HR Manual for Tata Power Community Development

HR Manual for Tata Power Community Development Tata Power Company (TPC), a pioneer in the power sector, has historically been committed to environment enrichment and community services. TPC has been undertaking corporate social responsibility (CSR) activities for decades. Tata Power Community Development Trust (TPCDT) is a CSR function deployment organization of TPC. With more and more power generation projects coming into existence, activities related to CSR have increased. TPCDT now required a set of policies and guidelines in order to have smooth governance and operational efficiency in terms of human resource management. Creating a well defined set of rules and policies in organizations manuals could help to avert problematic situations. Hence, the objective of this project is to frame a Human Resources (HR) Policy Manual for TPCDT. The project also aimed at studying various aspects of Human Resources in this organization, analyzing and incorporating HR Policies. Methodology for the project included examining the areas of operation of organization and assimilating information about its operational aspects. After gathering data, the policies that need be incorporated in Human Resource Manual of TPCDT were identified and an in-depth analysis of their applicability to the organization was carried out. The next step included study of various employee and labour related acts as a part of identification of compliance and necessary requirements. HR Manuals of similar organizations and trusts were also referred to understand overall content and flow of the HR Manual for TPCDT. Major observations were noticed in terms of organization structure and internal management systems. The whole structure of TPCDT was to be redesigned. Various grades and designations were identified in accordance with those of Tata Power Company Ltd. A whole new performance appraisal system was designed to suit the hierarchy of the organization. Further for smooth function within the organization an efficient grievance handling mechanism was also established. For all the above major observations certain policies and strategies were recommended by me. Majority of my proposals and recommendations were accepted viz. TPCDTs Organization Structure, Grades and Designations for TPCDT, Three Tier Performance Appraisal System, Looped Grievance Handling Mechanism, Recruitment Flowchart, Forms and Templates, etc. Various activities that were of assistance in formulating the HR Manual included literature review, discussions with HR personnel and higher management of the organization, organizational analysis of TPCDT, referring other organizations manual, academic study material and continuous guidance from company guide and faculty guide. Finally as an outcome of my summer internship project, an HR Manual has been drafted catering to the needs and requirements of TPCDT. This manual was accepted for implementation by the management of TPCDT. Introduction Tata Power Community Development Trust (TPCDT) is a registered trust under Bombay Public Trust Act-1950, dated 13th April, 2009, with its objects as relief of poor, advancement and propagation of education learning, for providing medical aid and relief and advancement of any object to general public utility without any profit. In the subsequent years after its registration, TPCDT started extending its area of operations with its objectives as follows: Relief of poor including the help from establishment and support of institutions, educational fund for the relief in poverty including relief of any distress Providing scholarships, prize and fellowships in any branch of science art or commerce, or in assisting students to study whether in India or aboard either by payment of lump sum or by payment of periodical sums or by giving interest free loans or at nominal interest Providing medical aid and relief, including for establishments, maintenance and support for hospitals, dispensaries, convalescent homes, rest-houses, recreation centers and institutions or funds for medical aid and relief for promotion of health and hygiene Providing aid to any charitable institutions Establishing, support, maintaining and for grant aid to goushalas and other institutions working for the protection and preservation of animals and birds. As the scope of operations increased, TPCDT required a set of policies and guidelines to have smooth governance and maintain its operational efficiency. In order to maintain quality of services rendered by TPCDT, it required hiring of personnel and various other personnel policies. There were some issues coming from various operational locations of TPCDT in context of organizations HR policies and guidelines. Issues like defining a link in between Tata Power and TPCDT in terms of management hierarchy were coming into picture. Other queries included: What can be the maximum permissible benefits and allowances can be granted for TPCDT employees? How the recruitment and selection policy will work? What will be the grievance handling procedure? What should be the procedure for performance appraisal? Solutions to such queries were required in form of written guidelines and policies clearly defining organizations human resource policies. For this vary purpose, TPCDT required an HR Manual to have all sets of policies and procedures compiled in a handbook format for smooth personnel management. Each employee should be able to use this manual as a guidebook when he or she needs to apply organization policy in a given situation. Every employee should feel free to consult this manual to assist in the interpretation of Human Resources policies. 1.1 About HR Manual: The Human Resources Policy Manual is Organizations Human Resources policies written into a usable guidebook for all employees. This manual not only outlines an organizations policy toward the various phases of the employer-employee relationship, but it also indicates how policy is to be administered. Consequently, each employee should be able to use this manual as a guidebook when he or she needs to apply organization policy in a given situation. Every employee should feel free to consult this manual to assist in the interpretation of Human Resources policies. Research and experience has shown that written policies promote consistency, continuity, and better understanding within an organization. When policies are put into writing in the form of a manual, they add a visual effect to their overall purpose. Moreover, written policies help management by eliminating the need for time-consuming and expensive memos, bulletins, and announcements. Written policies also aid supervisors and managers in consistently achieving fair and equitable interpretations of policy that require action on a regular, recurring basis. Moreover, fellow employees feel a deeper understanding of their role in the organization when they realize that policies are written and thereby uniformly administered. Employees feel more secure, confident, and more at one with the organization when policies are made clear for them. Policies promote a movement toward responsibility and accountability. Furthermore, the ever-present tendency to pass-the-buck is reduced to a great extent. Specifically speaking, this Policy Manual is designed to provide management with the following advantages. Understanding Written policy is one of the best antidotes in the Human Resources medicine chest for avoiding troubles an organizational grapevine may cause. Even though everything is subject to interpretation, the odds overwhelmingly favour the written word compared with the oral. Line of Authority Top management cannot make all the decisions that need to be made within the organizations community. Often at times they feel the need to because they are afraid to release that authority to subordinates. Naturally, it signifies that if top management felt that subordinates could make decisions like top management would, the reluctance would be reduced. The Policy Manual achieves this desired relationship. It thus results in a solid delineation of authority. Consistency Consistent application of organizations policies is constructive because it means employees will be treated equally. It prevents, to a great extent, the seepage of prejudice and bias in the decisions of supervision. The achievement of this one virtue takes a colossal step toward the maintenance of satisfactory employer-employee harmony. The need for a Policy Manual is very essential. In short, no organization is invulnerable to the fallacies of human nature. These written policies should increase understanding of organization-wide policy guidelines. Authority and Distribution- Every employee of organization has access to this Policy Manual in handbook format. This Policy Manual is up-to-date and contains the complete and accurate policies of organization as of the published date. Supplements to the Policy Manual- The policies, practices and guidelines in this Policy Manual will remain in effect until changes are considered necessary as a result of internal growth, competitive forces, or as a result of general economic conditions pertaining to higher education communities. However, any such change to be made on any Human Resources policy or practice will be made only after consideration is given to the mutual advantages, benefits, and responsibilities of such changes on supervisors or managers and on other employees of the organization. Management Rights- This manual describes general Human Resources policies and procedures for the organization. This manual should not be construed as, and does not constitute an offer of employment for any specific duration, nor is it intended to state any terms of employment. Organization reserves the right to change, modify, suspend, interpret or cancel its policies and practices at its sole discretion and without advance notice. This right extends to both published and unpublished policies. 1.2 Company Profile: Tata Power Company Limited Tata Power Company is a pioneer in the power sector has historically been committed to environment enrichment and community services. The company reaffirms commitment to the conservation of ecological systems and sustainable development through afforestation, water conservation, nurturing and developing of local species of flora and fauna in our areas of operation. As Indias largest private power utility, Tata Power has set the momentum of growth. In the quest to deliver sustainable energy, Tata Power is spreading its footprint nationwide, creating new benchmarks in operational efficiencies, investing in global resources and redefining paradigms. Tata Powers strength lies in fulfilling their commitments and their ability to manage well in the changing environment. Company takes pride in building lasting and trusting relationship with its customers along with a legacy of caring for communities in and around its areas of operations. Tata Power strives to lead the reform process for sustainable power with an aim to redefine the contours of Indian Power Sector. Tata Power Company and Tata Power Community Development Trust (TPCDT) TPC had undertaken the CSR activities for decades, reflecting the companys commitment towards sustainable energy generation without undue compromise to human and environmental development. These activities were undertaken as a voluntary initiative by the employees of TPC, and there was no separate CSR department. However, with large scale expansion, the need to have CSR as a separate entity was felt. The dilemma for the decision manager was whether to create a separate CSR department or continue with the existing set up. Other related issues needed to be addressed strategically as well as tactically to maintain a balance between shareholders interest and other stakeholders. Tata Power Companys CSR deployment bodies are: Project Description This project deals with preparing HR Manual for Tata Power Community Development Trust (TPCDT) and drafting various forms and templates for administrative purpose of TPCDT. 2.1 Methodology of Project Understanding Organization (Organizational Analysis) Identification of suitable policies for Tata Power Community Development Trust (TPCDT) Identification of related acts and regulatory framework Drafting HR Manual and designing forms and templates Review and feedbacks; incorporating corrections (if any) Step 1 Step 2 Step 3 Step 4 Step 5 Areas of operations Existing organizational structure Existing work mechanisms Through management discussions, analysis of other HR Manuals, etc Concluding policies for manual Identification of regulatory framework and compliance in terms of identified HR policies To compile policies in HR Manual and designing relevant forms and templates To make necessary changes as suggested by management Step 1 Understanding Organization (Organizational Analysis): Organizational analysis aims to generate an understanding of the organizational structure and culture of the system the project is looking at. This can help in understanding the ease or difficulty with which new strategies can be adopted. Organizational analysis is about organizations and people. This covers issues of structure and formality. But it also encompasses issues of process and, in a very fundamental way, change. The subject matter can be very abstract, and is drawn from literatures concerned with organizations in general rather than any specific sector. It also covers underlying change and organizational dynamics, which are common, to a great degree, to all organizations whatever sector they occupy. The project was started by referring literature on Corporate Social Responsibility (CSR) and operational activities of TPCDT. It comprised of data related to TPCDTs legal documents, field of operations and activities of TPCDT, about linkage between TPCDT and TPC, etc. The first and foremost task was to understand the organizational structure and its operational aspect. For this the TPCDTs Trust Deed was read by me in order to understand the purpose of incorporation of the trust. The next part included understanding the current organizational structure. Initially the structure of TPCDT was as follows: Some issues were identified in the existing organizational structure. These issues included: There were no clearly defined set of rules or regulations in terms of management, reporting and authority. A number of personnel from Tata Power were performing the roles of TPCDT personnel at some locations. There was no clear distinction as to differentiate the employment roles of the individual between TPCDT and TPC. In the absence of a HR Policy Manual, the management and decisions pressure was directed towards Head Office. Grades and designations were not defined. The link in between who is eligible for which designation from TPC to TPCDT was also missing. Guidelines for recruitment and other administrative procedures were also lacking to some extent. In order to understand the whole system properly, the current structure of company was examined again. On the basis of observations, a few organizational structures for TPCDT were proposed by me. The proposed structures were as follows: TPCDTs Organization Structure: Proposed Structure-1 TPCDTs Organization Structure: Proposed Structure-2 TPCDTs Organization Structure: Proposed Structure -3 TPCDTs Organization Structure: Proposed Structure-4 TPCDT CEO at Head Office (Support Departments: HR, Accounts, Operations, and Consultants.) Regional Head Assistant Regional Head Program Head Project Head Assistant Program Head Project Coordinator Assistant Project Coordinator Senior Field Worker Field Worker From the above proposed structures, the (proposed organizational structure -4) was finalized with some inputs from proposed structure 1, 2 and 3. The next important aspect in order to proceed towards framing policies and guidelines for TPCDT was Grades and Designations Structure. TPCDT required a clear structure in terms of designation on which a personnel is hired and also a framework for a TPC employee to join TPCDT and for which designation he is placed in TPCDT. After analyzing the organization and conducting necessary research, the following grades and designations table was proposed by me. The designations were identified in line with Tata Power Companys grades and designations. In the following a clear distinction is given between Tata Power Company and TPCDT. This also describes if a person moves from TPC to TPCDT, he will be placed on which designation. The table also highlights the linkage between Tata Powers employees and TPCDT employees Grade in TPCL Designations TPCL Designation TPCDT Grade in TPCDT Sub Grade/ Designation Sub Grades A General Manager CEO A/I ÂÂ   ÂÂ   B1 Deputy General Manager B2 Assistant General Manager C1 Chief Manager Chief Program Head B/II Regional Head B1/II A C2 Senior Manager Assistant Regional Head B2II B D1 Manager Program Head C/III Program Head C1/III A D2 Assistant Manager Project Head C2/III B Assistant Program Head C3/III C D3 Executive Project Coordinator D/IV Project Coordinator D1/IV A D4 Officer Assistant Project Coordinator D2/ IV B E1 Worker Field Worker E/V Senior Field Worker E1/V A ÂÂ   ÂÂ   ÂÂ   ÂÂ   Junior Field Worker E2/V B Note: This table was also accepted later on in grades and designation part. The grades were taken as I, II, III, IV and V. Step 2 Identification of suitable policies for Tata Power Community Development Trust (TPCDT) In this phase the key areas suitable for TPCDT were to be identified. Various discussions were made with Col Prakash Tewari (Retd) who is the Deputy General Manager of CSR and Rehabilitation Resettlement Department in Tata Power and Mr. Inam Mukkhadam who is accounts head for TPCDT. A few questions that came up in the early phases of the project were: What all will be the contents of the HR Manual that will be suitable with the area of operation of TPCDT? What can be the maximum permissible benefits and allowances can be granted considering the fact that TPCDT is a nonprofit organization? How to understand the working conditions of people in their work areas? How to distinguish between the working conditions of personnel among various areas of operations? In this phase of the project various secondary data sources were searched and data was collected. Most of the data was related to companies and profit making organizations. But as this organization is a nonprofit organization, financial freedom is limited as compared to a profit making entity. In the initial phase the following policies that could have been a part of manual were identified by me. A copy of the document first presented to management is attached in Attachment2. After a series of discussions and suggestions the content of HR Manual for TPCDT were finalized. The content comprised of the following: Introduction: It included a brief introduction and background of the organization. This is further continued by briefly describing organizations HR philosophy i.e. vision, mission, objectives and values of the organization and purpose of this HR manual. The literature for this part was identified from the trust deed of TPCDT and management guidelines. Employment: This consists of employment policy of TPCDT and policy of TPCDT to transfer or promote well performing and capable employees to fill vacancies so that employees are provided with opportunities to widen their exposure and further their career development within TPCDT. The next important aspect of employment block in a manual is describing the organization structure and hierarchy. Then is the categorization of employees and associates. It also describes the type and nature of employment in the organization. They were identified as the following in context with TPCDT: Probationer: An employee who has been hired for a permanent position and is put through a probation period Trainee: A person who is freshly qualified or new to the sector may be appointed Interns: Students undergoing practical training with the organization, as a part of their academic Curriculum for a period ranging from 1 to 6 months Fixed Term Employees: A person who has been hired on a Fixed Term Contract Employment based on an identified project, the duration of which may or may not last its specified tenure. Permanent Employee: A person who has successfully completed his/her probation period whose service has been confirmed in writing, by the Organization. Associates: These are experts from various fields having specialized knowledge and expertise Contract Staff: Employees belonging to contractors deployed in the Organizations premises by the Contractor Overseas Interns: Students from overseas universities who undertake practical training with the organizations. Recruitment and Selection End Job Analysis, Employee Qualification/ Specification Posting of Advertisement on Internet/ Newspaper, etc Confirmation Appointment, Induction, Training and Development Staffing: Tests/ Interviews Receiving and short listing of CVs Requirement at Location/Head Office CV s Selected Rejected No YesThis section was identified as one of the most important part of internal human resource management. In context of work areas and requirements a chart was proposed by me, which was finalized later with some minor adjustments in terminology for the purpose of recruitment and selection flowchart. The final chart is as follows: Following activities were of assistance in formulating this flowchart: Literature review Discussions with HR personnel and higher management of the organization Organizational analysis of TPCDT Code of conduct: This part of manual was designed on the lines of Tata Powers code of conduct. All Tata Group of Companies have their internal policy on code of conduct. In order to distinguish between Tata Power and TPCDT separate code of conduct was drafted in lines with Tata Code of Conduct which was having few elements from Tata Code of Conduct and other elements form the observations out of organizational analysis from Step 1 i.e. Organizational Analysis. Performance appraisal: The purpose of performance appraisal is for a supervisor and an employee to have a candid discussion about performance expectations and actual performance. The employees actual level of performance is compared to the expected level of performance using standards that were developed by the supervisor with input from the employee. The benefits to be gained from conducting performance appraisals include: Recognizing accomplishments Identifying newly acquired competencies Preparing employee development plans Planning improvement where deficiencies are found Goal-setting Communication between supervisors and employees A new performance appraisal system was identified for TPCDT. A new Three Tier Performance Appraisal System for TPCDT was suggested by me. In most of the Tata Group of Companies Performance Management System is followed. But after few discussions with higher management officials, it was accepted in the proposal phase itself. This system works as follows: The first phase will comprise of Self Performance Review. Employee will fill the form in 5 days of time. This has to be filled in by the candidate himself. He will be judging himself on parameters stating how the employee judges himself/herself. The employee should fill in the KRAs / Goals in the beginning of the year. HR dept will then provide employee a Subordinate Performance Appraisal Form. This form will deal with a critical evaluation of the employee by its other fellow employees or subordinates. This form is also on the same grounds measuring various parameters stating how employees subordinates judge him/her. The next and most vital part in the three phased appraisal system is IO-RO-SRO Performance Appraisal. This appraisal will have remarks from the immediate superior of the candidate who is known as IO and then it will be forwarded to the next superior RO of the candidate. Their remarks will be recorded in a form and along with all the three forms; the file is given to SRO for finalization of the result. (IO- Initiating Officer, RO- Reviewing Officer, SRO- Senior Reviewing Officer) [Note: An Initiating Officer (IO) is the immediate superior of an employee. A Reviewing Officer is the superior to Initiating Officer and a Senior Reviewing Officer (SRO) is the immediate superior to Reviewing Officer (RO)]. The evaluation aspect and flow will be as follows in all the performance appraisal forms with an exception to a final remark column in IO-RO-SRO Performance Appraisal Form by SRO Suppose there are 20 attributes on which self/subordinate/IO-RO have to give point out of five. Here five is the highest and one being the lowest. Based on the result employee will be rewarded. Based on this report, salary increments and/or shifting to next level or fresh contracts in the same/ higher grade are made. Promotion: For the purpose of promotion it was collectively decided by management to give on merit. It was decided that it can either be based on the report generated from Three Tier Performance Appraisal System or direct promotion from the Managing Trustee of the Organization. However, special consideration will be required for promotion over and above the post of Program Head. Promotion will be approved after approval from Managing Trustee. Salary payment: It was decided by management that the Managing Trustee of the trust must approve salary of all employees. Additionally, salaries above a threshold set by the Board of Trustees must be approved by the Compensation Committee of the Board of Trustees if required. It was decided that salary changes, including certain types of supplementary compensation and salary during a leave of absence, require the same authorization and review process. Employee recognition and awards: According to studies, employees work better if they are given incentives for good performance. People naturally love to be recognized so it is important that companies and organizations set up an awards giving system to keep its people inspired. In addition to my inputs in HR Manual for TPCDT, a new policy on having employee recognition and awards was proposed by me. This part of manual would define the procedures and guidelines related to Employee Recognition and Awards. This has also been accepted after a few discussions and awards like Karmyogi Award, Spot Awards, Best team Award, Best TPCDT Performer Award, and etc were introduced. Every individual will be evaluated by his/her Initiating Officer (IO) and Reviewing Officer (RO). This will be further reviewed by Senior Reviewing Officer (SRO). One of the differentiating fact about these awards is they are not single awards. As many competent employees can be nominated and given these awards. This decision was taken to increase team effort and motivate individuals. Job rotation: Job rotation is rotation from one function to another, in same location or different. Rotation is in the same function as well as different function but may be at various locations. It was collectively decided that the minimum period after which an employee may undergo job rotation should be 2yeras. Following activities were of assistance in determining the policy on job rotation: Management discussions Literature review Discussion with current employees of TPCDT Grievances redress mechanism: For any organization, a proper grievance handling mechanism is considered to be very important. For this purpose a new multi-loop grievances redress mechanism was proposed by me. It was designed considering the fact of small loops. Each loop is having a specific decision related to grievance handling outcome. This mechanism was also accepted by management as official grievance redress mechanism for TPCDT. This is in a flowchart structure and has different loops for different situations and grievance handling. Receipt of a Grievance in TPCDT (Within any department or area) Assessment of the Grievance by Grievance Officer (One at each location) Grievance sent to Head Office For review by (Chief Grievance Officer) Discussions within Board and Managing Trustee Grievance Resolved Sent to Chief Executive Officer Solution Sent to Origin Site Yes No Grievance Resolved Yes No Employment separation: This policy was drafted as per management directions. It included procedures related to job separation and types of job separation. The main focus while drafting this policy was on providing clear guidelines to employees about various policies, guidelines and formalities necessary at the time of job separation. Various types of job separation areas that were identified are as follows: Termination: Termination of employment is an inevitable part of personnel activity within any Organization. Resignation: An employee resignation policy is an organizations official instruction for initiating and processing the voluntary separation of an employee.ÂÂ  An organizations employee resignation policy officially compels staff to follow the procedures. Resignation procedures protect the employees rights and the organizations interest. Official forms confirm and document that the resignation adhered to policy. It was decided that an employee resignation may be submitted in writing to the Chairman/CEO 3 month prior to the final working date. Earned but un-used vacation time (pro-rated to the last day of employment) will be paid. Sick leaves left unused will not be paid. In order to have a smooth flow of instructions, a resignation process flowchart was proposed by me for TPCDT. The chart is as follows: Resignation Process Flowchart Initiating Officer Returns Resignation Letter to Officer Division inform HO regarding date of release Resignation Acceptance Letter Handed to Employee Approving Authority Approves Resignation END Reviewing Officer (RO) Returns Resignation to Officer Resignation Letter is handed back to Officer START Officer Submitting Resignation Letter Officer Reverses Decision Forward to HR at HO for further proceedings Reporting and Discussion with Initiating Officer Discussions Take Place Forward to CEO at HO for Further D

Concept Of Management And The Nature Of The Management Process Management Essay

Concept Of Management And The Nature Of The Management Process Management Essay Organisation and the management activities necessary for the funcyioning of the organisation, management can be defined as the process of planning, organizing, leading and controlling the resources of the organisation to predetermined stated organizational goals as productively as possible. A concise description of each of the fundamental management functions will further explain the concept of management and the nature of the management process. Planning Controlling Leading Organising Resources Human Financial Physical Information Performance Achieve goals Products Services Productivity Profit Figure 1 the four fundamental management functions constituting the management process. Planning Planning is the first step of the function of management. Planning involves deciding in advance what has to be done, when, how and whom it has to be done and how the results are to be evaluated. The function of what has to be done involves identification of alternatives and selection of one of them. According to Henri Fayol, it is the plan of action to be followed, the stages to go through and methods to use. It is kinds of future picture wherein proximate events are outlined with some distinctness whist remote events appear progressively less distinct. The term planning has been defined by different management thinkers who laid emphasis on the various features of planning. According to George R. Terry, planning is the selecting and relating of facts and the making and using of assumptions regarding the future in the visualization and formulation of proposed activities believed necessary to achieve the desired results. A plan is like a map, when you following a plan, you can always see how much you have progressed towards your project goal and how far you are from your destination. Knowing where you are is essential for making good decisions on where to go or what to do next. The most important is to ensure that everyone is clear of what to accomplish. Planning meaning is how setting goals and defining the actions necessary to achieve the goals, in light of the situation. That is situation must be analyses or understood and the appropriate goals and actions must be determined in order to take advantage of opportunities and how to solve problems. Basic management function involving formulation more than one detailed plans to achieve optimum balance of needs or demands with available resources. First, the planning process is identifies the goals or objectives to be achieved and formulates strategies to achieve them. The Managers throughout the organization must develop goals, strategy, and operational plans for their work groups that contribute to the success of the organizations as a whole. Planning is also crucial for meeting your needs during each action with your time, money, or other resources. With careful planning you often can see if at some point you are likely to face a problem. It is much easier to adjust your plan to avoid overcome crisis, rather than to deal with the crisis when it comes unexpected time. Example Adidas plans to expand their operations in Europe, so they need consultation to the Government of Germany. They have been granted landing rights in Paris and are awaiting further details from authorities. In addition, planning has a different meaning depending on the political or economic context in which it is used. Two attitudes to planning need to be held in tension on the one hand we need to be prepare for what may lie ahead, which may mean contingencies and flexible processes. There is one thing on which every manager can expect to be appraised, the extent to which he or she archives his or her units goals or objectives. Whether its a work team or a giant enterprise, the manager in charge is expected to move the unit ahead, and this means visualizing where the unit must go and helping het there. Organization exits to achieve some purpose, and if they fail to move forward and achieve their aims, to that extent they have failed. On the other hand, our future is shaped by consequences of our own planning and actions. Planning in organization and public policy is both the organizational process of creating and maintaining a program. In the psychological process of thinking about the activities required to set a desired target or goal on some scale. Business planning is like water to a thirsty plant. It keeps our business vital and thriving. Without business planning, business will never improve as it could may even lose or bankrupt. Many small business or stall owners dont have the business planning so they cant earn so much profit when their doing business. Their business unable further progress and stagnant it is, eventually as to close to close shop or behind other competitors. The rapid of change in todays business world and globalization is making it increasingly necessary that mangers keep their plans current. Strategic management is the application of the basic planning process at the highest levels of the organization. Through the strategic management process, top management determines the long-run direction and performance of the organization by ensuring careful formulation and strategies. For example, top management may ask middle and lower-level managers for inputs when formulation top-level plans. Once top-level plans have been finalized, different organizational units may be asked to formulate plans for their respective areas. A proper strategic management process helps ensure that plans throughout the different levels of the organization are coordinated and mutually supportive. The slogan of Adidas is Impossible is Nothing. The mission of Adidas is to be Best Sport Brand in the world. They had achieved the mission but in this strong competitive era, too many company like Nike and Reebok also want to be globally recognized brand, but they will work hard to achieve their goals. Adidas is passionate about all sports. So Adidas Company will try their best for planning new product or equipment for athletes. As part of their End-to-End Planning initiative they have been focusing on implementing an optimized demand planning process and system more than 20 countries in Europe. They standardized or partially automated certain planning functions to increase forecast accuracy. Following the first wave of implementations in 2008, they continued the roll-out to the remaining European countries throughout 2009. Hereafter, they expect to continue the roll-out to other key Adidas Group markets such as Japan. They set up a dedicated Profitability Management department to monitor macroeconomics trends, forecast the impact on product and supplier cost and devises their supply chain profitability strategy. In addition, their development teams also contribute significantly to this initiative by engineering their products with a stronger focus on price. Throughout 2009, they also engaged their supplier with the aim to increase transparency and predictability in costing. Such as, they closely tracked raw material costs and leveraged this information to consolidate volumes. This allowed them to negotiate more effectively and offset cost increase. In addition, their sourcing teams improved their products allocation process to better utilize suppliers capabilities and take into account of total supply chain cost. Moreover, by leveraging consolidated volumes, their Transport and Customs team success negotiated reduced transportation cost and optimized shipment routes with their service providers. In conclusion, planning is very important for the organization to achieve their goals and objectives. Especially, current globalization every organization is using latest technology to do the market analyses and compressive planning. In order for them to survive and growth, for example, in year 2008, financial crisis, many banks in US will decline bankrupt. This is because of their poor planning. Organising According to Haimann, Organisation is the process of defining and grouping the activities of the enterprise and establishing the authority relationship among them. The study of Haimann mentioned definitions makes it clear that on the one hand organization is a process to define and classify the functions to be performed for the attainment of the objectives of the organization, and on the other hand, it is an art to establish sweet relationship among different persons. There are two concept are prevalent about organisation. In the other word, there are two meaning of organization: Organising as process: According to the first concept of organising, it has been considered as a process. In the other word, organising is not function that can be performed at a single stroke, but it is a chain of various functions. It includes getting information about objectives, deciding various activities and grouping them, determining important activities, allowing authority and responsibility, etc. Organising is related to human being and human conduct is deeply affected by the condition of work, their competency and capability, changes in the internal and external environment of the organisation. The process of organising has also to be changed according these changes. Hence, organising as a process can also be described as dynamic element. Organising s a structure of relationship: according to this concept, organising is treated as a structure of relationship. Under this various posts are created or established and the mutual relationship of employees working on various posts, their authorities and responsibilities are defined. Relationship lays down as to who is the superior and who is the subordinate. Various posts in different department of the organization are mostly permanent. Therefore, organisation as a structure of relationship is called static element. So far as the similarity between both the concepts is concerned, business organisation is looked upon as a group of different parts under both the concepts. These parts are both tangible (like human, material, machine and money) and intangible (like authority, responsibility, function and objective). Both the concepts lay stress upon the establishment of relationship between these two parts. On the contrary, there are certain different between both the concepts. According to the concept of organizing as a process organizing is that function which continues throughout the existence of the enterprise and changes go on taking place in it. Herein man is the central point. In the other words, many factors affect them and changes have to be introduced accordingly. On the other hand, according to the concept of organizing as a structure of relationship, posts are established in the organisation and authorities and responsibilities of each post are determined. Therefore, here under this concept more attention is paid to post which are stable than to men. In order to complete the organisation function of management, there are eight steps had to be taken. First step in the process is to know about the objectives of the enterprise. Although the determination of the objectives of an enterprise is done under the first function of management, i.e., planning but before commencing the process of organizing clear and detailed information about these objectives has to be obtained. On the basis of the information about objectives various function designed to achieve these objectives are determined. For example, it can be the objectives of the enterprise to produce sport shoes. In this respect information about how many types of sport shoes will be manufactured, whether the necessary parts required for manufacturing sport shoes will be manufactured or purchased, how extensive the sales area will be or will it be state, the whole country or international? After having learnt about the objectives of the enterprise, necessary functions to achieve t he objectives are determined. For example, a sport shoes manufacturing company can have different activities like purchase of raw material, purchase of manufactured parts production, advertisement, sales, arrangement of finance, research, accounts, correspondence, keeping stock of material, recruitment of employees, etc.. In order to achieve the objectives of the enterprise grouping of various activities is done. Under the grouping of activities all the similar type of activities are given to one particular department. For example, the activities like the purchase of raw material, purchase of ready-made parts, production, stocking the material, research, etc., are assigned to the production department. Similarly, advertisement and sales activities are given to the marketing department and department of finance take care of finance account and correspondence. Taking into consideration the importance and quantity of the work a department can be further divided into many branches or su b-departments. For example, under the department of production, purchase department, stock department, and research department, etc., can be established to get the benefit of specialization. It makes the optimum utilisation of human and material resources possible. The forth step is to key activities signifies the importance of this activities in the achievement of the objectives of the enterprise. Such an activities needs special attention. It depends on the objectives on the organisation to determine the key activity. Thus, in all the organisations key activity can be different. In the given example of a sport shoes producing company the main problem can be of advertisement because only an effective and large scale advertisement can attract the attention of the consumers toward a new product. Thus, in this case the function of advertisement is the key activity which requires the utmost attention. For this purpose the advertisement activities should be taken from the purview of the marketing department and be handed over to the newly created advertisement department so that the superior officers are in direct touch with this department and they are in a position to pay full attention to it. After dividing the various activities into different departments and sub-departments and having determined the activity that each individual is expected to perform, his responsibility is fixed. In other words, what they are to do for the attainment of the objectives of the organisation. For example, the purchase manager will be given the responsibility for the purchase of goods; the sales manager will be responsible for sales; the advertising manager will be responsible for advertisement and the finance manager will take care of the responsibility of finance. A person who is saddled with a responsibility must be given some authority too. Authority means the freedom of taking decision, guiding the subordinates and the freedom of supervising and controlling. Authority should be in proportion to the work or responsibility. If the responsibility is greater than the authority given and is insufficient to cope with the responsibility, the responsibility cannot be discharged successfully. F or example, if a marketing manager is assigned the responsibility of increasing sales but has not been given the authority to appoint sales representatives needed for the job, the increase in sales cannot be expected. So long as the authority to discharge the responsibility is not given, the person concerned has not accountability. It is only after granting authority to an individual that he can be made accountable. When two or more than two persons work for the attainment o common goals their inter-relationship must be defined very clearly. Everybody should know who is his superior and who is his subordinate? For example, the purchase manager will be the superior for all the employees of the purchase department. They will receive order from him and will also be responsible to him. On the other hand, the purchase manager will get orders from the General Manager and will be responsible to him. Defining clearly the inter-relationship helps in establishing coordination. Organising it i s not only such an activity which includes determination of activities; and the defining of the inter-relationship but it also ensure the optimum utilisation of human resources by providing physical resources and the right environment. For example, the factory and the office should be located at a proper place so that the employees can perform better. Leading Leading is influencing people so that they will contribute to the organisation and group goals; it has to do predominantly with the personal aspect of managing. All managers would agree that their most important problems arise from people-their desires and attitudes, their behavior as individuals and in groups-and those effective managers also need to be effective leaders. Since leadership implies followership and people tend to follow those who offer a means of satisfying their own needs, wishes, and desires, it is understandable that leading involves motivation, leadership styles and approaches, and communication. The leading and motivation function is concerned with the human resources within an organisation. Specifically, leading is the process of influencing people to work toward a common goal. Motivation is the process of providing reasons for people to work in the best interest of an organisation. Together, leading and motivation are often referred to as directing. We have already noted the importance of an organisations human resources. Because of this importance, leading and motivation are critical activities. Obviously, different people do thing for different reasons, then they have different motivation. The managerial function of leading is management as the art of getting things done willingly, with and through other individuals. Management is concerned with two key aspects, i.e., material resources or things and human resources or individuals. Material resources are susceptible to scientific or mechanical treatment because they are subject to the laws of mechanics. However, human resources cannot be subject to such treatment. Through the power of leadership and the science of cooperation, managers may evolve n effective method of integrating the interest of individuals and the business organisation. Management can expert its power with or through individuals but never over them. In other words, authority may be imposed by managers from above. However, authority should be supported, nourished and recognized by the subordinates from below for it to be meaningful and for it to work smoothly. The source of the power of management is the medium of leading, motivating, teaching, influencing, counseling, coaxing, delegating, and setting an example. Therefore, a manager plans, organizes, leads and control the employees working with him or her. The master key which leads to successful management of any business enterprise is motivation. It is responsible for ensuring productivity of human resources. It can make an individual carry out a particular activity. Thus, it assumes great importance in modern business management. Employees can be motivated by financial and non-financial incentives. The power of motivation is enhanced by effective communication and participation. Proper informatio n feedback is essential for effective motivation and leading. Leaders are a special breed of individuals who can move people toward the attainment of established goals. Such power does not emanate through the use of force or fear. Only dictators apply force in order to lead people. True leaders inspire and motivate people to perform activities in line with the objectives of the organisation. According to the theory of Fiedler. Fred Fiedler assumes that it is not easy for managers to be flexible in their styles of management, especially if their management style has made them successful as managers. He argues that any attempt to change a managers style of leadership to conform to changing situation that help in determining effective leadership. These are leader-member relation. This is the most important influence on the effectiveness of the manager. A manager who is loved and respected by his subordinate can easily motivate them to accomplish the tasks. He does not have to use formal power on his subordinates. The task structure can clear-cut and step by step procedures or instructions on tasks provide the manager a greater authority. He can measure work performance. On the other hand, if instructions are not clear, subordinates may disagree or question such instructions. Besides that, the position power also important. The power of the person depends on his position. Fo r example, presidents of corporations, comptrollers, or budget officer have high-position power. According to Path-goal theory, this was formulated by Robert House and Martin Evans. This theory is based on the expectation of rewards. The source of reward is the leader. Thus, the manager should reward the good service or performance of his subordinate in order to influence them to work on established objectives. Rewards are in the form of promotion and pays, as well as support, encouragement, security, and respect. However, leadership styles also affected the motivation to the subordinate. A leader using the suitable leadership styles will lead the subordinate makes a best task performance. Leadership was viewed as a combination of personality traits, such as self-confidence, concern for people, intelligence, and dependability. Achieving, a consensus on which traits were most important was difficult, however, and attention turned to styles of leadership behavior. In the last few decades, several styles of leadership have been identified: authoritarian, laissez-faire, and democratic. The authoritarian leader holds all authority and responsibility, with communication usually moving from top to bottom. This leader assigns workers to specific tasks and expects orderly, precise results. At the other extreme is the laissez-faire leader, who gives authority to employees. With the laissez-faire style, subordinate are allowed to work as they choose with a minimum of interference. Communication flows hor izontally among group members. Leader as laissez-faire style gives employees as much freedom as possible to develop new products. The democratic leader holds final responsibility but also delegates authorities to others, who determine work assignments. In this leader styles, communication is active upward and downward. Employee commitment is high because of participation in the decision-making process. This style of leadership use to encourage employees to become more than just rank-and-file workers. The Adidas set up their way to leading the company, to make their brand more famous. They improve their quality of product to make their product become more good and durable. Adidas not only manage their way on production, they also have a good leading in the process of production. Adidas patient to leading and manage their employee from first step process of production to the end process of production, this cause their company can become so success and famous in today. If have a good leader, is the reason that will success in any condition .So that, if need a good team is cant without a good leader. Conclusion, leading is a process that cant less when doing anything. Leading is important to organizing a group of people to achieve a common goal. When have a good leader, the team have the high percentage to complete their work with nicely or perfectly. Controlling Controlling is the process of evaluating and regulation ongoing activities to ensure that goal are achieved. To see how controlling works, consider a rocker launched by NASA to place a satellite in orbit. Do NASA personnel simply fire the rocket and then check back in a few days to find out whether the satellite is in place? Of course not. The rocket is monitored constantly and its course is regulated and adjusted as needed to get the satellite to its destination. Controlling is the management function in which performance is measured and corrective action is taken to ensure the accomplishment of organizational goals. It is the policing operation in management, although the manager seeks to create a positive climate so that the process of control is accepted as part of routine activity. Controlling is also a forward- looking process in that the manager seeks to anticipate and prevent it. The manager initiates the control function during the planning phase, when possible deviation is anticipated and policies are developed to help ensure uniformity of practice. During the organizing phase, a manager may consciously introduce the deadly parallel arrangement as a control factor. Close supervision and a tight leadership style reflect an aspect of control. Through reward and positive sanctions, the manager seeks to motivate workers to conform, thus limiting the amount of control that must be imposed. Finally, the manager develops specific control tools, such as inspection, visible control charts, work counts, special reports, and audits. The basic control process involves three phases that are cyclic: establishing standards, measuring performance, and correcting deviation. In the first step, the specific units of measure that delineate acceptable work are determined. Basic standards may be started as staff hours allowed per activity, speed and time limits, quantity that must be produced, and number of errors or rejects permitted. The second step in the control process, measuring performance, involves comparing the work (i.e. the goods produced or the service provided) against the standard. Employee evaluation is one aspect of this measurement. In manufacturing, inspection of goods is a routine part of this process; studies of client satisfaction are key elements when services are involved. Finally, if necessary, remedial action is taken, including retraining employees, repairing equipment, or changing the quality of the raw materials used in a manufacturing process. Several features are necessary to ensure the adequacy of control process and tools: Timeliness: The control device should reflect deviation from the standard promptly, at an early stage, so there is only a small time lag between detection and the beginning of corrective action. Economy: If possible, control devices should involved routine, normal processes rather than special inspection routines at additional expense. The control device must be worth their cost. Comprehensiveness: The controls should be directed at the basic phases of the work rather than later levels or steps in the process; for example, a detective part is best inspected and eliminated before it has been assembled with other parts. Specificity and appropriateness: The control process should reflect the nature of the activity. Proper laboratory inspection methods, for example, differ from the financial audit and machine inspection processes. Objectivity: The process should be grounded in fact, and standards should be known and verifiable. Responsibility: Control should reflect the authority- responsibility pattern. As far as possible, the worker and the immediate supervisor should be involved in the monitoring and correction process. Understandability: Control devices, charts, graphs, and reports that are complicated or cumbersome will not be used readily. Controlling activities require the manager to maintain a mindset that continually looks for ways to improve individual, team, and organizational performance. Performance standards describe a model of excellence for work activities and serve as the basis of comparison between actual and desired work performance. Other important controlling functions are continual employee feedback and employee performance appraisal activities. Employee performance appraisals must be ongoing objective and based on established performance standards. A manager should never wait until the annual performance review to discuss problem or deficiencies with a staff member. Consistent, day-to-day feedback and coaching about job performance clarifies expectations, improves the quality of work, and allows the manager to correct problems before they become serious. Coordination of elements of a system is one aspect of managerial control to reach effective outcomes. Other managerial control elements are financial management, compliance, quality and risk management, feedback mechanisms, performance management, policies and procedures and research and trend analysis. These elements are used by managers to communicate to reach a goal, track activities toward the goal, guide behaviors, and coordinate efforts and decide what to do. Managerial coordination and control are important to the success of any organization (McNamare, 1999a, 1999d). Ongoing, careful review using standardized documents, informatics systems, and standardized measured avoids drift and the waste of time and resources that occur when direction is vague. Well-exercised, managerial control is flexible enough to allow innovation yet present enough to effectively structure groups and organization toward goal attainment. The management function of controlling involves the feeding back of information about the result and outcomes of work activities, combined with activities to follow up and compare outcomes with plans. Appropriate adjustments need to be made wherever outcomes vary or deviate from expectations (Hersey et al., 2001). The controlling function of management has been described as a constant process of reevaluation to see whether what is the currently occurring meets needs, plans, and standards, as well as to identify where improvement might be a benefit (Kepler, 1980). According to the case study, Adidas has strongly controlled over its own distribution channel. Adidas policy is to control and monitor hazardous substance to protect human health and environment one of those is to eliminate PVC making progress in finding substitutes like polyurethane, ethyl vinyl, silicones thermoplastic rubber. Adidas also provide training sessions on employment standards and HR systems, health and safety is important for company. Establishing teams to manage and monitor SARS in Asia factory, washing station, disinfectant units. Finally Adidas protects and support the rights of its employees by following all the current employment laws. (Jaswinder, S, 2009) Adidas strategy emphasized Total Quality Management, supported by integrated software systems and code-related labour practices as devices used to reduce unit costs. Monthly monitoring of machine usage and waste reduction, machine leasing and sharing across the five components that comprise the group and a more transparent purchasing system also helped to increase recent performance. (Lopito, 2008)

Service Before Self Essay -- essays research papers

English Composition/ Illustration Final Draft Service Before Self   Ã‚  Ã‚  Ã‚  Ã‚   The day I finally signed up to join the military I knew I was taking a risk in my life. I believe it is important to take risks in life to move into bigger and better things. My first huge decision I made in my life was to join the United States Air Force. My uncle was the main person who put the idea in my head but my dad and my friends thought otherwise. My parents were divorced when I was young and I didn’t have the privilege like my friends to sit down and talk their decisions through with their parents. My father was excluded from the draft after he got in a car accident and I didn’t think he was very fond me joining the service. The main reason I chose to go into the military was make the decision to serve the country. The major decision in my opinion to join the military is to have the privilege to serve your own country. I think it is great for young men and woman to have ambition to serve and it can change a person into very motivated individual. My uncle and m y mother were both very supportive and thought that it would be a great decision in my life. My mother knew it would be hard to see me go but was excited of the idea of coming to visit me in Texas. I knew it was taking a big risk being on my own but I was ready for a change in my life.   Ã‚  Ã‚  Ã‚  Ã‚  With all the excitement of joining the military everything was moving very fast and it was already S...

Superficial Love in Shakespeares A Midsummer Nights Dream :: Midsummer Nights Dream Essays

Superficial Love in Shakespeare's A Midsummer Night's Dream In the first soliloquy of Shakespeare's A Midsummer Night's Dream, Helena talks primarily of a love that contains depth, a love that looks at who a person is, personality-wise, as opposed to nothing more than their appearance. Helena explains, "Love looks not with the eyes, but with the mind" (1.1.240). In the shallow culture in which Helena lives, and even in today's society, it is difficult for people to look beyond the outer shell and follow a deeper perception. The concept of the "perfect person" is constantly drilled into people's minds. In Helena's day, this was through expectations taught to children from their parents. Today, this type of expectation comes mostly from the media and entertainment industries. Helena describes love as the "admiring of his qualities" and as possessing the ability to "transpose to form and dignity" (1.1.238-240). Though this may not be a perfect definition, it is much closer to the Biblical definition as described in 1 Corinthians than most common definitions of Helena's day. Because of the strong influence of the shallow culture in which Helena lives, she, too, finds it difficult to keep society's pressure out of her definition of love. One of the first things she points out in her soliloquy is the fact that, "Through Athens I am thought as fair as she" (1.1.232). She then proceeds to explain how she wishes Demetrius would think she is as fair as Hermia. If Helena believes so strongly in love coming as a result of admiration of one's personality, one must question why she loves this man who focuses merely on the appearance of women and pays no regard for who they are as a person. Then, again, the number of men in her day who didn't found their love on such superficial characteristics was probably pretty low, if not zero. Either way, Helena's perception of love is not perfect, her thoughts are still influenced by the surrounding culture. Helena's interpretation of love, as a deep, powerful emotion is virtually unseen in the rest of the play. Rather, the opposite, superficial love, plagues most characters of the play. Theseus, Demetrius, and Lysander constantly offer comments about females. Rather than focusing on who these women are, these comments pertain to the appearance of the women.

Alberti and urban context

Among Renaissance architects, Leon Batista Alberti was perhaps the most visionary authority on urban context and city planning.Though he was not an urban planner in the modern sense, he had a keen understanding of the city as an integrated, organic whole, and his designs and writings reveal his view that cities should be well-ordered and buildings should integrate themselves smoothly into that overall fabric.   In this regard, he was well ahead of his time and anticipated the ideas of urban context that exist today.Despite his visionary skill and prowess at architecture, Alberti (1404-72) was actually not a professional architect and seems never to have actually even supervised the construction of any of his works.   He was a polymath, or â€Å"Renaissance man† – cultured, well-educated, and well-versed in various academic fields, from art and religion to science and mathematics.According to art historians Ludwig Heydenreich and Wolfgang Lotz, Alberti â€Å"remaine d to the end the adviser who laid down the general lines and occasionally gave instruction for details . . . but he never set one stone on another.†[1]   Biographer Anthony Grafton’s description is even more to the point – â€Å"an impresario of society and space.†[2]Indeed, Alberti lacked the practical building experience most contemporary architects had, mainly because he was trained to advise and administer rather than actually build.   Born illegitimate but privileged in Genoa, he was well-educated as a youth and in 1428 took both a degree in canon law and orders in the Catholic Church.For much of the remainder of his life, Alberti served as an administrator and advisor to the popes, most notably Nicholas V, a friend from youth, who hired him to consult on major building projects in Rome.   Though mostly a career church administrator, Alberti pursued a wide array of intellectual interests and â€Å"presented himself as a master of all the ration al arts of living upon which his contemporaries set great store.†[3]In accordance with the Renaissance’s reverence for ancient Greek and Roman models, Alberti drew heavily from antiquity – not merely for decoration (which he believed should be used sparingly and tastefully, not simply for the sake of decoration alone), but for proportion and, more importantly, placement within a given physical and historical context.For example, in one of his first major works, the church of San Francesco at Rimini (whose renovation and redesign he supervised around 1450), Alberti used exterior motifs drawn from the area’s ancient monuments, varying these to suit the building itself and thus let it reflect the local architectural, cultural, and political contexts.The church’s faà §ade uses simple forms and a scale suited to the buildings around it, because, says Heydenreich, no single person’s vision would dominate that setting: â€Å"[It] was the product o f a collaboration between patron, adviser, and working architects. . . . ‘Local styles’ of this kind occasionally appear, but only where the political structure of the region favours them. . . .†[4]   In this sense, he heralded the post-modernists of the late twentieth century, who believe in urban fabric and context rather simply in designing buildings with no relationship to their surroundings.Alberti’s works in Florence between 1455 and 1470 demonstrate, in Heydenreich’s words, â€Å"[how] deeply the traditional forces in a city can influence the idiom of an architect.†[5]   There, his church of Santa Maria Novello draws heavily from local Tuscan styles and fuses them with a large Roman scale (as mandated by the Pope), making a distinctive building that fits with its prominent neighboring structures.(Though he used local elements freely, Alberti rarely directly imitated other buildings; when he borrowed forms or elements, he tended to f use them with those on nearby structures.)   Also, and perhaps more importantly, it embraces a unity of design, both within itself and in relation to the buildings around it, so that it does not appear incongruous or artificially imposed on its immediate context.Alberti also aimed to site buildings according to surveys he conducted, in keeping with his mathematical and cartographic skills.   Using a measuring disk he created, his survey of Rome (conducted around 1444, when he first entered architecture) â€Å"allowed him to establish the radial coordinates of Rome’s main churches and the towers on the city walls and to plot those in plan.†[1] L Heydenreich & W Lotz, Architecture in Italy, 1400 to 1600, Penguin, London, 1974, p. 27. [2] A Grafton, Leon Batista Alberti, Hill & Wang, New York, 2000, p. 263. [3] Grafton, p. 21. [4] Heydenreich & Lotz, p. 32. [5] Heydenreich & Lotz, p. 33. [6] R Tavernor, On Alberti and the art of building, Yale University Press, New Ha ven, 1998, p. 13.

Accrual Swaps

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