e st ) 1 = ?  µ and the forward Libor value is (4. 6e) (4. 6f) L? L (tex , x; ? st ) k  µ Dst Dend 1 = ? The only remaining issue is calibration. For range notes, we should use constant mean reversion and calibrate along the diagonal, exactly as we did for the cancelable accrual swaps. We only need to specify the strikes of the reference swaptions. A good method is to transfer the basis spreads and margin to the coupon leg, and then match the ratio of the coupon leg to the ? oating leg. For exercise on date tk , this ratio yields (4. 7a) n X ?k = ? j Rf ix DA (tj )e (tj ? tk ) Mj Kk DA (tk ) j=k+1 (?  ¤ ?  ¤ 1 + (Rmax ? 1 ? ) B1 (? st ) ? 1 + (Rmax + 1 ? ) B1 (? st ) 2 2 ? [1 + L0 (? st )] ? st =tj? 1 +1 )  ¤ ?  ¤ ? 1 + (Rmin ? 1 ? ) B3 (? st ) ? 1 + (Rmin + 1 ? ) B3 (? st ) 2 2 ? 1 + Lf (tex , x; ? st ) k tj X + DA (tn )e (tn ? tk ) Kk DA (tk ) 18 As before, the Bj are dimensionless Black formulas, (4. 7b) B? (? st ) = K? N (d? ) ? L0 (? st )N (d? ) 1 2 d? = 1,2 log K? /L0 (? st )  ± 1 ? 2 (K? )tf ix 2 imp v ? imp (K? ) tf ix K3,4 = Rmin  ± 1 ? , 2 (4. 7c) (4. 7d) K1,2 = Rmax  ± 1 ? , 2 and L0 (? st ) is today’s forward rate: Appendix A. Calibrating the LGM model. The are several methods of calibrating the LGM model for pricing a Bermudan swaption. The most popular method is to choose a constant mean reversion ? , and then calibrate on the diagonal European swaptions making up the Bermudan. In the LGM model, a â€Å"constant mean reversion ? † means that the model function h(t) is given by (A. 1) h(t) = 1 ? e t . ? Usually the value of ? s selected from a table of values that are known to yield the correct market prices of liquid Bermudans; It is known empirically that the needed mean reversion parameters are very, very stable, changing little from year to year. ? 1M 3M 6M 1Y 3Y 5Y 7Y 10Y 1Y -1. 00% -0. 75% -0. 50% 0. 00% 0. 25% 0. 50% 1. 00% 1. 50% 2Y -0. 50% -0. 25% 0. 00% 0. 25% 0. 50% 1. 00% 1. 25% 1. 50% 3Y -0. 25% 0. 00% 0. 25% 0. 50% 1. 00% 1. 25% 1. 50% 1. 75% 4Y -0. 25% 0. 00% 0. 25% 0. 50% 1. 00% 1. 25% 1. 50% 1. 75% 5Y -0. 25% 0. 00% 0. 25% 0. 50% 1. 00% 1. 25% 1. 50% 1. 75% 7Y -0. 25% 0. 00% 0. 25% 0. 50% 1. 00% 1. 25% 1. 50% 1. 75% 10Y -0. 25% 0. 0% 0. 25% 0. 50% 1. 00% 1. 25% 1. 50% 1. 75% Table A. 1 Mean reverssion ? for Bermudan swaptions. Rows are time-to-? rst exercise; columns are tenor of the longest underlying swap obtained upon exercise. With h(t) known, we only need determine ? (t) by calibrating to European swaptions. Consider a European swaption with noti? cation date tex . Suppose that if one exercises the option, one recieves a ? xed leg worth (A. 2a) Vf ix (t, x) = n X i=1 Rf ix cvg(ti? 1 , ti , dcbf ix )Z(t, x; ti ), and pays a ? oating leg worth (A. 2b) Vf lt (t, x) = Z(t, x; t0 ) ? Z(t, x; tn ) + n X i=1 cvg(ti? 1 , ti , dcbf lt ) bsi Z(t, x; ti ). 9 Here cvg(ti? 1 , ti , dcbf ix ) and cvg(ti? 1 , ti , dcbf lt ) are the day count fraction s for interval i using the ? xed leg and ? oating leg day count bases. (For simplicity, we are cheating slightly by applying the ? oating leg’s basis spread at the frequency of the ? xed leg. Mea culpa). Adjusting the basis spread for the di? erence in the day count bases (A. 3) bsnew = i cvg(ti? 1 , ti , dcbf lt ) bsi cvg(ti? 1 , ti , dcbf ix ) allows us to write the value of the swap as (A. 4) Vswap (t, x) = Vf ix (t, x) ? Vf lt (t, x) n X = (Rf ix ? bsi ) cvg(ti? 1 , ti , dcbf ix )Z(t, x; ti ) + Z(t, x; tn ) ? Z(t, x; t0 ) i=1 Under the LGM model, today’s value of the swaption is (A. 5) 1 Vswptn (0, 0) = p 2 (tex ) Z e? xex /2? (tex ) 2 [Vswap (tex , xex )]+ dxex N (tex , xex ) Substituting the explicit formulas for the zero coupon bonds and working out the integral yields (A. 6a) n X (Rf ix ? bsi ) cvg(ti? 1 , ti , dcbf ix )D(ti )N Vswptn (0, 0) = where y is determined implicitly via (A. 6b) y + [h(ti ) ? h(t0 )] ? ex p ? ex i=1 A A ! ! y + [h(tn ) ? h(t0 )] ? ex y p ? D(t0 )N p , +D(tn )N ? ex ? ex A ! n X 2 1 (Rf ix ? bsi ) cvg(ti? 1 , ti , dcbf ix )e? [h(ti )? h(t0 )]y? 2 [h(ti )? h(t0 )] ? ex i=1 +D(tn )e? [h(tn )? h(t0 )]y? [h(tn )? h(t0 )] 1 2 ? ex = D(t0 ). The values of h(t) are known for all t, so the only unknown parameter in this price is ? (tex ). One can show that the value of the swaption is an increasing function of ? (tex ), so there is exactly one ? (tex ) which matches the LGM value of the swaption to its market price. This solution is easily found via a global Newton iteration. T o price a Bermudan swaption, one typcially calibrates on the component Europeans. For, say, a 10NC3 Bermudan swaption struck at 8. 2% and callable quarterly, one would calibrate to the 3 into 7 swaption struck at 8. 2%, the 3. 25 into 6. 5 swaption struck at 8. 2%, †¦ , then 8. 75 into 1. 25 swaption struck at 8. 25%, and ? nally the 9 into 1 swaption struck at 8. 2%. Calibrating each swaption gives the value of ? (t) on the swaption’s exercise date. One generally uses piecewise linear interpolation to obtain ? (t) at dates between the exercise dates. The remaining problem is to pick the strike of the reference swaptions. A good method is to transfer the basis spreads and margin to the coupon leg, and then match the ratio of the coupon leg to the funding leg to the equivalent ratio for a swaption. For the exercise on date tk , this ratio is de? ed to be 20 n X ? j D(tj ) (A. 7a) ? k = Mj D(tk ) ? j=k+1 D(tn ) X D(ti ) + cvg(ti? 1 , ti )(bs0 +mi ) ? i D(tk ) i=1 D(tk ) n  ¤ ?  ¤ 1 + (Rmax ? 1 ? ) B1 (? st ) ? 1 + (Rmax + 1 ? ) B2 (? st ) 2 2 ? [1 + L0 (? st )] st =tj? 1 +1  ¤ ?  ¤ ? 1 + (Rmin ? 1 ? ) B3 (? st ) ? 1 + (Rmin + 1 ? ) B4 (? st ) 2 2 ? ? [1 + L0 (? st )] tj X ? where B? are Black’s formula at strikes around the boundaries: (A. 7b) B? (? st ) = ? D(? end ) {K? N (d? ) ? L0 (? st )N (d? )} 1 2 d? = 1,2 log K? /L0 (? st )  ± 1 ? 2 (K? )tf ix 2 imp v ? imp (K? ) tf ix (A. 7c) with (A. 7d) K1,2 = Rmax  ± 1 ? , 2 K3,4 = Rmin  ± 1 ?. 2 This is to be matched to the swaption whose swap starts on tk and ends on tn , with the strike Rf ix chosen so that the equivalent ratio matches the ? k de? ned above: (A. 7e) ? k = n X i=k+1 (Rf ix ? bsi ) cvg(ti? 1 , ti , dcbf ix ) D(ti ) D(tn ) + D(tk ) D(tk ) The above methodology works well for deals that are similar to bullet swaptions. For some exotics, such as amortizing deals or zero coupon callables, one may wish to choose both the tenor of the and the strike of the reference swaptions. This allows one to match the exotic deal’s duration as well as its moneyness. Appendix B. Floating rate accrual notes. 21 How to cite Accrual Swaps, Papers

Improving Academic Achievement And Through -Myassignmenthelp.Com

Question: Discuss About The Improving Academic Achievement And Through? Answer: Introduction Students enrolled in higher education face several learning challenges. For some students, time management is an issue in learning and for others, coping with reading materials become challenging. Despite these issues, one of the main limitations found in student is that they are not active learners. This can be said because they have little interaction with their peers or tutor during the learning process (Fook Sidhu, 2015). Hence, collaborative approach is needed so that students excel in the process of learning. Due to this limitation, active learning approach has been recommended to students so that they deeply engage in their education for different purpose. Students can achieve this by different management such as peer-to-peer interaction, group discussion, collaborative assignment and group participation. This kind of participation in active learning activities can decrease failure rates and increase enthusiasm for learning among students (Armbruster et al., 2009). Although m any types of active learning methods has been mentioned, however the main focus of this report is to evaluate the role of peer-to-peer learning in promoting students independence. Exploration in this area will help to determine how peer-to-peer learning can influence key learning outcome of students. Findings and discussion Peer-to-peer learning is the process of two-way or reciprocal learning activity, which involves mutual sharing of ideas, knowledge and experience between participants. This form of participative learning with peers helps students to develop the skills of working collaboratively and receiving valuable feedback regarding their knowledge from peers (Kelly Katz, 2016). Peers may be anyone with good expertise in the area of study or little experience in study. However, people in the role of a tutor or expert practitioner cannot be regarded as peers. Hence, seniors, classmates or colleagues can regarded as peers in the learning process. Peer-to-peer learning is highly encouraged for students because it plays a major role in promoting students independence. This can be said because peer learning increases confidence of students and develop their competence in area of study or practice. The research study by Stone, Cooper Cant, (2013) also supports the fact students who want to acquire pro fessional skills should engage in peer learning. The author proved this point by exploring the role of peer learning on developing communication and critical thinking skills in undergraduate student nurses. The evaluation of effects of peer learning on students revealed that peer learning promotes independent study and problem solving skills in students. Such students developed sense of autonomy in learning and understood their responsibility in education. This proved that students get benefit from peer learning process. Another argument regarding the benefits of peer-to-peer learning process for student is that they easily accept information in this process of learning. This can be said because they can always approach their peers for advice and guidance. Peer-to-peer learning process also has the advantage of addressing barriers in the learning process. For instance, when students engage in the process of learning, they often develop anxiety when fail to interpret new concepts in their subject. Stress, anxiety and shame are some negative emotions that students develop during the learning process. This happens both for novice as well as proficient learners. However, when such students get the opportunity of peer tutoring, they develop a sense of belonging and their anxiety is reduced too. Interaction with peer and students enhance the knowledge acquisition process. This clarity in learning process opens opportunities for independence in their future professional careers too. After placement in their desired professional post, they are able to solve their problems independently (Stone, Cooper Cant, 2013). Peer learning is also favored by many educators because it has an impact on developing social skills among students. This is also related to promoting independent in students because social skills facilitate good academic and employment performance. Poor social skill is the reason for adverse learning experiences among students. They develop low self-esteem when they face challenges in the acquiring new ideas. However, Mellado, Valdebenito Aravena, (2017) argued that cooperative learning methods like peer tutoring is an effective strategy that can enable students to achieve academic objectives. Investigation on skill development in children after the peer tutoring program has showed that students have made improved their social skills and developed confidence in expressing themselves. This proved that peer-to-peer learning is an innovative learning framework as contributes to personal development of students. Social skill is necessary for students today due its future implications i n their job role. Hence, getting to closely interact with peers brings changes in students behavior and this process facilitates development of communication skills. When they discuss with peers regarding their lessons, they ask questions as well defend different point of views. This helps students to resolve their cognitive challenges (De Backer, Van Keer Valcke, (2015). Therefore, peer tutoring has become a powerful resource in education context as it improves social skills and enhances satisfaction in the learning process too. Students should embrace peer tutoring because of its ability to develop communicative and collaborative behavior in patient. The main problem for students who are enrolled in higher education is that they become disengaged from their study once they submit their assessment work. They do not make any judgment about what issues they faced during completing the work or miss opportunities for crucial learning (Thomas, Martin Pleasants, 2011). In short, it can be said that they become passive recipient of assessment outcomes. The review of literature has revealed students motivation is an important criterion in effective learning. However, lack of motivation in some students is the reason for poor accounting achievement. In the context of this problem too, peer learning is found beneficial in developing learning outcome and facilitating meaningful learning among students. Razak See, (2010) has explained that peer learning is one tool that promotes attainment in students and increases their motivation to review and comprehend lessons. The benefits of peer learning can also be explained by theoretical model of social constructivism, which states that progress in learning occurs under the guidance of others (Kiraly, 2014). Hence, this element is provided by peer learning within groups, as students can get help from experts and assimilate ideas that cannot understand on their own. Students get the opportunity to actively convey ideas to their peers and solve their problems under the guidance of suitable peers. The benefits of peer learning for promoting students independence are understood from discussion on four valid points that improves learning outcome of students. Peer learning process has also found favor according to different social learning theories. For instance, Vgygotskys social constructivist theory has importance implications for peer learning. He gave idea about a zone of proximal development (ZPD) which showed that potential development of a person occurs when they engage in problem solving under experts guidance (Kiraly, 2014). This implies that effective knowledge acquisition can occurs under the guidance of experts. This further emphasizes the if students collaborate with their peers, they can become confident learners. Conclusion The report gave an insight into the common issues faced by students when they learn in a isolated environment without any interaction with others. With evidence regarding the need for active learning process among students to engage in learning, the report proceeded with the discussion on peer-to-peer learning method as an active learning activity. Four comprehensive arguments were presented evidence based findings that proved how peer-to-peer learning can promote students independence. The four role of peer learning on students learning included creating sense of belonging and reducing anxiety among students, improving social skills and confidence in students, promoting independent study and critical thinking skills and facilitating meaningful learning. The importance of peer learning is also proved by the theoretical model of social constructivism. Hence, all these points justified the effectiveness of peer learning method on promoting students independence in learning. This report has implications for education sector and universities so that institutions increase the opportunity for students to engage with their peer to achieve academic success. Recommendation Based on reviewing the findings on the impact of peer-to-peer learning on students independence, it can be said that learning organizations can play a key role in encouraging students to engage in collaboration with peer. Hence, some key recommendation to engage students in collaborative learning by interaction with peers includes the following: Higher universities should implement peer learning programs in universities so that group discussions enhance meaningful learning process. Motivation of the students in the learning process should be encouraged by developing assessment process for students. This will help to identify glitches or barriers that impede students from learning new concepts in an effective manner. While implementing peer learning programs, peers should give sufficient time to students so that factors contributing to anxiety in student are identified and proper solution is given to students to address conflicts during the learning process. The interest of students in peer learning process should aroused by implementing short educative sessions in universities. In addition, participation in peer tutoring should be encourages by strategies like rewards for academic achievement. References Armbruster, P., Patel, M., Johnson, E., Weiss, M. (2009). Active learning and student-centered pedagogy improve student attitudes and performance in introductory biology.CBE-Life Sciences Education,8(3), 203-213, doi:10.1187/cbe.09-03-0025 De Backer, L., Van Keer, H., Valcke, M. (2015). Exploring evolutions in reciprocal peer tutoring groups' socially shared metacognitive regulation and identifying its metacognitive correlates.Learning and Instruction,38, 63-78, available at: https://doi.org/10.1016/j.learninstruc.2015.04.001 Fook, C. Y., Sidhu, G. K. (2015). Investigating learning challenges faced by students in higher economics.Procedia-social and behavioral sciences,186, 604-612, doi: 10.1016/j.sbspro.2015.04.001 Kelly, P., Katz, L. (2016). Comparing Peer-to-Peer and Individual Learning: Teaching Basic Computer Skills to Disadvantaged Adults.International Journal of Adult Vocational Education and Technology (IJAVET),7(4), 1-15, DOI:10.4018/IJAVET.2016100101 Kiraly, D. (2014).A social constructivist approach to translator education: Empowerment from theory to practice. Routledge, available at: https://books.google.co.in/books?hl=enlr=id=mcoJBAAAQBAJoi=fndpg=PP1dq=A+social+constructivist+approach+to+translator+education:+Empowerment+from+theory+to+practice.+ots=d6VaPJ2lE4sig=5QtRjsGLMNmlQqKay4-kbWf1XfA#v=onepageq=A%20social%20constructivist%20approach%20to%20translator%20education%3A%20Empowerment%20from%20theory%20to%20practice.f=false Mellado, M. E., Valdebenito, V., Aravena, O. (2017). Peer tutoring to develop social skills among university students.Int. J. of Pedagogies Learning,12(2), 147-159, available at: https://www.adamhousepress.com.au/wp-content/uploads/2017/12/5Mellado.pdf Razak, R. A., See, Y. C. (2010). Improving academic achievement and motivation through online peer learning.Procedia-Social and Behavioral Sciences,9, 358-362, doi:10.1016/j.sbspro.2010.12.164 Stone, R., Cooper, S., Cant, R. (2013). The value of peer learning in undergraduate nursing education: a systematic review.ISRN nursing,2013, available at: https://dx.doi.org/10.1155/2013/930901 Thomas, G., Martin, D., Pleasants, K. (2011). Using self-and peer-assessment to enhance students future-learning in higher education.Journal of University Teaching Learning Practice,8(1), 5, Available at:https://ro.uow.edu.au/jutlp/vol8/iss1/5

Travelling Through the Dark by William Stafford

Visual Analysis of a Beuford Smith Photograph essays

Visual Analysis of a Beuford Smith Photograph essays The piece which I will analyze was shot by Beuford Smith, and is titled These Colors Dont Run. It was taken in 1999 and is a silver gelatin print. It is displayed in Robert B. Menschel Photography Gallery in the Schine Student Center at Syracuse University. The issue at the heart of Smiths print appears to be race and racial tensions, as with many of the other works on display. However unlike some of the others it was not taken in the period during and before the sixties. Rather than this time, which we view as the era of racial change and the peak of racial tension in America, it was taken in 1999 a time were less attention is given to any remaining tension. Nor does the print depict an actual event which we view as a symbol of racial tension, like Million Man March in Washington D.C. and Three Placards, June 14, Anti-Apartheid Rally, Central Park, New York New York City. How exactly is Smith commenting on our current situation and what is he saying? The print is done in black and white making one think that it was in fact taken during the civil rights movement. However this assumption is quickly found to be wrong as one investigates the boys shirt which has the words Operation Desert Storm. By having viewers make this assumption and then find out that it is not so the color serves to emphasize that this is about a current issue. Also the black and white coloring makes it feel more like a factual documentation. The two focal points of the piece are the young African American boy on the out side of the shop and the elderly white woman seated at the counter looking through the glass. The boy has a sad look on his face and has his hands on to caf style tables on the sidewalk. His stance seemed determined his gaze is fixed on something in the street. Although we can assume the boy is not actually thinking of racial issues his appearance may be intended by Smith to represent the larger African American society, un...

Merchants PLC Essay Example | Topics and Well Written Essays - 3000 words

Merchants PLC - Essay Example The capacity of raising funds depends on the resources from which funds might be accessible. The company forms of partnership and sole proprietor have limited chances for raising funds. Banks will generally only desire to finance businesses where there is a revenue stream or short risk assets i.e., assets which can safeguard loans. The company can raise money by several ways by increasing short- term and long-term capital; they can also include -issue of shares and debentures, loans from financial institutions, loans from banks, public deposits. â€Å"Capital may also be raised by development taxes levied on either developers or households, or both. For example, a value-capture tax may be levied on the estimated increase in land value due to the development of related infrastructure.† http://www.appliedeconomics.com.au/pubs/papers/pa03_trans.htm The main benefits of the sources of finance for the company includes decreasing the reliance on outside sources of finance; it raises the credit value of the company, allowing the company to withstand complicated circumstances and allowing the company to accept a constant dividend policy. â€Å"Merchants has for many years focused on a simple proposition, to deliver a high and rising income stream, together with long term capital growth through investing predominantly in large UK companies.† http://www.alliancetrustsavings.co.uk/resource/taking-stock/june2012/merchants-trust-plc.htm

Impact of Job Satisfaction on Staff Turnover Literature review

Impact of Job Satisfaction on Staff Turnover - Literature review Example Nature of work is one of the primary factors which help in the development of job satisfaction among the employees. The interest level of the employees in the nature of job which is imposed by the employer guides the job satisfaction. If the employees face person role conflict, then the job satisfaction of the employees will never be high. Person role conflict deals with the fact that employees have to undertake tasks which does not suit them and they are either under qualified or overqualified for the job. The role play of an employee in the job should also provide opportunity for them to utilize their skills which they have learned over the years. The employees should be provided with autonomy in carrying out their jobs. Another important job related factor which helps in the building of the satisfaction is the clarity in the role to be played by the employees. If the employees are made clear regarding the roles which the organization expects from them in carrying out, then a signi ficant amount of satisfaction is generated within them. Proper training regarding the jobs also helps in the development of job satisfaction among the employees as they generate confidence within them through the training process. The employee participation has been one of the crucial aspect in modern times and effective participation of employees in the decision making process of the organization generates a feeling within the employees that they belong to the organization and are an important part of it. This feeling within them in turn imbibes satisfaction regarding their job (Lee, 1991, p.9). The work pressure which an employee has to experience generates the satisfaction level within them. Often employees are dissatisfied with their job owing to the excessive job pressure which they have to face (Spector, 1997, p.24). The above study does not incorporate the factor of place of work and the ambience of the workplace which also often can raise dissatisfaction among the employees in the workplace. Pay package: Pay package is the most important criteria which dominates the level of job satisfaction among the employees. Employees always look for a better pay package and the pay package often helps in balancing the other limitation which an employee may face in an organization. The inclusion of additional facilities in the pay package helps to raise the satisfaction level of the employees and the fulfillment of the demand for increment in the salary structure makes them satisfied in their job. Other than the salary which the employees receive, often employees

CSR & Ethical Practice Essay Example | Topics and Well Written Essays - 1000 words

CSR & Ethical Practice - Essay Example One advantage of business ethics is that it helps an organisation to achieve competitive advantage (Shaw, Barry, & Panagiotou, 2010). A company that engages in ethically sound business practices improves customer loyalty and trust. In this case, the consumers become loyal to the brand even if the company is facing difficult financial times. Therefore, companies will always set their ethical standards depending on organizational values. In the long run, consumer confidence increases with ethical responsibility. According to Carroll (2013), the underlying assumption is that business ethics benefits the society since it is the main basis for social responsibility among organisations. On the other hand, business ethics leads to more accountability and integrity in the organisation. The implication is that sound business ethics obligate company employees to become more responsible in certain operations like financial reporting (Choi, & Pae, 2011). Another remarkable strength of business ethics is that it makes organisations realise that their success is more than profitability (Carroll, 2013). Some of the models of business reporting like the triple bottom line approach came up as a result of business ethics and corporate social responsibility. In this case, companies focus their reporting on people, planet, and profit (Slaper, & Hall, 2011). Therefore, ethically companies have the obligation to report their financial performance, environmental as well as social performance. The triple bottom line approach recommends that company survival depends on their ability to make profits, encourage sustainable and ethical business conducts (Henriques, & Richardson, 2013). The underlying assumption is that business ethics is a prerequisite for sustaining an investment. Consumers have confidence on the company that protect environment and contribute to the well-being of the society (Choi, & Pae, 2011). On the contrary, business ethics has a negative effect

Building Materials Used in Construction

Building Materials Used in Construction In the world of construction, the king of building materials is concrete. It is the most common material and it constitutes the base of a lot of constructions like buildings, roads, bridges, tunnels, water pipes dams etc. It is an absolutely indispensable tool of civil engineers that is used for more than a century. Before its existence the largest portion of constructions was covered by two other materials, wood and stone, while the last years is very widespread the use of the reinforced concrete; which is concrete with bars of steel in it. Concrete Analysis We can say that concrete is a type of an artificial stone, a mixture of four elements; cement, water, sand and aggregates that is succeed by the process of hydration. This process converts the slushiness mixture into an artificial stone just in a few hours. The tight concrete continue to harden for many years but in the seven fist days has already taken more than 70% of its total resistance and in 28 days will have take practically its total resistance. Making the concrete we should be very careful to the proportioning. Cement and water are the two chemically reactive elements while sand and aggregates are chemically inactive. The proportions of each material in the mixture affect the properties of the final hardened concrete. As the cement (created by crushing up clay and limestone together and roasting it in a kiln) content increases, so does the strength and durability of the concrete (a good rate is 12%), the water should be pure and not overtop the 17% because the mixture will b e weak and as it concerns the aggregates, too much fine aggregate gives a sticky mix while too much coarse aggregate gives a harsh mix. All these materials together give to concrete some properties and the basic property is that concrete has a very high resistance in compression but low resistance in tension. For this reason, as it mentioned before, we use a lot the reinforced concrete; because the steel bars can handle the concrete in tension. As exists the term of resistance so on the opposite will be another term, this of fatigue. Explanation of Fatigue Fatigue is a process of progressive, permanent structural change occurring in a material which is subjected to conditions that produce time fluctuating stresses and strains. The structural changes appeal in cracks or complete fracture after a sufficient number of fluctuations. The fatigue process occurring in concrete has been under investigation since about 1900 with the majority of the significant work having been done during the past twenty years. This process has been observed in concrete under repeated compressive and flexural loading and small amounts of experimental work show that it also occurs under reversed flexural loading and repeated tensile loading. Reasons of damage on concrete In perfect conditions, that means an artificial environment of a laboratory and without considering the human mistake, concrete can last without any corrosion about 50 years. But in our environment the fatigue of concrete will start to appear in 30 years and after that the construction need to be watched and maintained. We can examine the fatigue of concrete with two different points of view; macro scale and micro scale. As it concerns the fatigue of the concrete from a macro scale view we can refer the following reasons. Concrete is a mixture of materials that is decayed in time so the first reason of the fatigue of the concrete is because of its old. Another cause is the intense alkaline environment of the atmosphere and this phenomenon is appeared especially at the urban centers because of the exhausts that are coming out from vehicles and factories. The proximity of the concrete in the water environment, such as big amounts of salt in the air and big amounts of moisture, is also an erosive factor. Furthermore, the boisterous change of the temperature as much as in winter and summer months in combination with the high amounts of moisture cause intense shrinkages and expansions on concrete. It is characteristically observed that the boisterous change of temperature per 10Π¿ C doubles the velocity of corrosion in concrete. Earthquake is the most dangerous reason of all. It can affect the concrete less or too much. It can cause a damage that is invisible but this may affect the building for years creating a bigger damage later. But the earthquake also can cause a complete fracture of concrete. The last important reason tin the macro scale view is the fire. During the fire the most common problems that are presenting are firstly the fracture part of the concrete because of the violent development of pressures from the evaporation of the water contain in concrete, secondly the thermal expansion of the concrete which also leads to fracture and finally the various change of stresses on concrete because of the abrupt freezing extinguishing of fire. If we want to study it from a micro scale view we could refer the following causes of the fatigue on concrete : Sulfate Deterioration. Sodium, magnesium, and calcium sulfates are salts These sulfates react chemically with the hydrated lime and hydrated aluminate in cement paste and form calcium sulfate and calcium sulfoaluminate. The volume of these reaction byproducts is greater than the volume of the cement paste from which they are formed, causing disruption of the concrete from expansion. Alkali Aggregate Reaction. Certain types of sand and aggregate, such as opal, chert, and flint, or volcanics with high silica content, are reactive with the calcium, sodium, and potassium hydroxide alkalies in portland cement concrete. These reactions, though observed and studied for more than 50 years, remain poorly defined and little understood. Some concrete containing alkali reactive aggregate shows immediate evidence of destructive expansion and deterioration. Other concrete might remain undisturbed for many years. Petrographic examination of reactive concrete shows that a gel is formed around the reactive aggregate. This gel undergoes extensive expansion in the presence of water or water vapor (a relative humidity of 80 to 85 percent is all the water required), creating tension cracks around the aggregate and expansion of the concrete. If unconfined, the expansion within the concrete is first apparent by pattern cracking on the surface. Deterioration Caused by Cyclic Freezing and Thawing. Freeze-thaw deterioration is a common cause of damage to concrete constructed in the colder climates. For freeze-thaw damage to occur, the following conditions must exist: The concrete must undergo cyclic freezing and thawing. b. The pores in the concrete, during freezing, must be nearly saturated with water (more than 90 percent of saturation). Water experiences about 15 percent volumetric expansion during freezing. If the pores and capillaries in concrete are nearly saturated during freezing, the expansion exerts tensile forces that fracture the cement mortar matrix. This deterioration occurs from the outer surfaces inward in almost a layering manner. The rate of progression of freeze-thaw deterioration depends on the number of cycles of freezing and thawing, the degree of saturation during freezing, the porosity of the concrete, and the exposure conditions. Acid Exposure. The more common sources of acidic exposure involving concrete structures occur in the vicinity of under-ground mines. Drainage waters exiting from such mines can contain acids of sometimes unexpectedly low pH value. A pH value of 7 is defined as neutral. Values higher than 7 are defined as basic, while pH values lower than 7 are acidic. A 15- to 20-percent solution of sulfuric acid will have a pH value of about 1. Such a solution will damage concrete very rapidly. Acidic waters having pH values of 5 to 6 will also damage concrete, but only after long exposure. Construction Defects. Some of the more common types of damage to concrete caused by construction defects are rock pockets and honeycombing, form failures, dimensional errors, and finishing defects. Honeycomb and rock pockets are areas of concrete where voids are left due to failure of the cement mortar to fill the spaces around and among coarse aggregate particles. Solutions Generally when a crack affects the performance of the structure, then we will repair it to restore its structural properties. Epoxy injection is typically the basis for this type of repair, with or without added reinforcement. The injected epoxy is actually stronger than the concrete and can restore the concrete strength. To use epoxy injection to repair a crack, the crack is first cleaned by vacuuming or flushing with water to get out any dirt or contamination. The cracks on the surface are then sealed with an epoxy gel to prevent the injected epoxy from running out. Injection and venting ports are installed and the epoxy is injected. High pressure is not used since that could actually widen the cracks. Once the cracks have been filled, the ports and surface seals are removed, typically by grinding the surfaces flush with the concrete matrix. When concrete is too deteriorated for epoxy injection, then all unsound concrete is removed and new concrete is placed. (http://www.concretene twork.com/concrete-repair/structure.html) (accessed 17/04/2011) Portland cement mortar may be used for repairing defects on surfaces not prominently exposed, where the defects are too wide for dry pack filling or where the defects are too shallow for concrete filling and no deeper than the far side of the reinforcement that is nearest the surface. Repairs may be made either by use of shotcrete or by hand application methods. Replacement mortar can be used to make shallow, small size repairs to new or green concrete. Surface grinding can be used to repair some bulges, offsets, and other irregularities that exceed the desired surface tolerances. Excessive surface grinding, however, may result in weakening of the concrete surface, exposure of easily removed aggregate particles, or unsightly appearance. The dry pack concrete repair technique shall be limited to areas that are small in width and relatively deep, such as core holes, holes left by the removal of form ties, cone-bolt and she-bolt holes, and narrow slots cut for repair of cracks. Epoxy bonded dry pack shall be used for critical repairs or for repairs expected to be exposed to severe service conditions. Dry pack mortar shall consist of type I or II Portland cement, clean sand that will pass a 1.18-mm sieve, and clean water. Epoxy-bonded concrete is defined as freshly mixed Portland cement concrete that is placed over a fluid epoxy resin bond coat on hardened existing concrete. Epoxy-bonded concrete repair may be used when the depth of repair is 30 cm à ¢Ãƒ ¢Ã¢â‚¬Å¡Ã‚ ¬ 50 cm or greater. Resin injection is used to repair concrete that is cracked or delaminated and to seal cracks in concrete to water leakage. Two basic types of resin and injection techniques are used to repair Reclamation concrete. Epoxy Resins and Polyurethane Resins. Conclusion Without concrete, the history of constructions would not be the same and our concern must be to make it stronger and friendlier to the environment and make even more impressive constructions.

The American Christian Worldview :: essays research papers

The American Christian Worldview All across the United States Christians are talking about this term called Worldview. What is it anyway? Many times, we release our guard and end up allowing society to change our thinking into what the rest of the â€Å"popular culture† thinks of our very being. As Christians, we should be giving scriptural backup for whatever conclusions one makes about this culture.   Ã‚  Ã‚  Ã‚  Ã‚  Every society has a culture. Each culture has a different method of thinking. One of the major issues each culture eventually deals with is their basic theology. If I were to ask someone who God was, the answer would vary depending on which part of the country I was in. This is where the development of worldview begins. People within that culture begin to migrate towards those who have the same beliefs in fellowship. Those people who have the same beliefs begin to form a culture. After a culture is formed, cultural studies begin taking form. In a religious community, the members of that community begin to form a standard of ethics to live by. After the individuals form a religious community, start a culture that culture begins to do cultural studies. Those cultural studies are a basis for the individuals to set boundaries of accepted ways to produce or consume culture in their community. The next step in this process deals with aesthetics. Aesthetics are the ways in which the culture communicates their beliefs and values. After all these concepts have taken their course, the individual has developed a worldview. Starting back at the very beginning of this process is the most dangerous aspect of this entire process we follow to gain a worldview. In today’s society there is a variety of versions of â€Å"God.† Depending on which God you believe in, your community and culture could be very far fetched from what the truth is. The overlying theme behind every formation that coincides with any worldview can be asked in one question. What is the purpose of my life? As Christians, we should be involved in society’s version of â€Å"popular culture.† We are called in the Bible to be the salt of the world, as the salt we shouldn’t be merely consuming the culture in which we live in, we should be part of it, adding everything we can.