Climate Change Problems For The Fiji Islands Environmental Sciences Essay

This paper explores the hazards that climate alteration airss to the touristry development in Fiji islands. It shows the inauspicious effects of the altering clime and the dangers pose by the touristry activities and besides pose a major jeopardy for the local people in the part. It besides deals with the unsafe C emanations and CO2 consequence on the landscape, nutrient, H2O, energy. Cardinal words: Pacific, clime alteration, C and CO2 emanations.IntroductionThe Pacific is the world`s largest ocean with a surface country of 175 million sq kilometer and constitutes for 40 % of the planet`s Waterss. Located in the tropical latitudes, it covers more than half the globe`s perimeter. Temperature of the surface H2O in the western tropical parts is ever more than 28 ISC over a deepness of several hundred metres. This makes up the world`s storage of thermic energy for exchange with ambiance. Here the interaction between ambiance and ocean is most utmost and influences the clime non merely regionally but planet-wide. The states of the Pacific are obscured human colonies absorbed in this huge fluid existence. The ocean is the most of import factor commanding the environment and life. Hence any alteration in pelagic conditions and climatic alterations are of import for environment and life ( Philander, 1990 ) . The average clime of a part is defined by the mean conditions observed over 3 decennaries or more, sing all features that makes conditions felt by everyone and predicted by meteorologists. The chief features are temperature, air current conditions, atmospheric force per unit area, cloud screen and rainfall. The south Pacific is bounded by an country of low force per unit area near the equator and high force per unit area around 30 S analogue. North-south force per unit area creates the regular air currents in these two countries known as the south-east trades ( Philander, 1990 ) . Any alteration in earth`s clime has an impact on world, biodiversity, wellness and services provided by ecosystems worldwide. For accommodating to such clime alterations it is necessary to understand fluctuation of clime, why and how the clime alterations, and how it impacts the earth`s ecosystems. Climate manner is an of import manner of understanding clime variableness, alterations and impacts. Earth`s clime is altering and such alterations tend to take topographic point with different forms which may be characterised by one or many manners of the clime systems ( Philander, 1990 ) . Fiji is the largest touristry finish in the south Pacific but international reachings are unstable over the last 5 old ages because of harmful events like political putsch in Fiji in 2000, terrorist onslaughts in United States on 11th September 2001, the Bali onslaught in 2002, and terrible acute respiratory syndrome eruption in Asia in 2003. Tourism is endangered to natural jeopardies and catastrophes like temblors, tsunamis, inundation, drouths, and cyclones. Climate alteration is an of import feature in catastrophe direction as it is likely to impact Fiji through sea degree rise and storm rush, altering temperature and utmost conditions events ( Wilbanks, 2003 ) . About 400,000 tourer visited Fiji in 2002 with an mean length of stay of 8 yearss. While most visitants come for remainder and relaxation linked to beach environments, current selling runs aim to switch the image from pure beach publicity to a wider experience ( Ministry of tourism-Fiji, 2003 ) . The chief purpose of this journal paper is to analyse effects of clime alteration in Fiji islands and accommodating and minimising clime alteration by the tourer resorts. The ground for behind this is that studies and interviews were undertaken ensuing in many operators already prepared for clime related alterations and adapt to possible impacts ensuing in clime alteration.MethodTourism in Fiji is mostly based on resorts therefore adjustment sector is outstanding touristry sub-sector. Tourists spend most of their clip at the resorts. For the above grounds it was appropriate to concentrate on this analysis of adjustment.Effectss of clime alteration on touristry in FijiTourism operators are familiar with ecological factors like strong reefs and plain H2O crucial for touristry in Fiji. Operators were witting of the clime alteration associated impacts like cyclones, the walloping of coral reefs and deluging. Contaminated H2O was related to mounting H2O temperature and clime alteration. I ncreasing sea degrees were mentioned by three concerns, two of which lie in low lying Mamanuca Islands. Generally adjustment concern had experienced at least one of the clime related impact. The most common impacts were eroding, H2O handiness, and break of electricity. Many resorts were affected by cyclones ensuing in coral bleaching and belongings injury ( Short, 2004 ) . Climate related impacts antecedently experienced Frequency out of 25 Remarks by respondents Shore line/beach eroding 9 Banks on border property/beach give manner Decreased H2O handiness 9 In recent drouths Interrupted supply concatenation 8 Power cuts Coral bleaching 8 Noticed by tourers, snorkelling affected Damage to belongings 5 From sea rush Sea degree rise 3–Storm frequence and strength 3 Care of gardens ( Short, 2004 ) . Eight concerns said that they were non affected by any of the factors listed. There are five countries located in the Mamanuca Islands, which are comparatively exposed to climate alteration due to the hazard of cyclones, sea degree rise, hapless H2O quality, vanishing corals, and unequal H2O handiness. There is demand of apprehension of clime alteration jobs and the directors do non portion the jobs faced by clime alteration ( Short, 2004 ) . Tourist adjustment uses big assortment of energy resources with electrical energy created from hydropower or Diesel generator being most of import for energy usage. Petrol and Diesel is use for concern vehicles and other intents. Besides liquefied crude oil gas is used by most concerns largely for cookery, hot H2O and in wash. Energy use and nursery gas emanations differ loosely for diverse concerns. The criterion of adjustment and geographical location are the two factors that have major influence on energy ingestion and carbon-dioxide emanations ( Becken, 2002 ) . Tourist adjustment in the Mamanuca Islands is about 2-3 times every bit carbon-intensive as that in Viti Levu. The key cause for this is in electricity coevals, which is to a great extent less carbon-intensive in Viti Levu. Because of the high measure of renewable energy beginnings ( hydro and bagasse ) compared with Diesel production on islands with its natural insufficiency ( about 65-70 % of energy input is lost during the procedure of coevals ) . Resorts on distant islands run more or less self-sufficiently, and therefore have auxiliary energy demands ( e.g. , sewerage intervention, stop deading trash ) . Transport energy use is besides high given that non merely do tourers hold to be transported to and from the resort, but so besides do nutrient supplies, energy ( Diesel and gas ) , H2O and other devices required for runing the resort ( Becken, 2002 ) . The unsmooth executable estimation is obtained of energy usage and C monoxide emanations related with touristry for the Fiji. The entire figure of visitor darks spent in Fiji was 2,891,295 in 2002 ( Department of energy, 2003 ) . 82 % of visitor-nights were spent in hotels, 13 % in backpacker/budget adjustment, and the staying darks were being spent in motels, on boats or in other signifiers of commercial and non-commercial adjustment. Total energy used due to tourist adjustment was calculated at 1,078,373,475 MJ per annum which is tantamount to national energy usage of 6.5 % . in footings of C monoxide the adjustment industry emits 68,219 metric tons per annum. ( Department of energy, 2003 ) Tourism in Fiji is highly exposed to climate alteration related jeopardies such as cyclones, deluging and storms, sea degree rise, eroding, conveyance and communicating break, and momently less H2O handiness. Another most of import apprehensiveness for the touristry industry is the want of natural systems, such as coral reefs and forest ecosystems, farther exasperated by clime alteration. Tourism concerns in common are affected in the signifier of physical harm from a cyclone or storm rush, eroding, and coral bleaching. In malice of the high hazard linked with tourer installations built on the waterfront, most new developments spotlight on coastal countries. Mangroves are been cut down in big graduated table who in bend Acts of the Apostless like a protection against clime related alterations ( Jones, 2003 ) . Tourist adjustment suppliers adapt to climatic conditions that may impact their concern, and in making so they are besides prepared for impacts that may ensue from a altering clime. Typically, operators focus on comparatively concrete and foreseeable bad impacts, such as cyclones and storm rushs, for illustration by cyclone-proofing their constructions and raising breakwaters. A figure of adjustment suppliers have insurance screen against cyclones and storm rushs. By and large, it seems that the hazard of accumulative impacts or more abstract impacts are less recognized and addressed. Pollution control, sewerage intervention, and H2O direction are illustrations of this. The exposure to extreme climate-related events can be reduced when clime alteration version is integrated in the development procedure from the earliest phases ( Jones, 2003 ) . The exact location of the development and design such as constructing stuff, orientation, constructions and landscaping aids in cut downing the exposure. There is a chance to alter touristry development in at finishs less vulnerable to climate alteration, with current efforts to diversifying Fiji`s touristry merchandise in relation to ecotourism. New and alone touristry merchandises can be developed in Fiji on high land countries utilizing Nipponese construct of shakkei ( borrowed landscape ) , where hotel layout, garden landscape gardening and scenery are assorted together into an overall experience of ecosystem that is different from the typical beachfront ( Ayala, 1995 ) . A figure of nursery gas moderateness processs are in topographic point such as accommodating generator sizes, exchanging off visible radiations, energy efficient visible radiation bulbs and solar hot H2O. There is a immense potency for solar energy and wind-generated power particularly on the Coral Coast, the Mamanuca Islands, and Sonasavu, these engineerings are taken up easy, inhibited by deficiency of cognition, capital, capacity and authorities inducements. Often, the energy demand of a individual tourer resort is excessively little to warrant investing in a air current turbine. The policy focal point and involvements of resort operators in Fiji are development-driven, although there is a strong acknowledgment of the construct of sustainable development. Climate alteration is chiefly seen from the position of touristry ‘s exposure and version. Extenuation seems to be less pressing, although in the average term increasing nursery gas emanations ( e.g. , as a consequence of i ncreasing tourer reachings ) could sabotage Fiji ‘s credibleness in international dialogues on clime alteration. The above order of Government and industry precedences has to be recognised when seeking to implement any climate-change-related steps ( Ayala, 1995 ) . Climate alteration can be assorted with sustainable development by placing cardinal jobs and so associating those to climate alteration. In the instance of Fiji touristry these major local jobs are land usage issues, old stock adjustment, deficiency of new capital and investing, limited air capacity, dependance on air travel, economic escape, deficiency of alone merchandising point, environment debasement and political instability ( Narayan, 2000 ) . Environmental jobs like pollution, deforestation and inordinate usage of resources are to be considered. Potential issues in add-on to these jobs are more likely to be funded by giver bureaus, stakeholders and industry members ( Hay et al. , 2003 ) . Acknowledging co-benefits of clime alteration policies is every bit of import as its consequence, for illustration, heavy usage of air conditioning leads to increase in nursery gas emanations or the resettlement of sand adds to local environmental impacts. Future work would necessitate to take into history technological and economic facets, every bit good as the expected sum of decreased or increased nursery gas emanations ( Dang et al. , 2003 ) . Energy is a major cost driver for the operation of a touristry adjustment concern, particularly when energy is derived from fossil fuels either for conveyance or electricity coevals. The operation of Diesel generators is dearly-won, because of inefficiencies, transit costs ( diesel cargo ) , care, and wages for powerhouse staff. Therefore, directors have an economic involvement in maintaining electricity ingestion depression. The Southern Cross with Diesel generators, nevertheless, is that one time a generator is purchased, the optimal scope of electricity coevals is determined at approximately 80 % of the maximal public presentation. Mini hydropower strategies are less relevant for coastal resorts, but could be an option for touristry ventures operated in inland communities ( referred to as ecotourism operators by the Fiji Ministry of Tourism and Visitor Bureau ) . The capital costs are really high, nevertheless, and accordingly the consumption is minimum. The Department of Energy p resently assesses possible sites for mini hydropower strategies, and it is besides researching possible for geothermic electricity coevals on Vanua Levu, the 2nd largest island of Fiji. Wind energy is non widely used in Fiji, but the Coral Coast, Mamanuca Islands, and Sonasavu are assuring locations for wind-powered coevals. Wind energy systems are available at different graduated tables, runing from little 1-kW 1s to 100-700 kilowatt strategies ( medium graduated table ) , or even larger 1s ( UNEP, 2003 ) . Tourist resorts would necessitate small- to medium-scale air current systems if they want to run into their whole electricity demand by wind power. Small islands are improbable to raise air current turbines because of deficiency of infinite and noise pollution. Resorts on larger countries are in a better place to prosecute wind energy. No renewable energy beginnings are presently earnestly discussed for conveyance, although one resort looked into wind-driven boats, and there are geographic expeditions into replacing fossil fuel with bio-fuel, for illustration derived from coconut ( copra ) oil ( Sopac, 2004 ) .Stairss to minimise the consequence of clime alteration in FijiReforestation is the most of import agencies of cut downing clime alteration. Trees minimizes vulnerable nature of cyclones, better microclimate and enhances landscapes which are used in touristry activities. Trees cut down C content in the air and are utile in adaptative steps like eroding control and watershed direction. Forest protection and plantation should be done under adaptation policies. Developing little graduated table engineerings for air current and solar energy on the distant island would assist cut down the dependence on imported dodo fuel and economic escape ( Dang et al. , 2003 ) . Adaptation Impact on extenuation Impact on environment Tree plantation Reduces net CO2 emanations through C sinks Benefits biodiversity, H2O direction, dirts Water preservation Reduces energy costs for providing H2O Positive in countries where H2O is limited Renewable natural resources Reduces CO2 emanations Overall, less fouling than fossil fuels Natural edifice stuffs Small C footmark for locally produced stuffs Depends on sustainability of plantations Reducing H2O pollution Increased energy used for sewerage intervention Positive for coral reefs and marine life Marine protection Impersonal Positive for Marine biodiversity Rain H2O aggregation Saves transport energy for providing H2O Possibly interrupts the natural H2O rhythm Guest instruction Impersonal Additions consciousness Puting back constructions Impersonal Positive when constructions built off from beachfront Diversifying markets Positive if markets are eco-efficient Depends on environmental impacts of new markets Weather proofing tourer activities Depends on the type of activities Depends on the type of activities Water desalination High energy costs Returns force per unit area off fresh water resources Increasing beach conditioning Additions CO2 emanations Air pollution in instance of Diesel coevals Beach nutriment Energy usage for excavation and transit Disturbs eco systems Reducing beach eroding with sea walls Impersonal Disturbs natural currents and cause eroding ( Dang et al. , 2003 ) . There is no common scheme to turn to interactions between clime alteration and touristry in Fiji, nor is at that place a sector-wide industry association that could advance any climate-change-related enterprises. However, there are stray illustrations among industry members that reveal a high apprehension and advanced usage of engineering and direction to turn to climatically unfavorable conditions. Those operators are besides best prepared for increased hazards ensuing from clime alteration. Besides, a figure of operators engage in wider environmental direction, energy preservation, and hence climate alteration extenuation, although the nursery gas emanation facet is seldom the ground for the mitigating steps undertaken Overall, there is a demand for tourism-specific information on what clime alteration is, how it will impact touristry, and what operators could make to accommodate and extenuate. In the medium term it would besides be of import to include climate alteration in the course of study of third instruction for pupils in the field of touristry, resource direction technology and architecture. Since the range and costs for many version and extenuation steps are mostly determined by the design of tourer installations, the incorporation of these facets into architectural classs is peculiarly of import. Alongside information and instruction enterprises, the Government could help concerns in set abouting energy audits, easing the execution of Environmental Management Systems ( e.g. , Green Globe 21 ) , and supplying inducements, for illustration for the consumption of renewable energy beginnings. Climate alteration could organize portion of a wider hazard direction program for touristry. Such an enterprise is presently being discussed between the Ministry of Tourism and the Disaster Management Office. A two-level attack could be possible, where guidelines are provided for touristry operators to develop their ain hazard or catastrophe direction program at the concern degree, while Government screens wider issues beyond single concerns, such as touristry substructure and larger emptying programs. The current effort by the Fiji Visitor Bureau to diversify the merchandise could be seen as portion of national-level hazard direction, as they attempt to distribute hazard across different markets ( e.g. , event touristry, athletics touristry, nature touristry ) and seasons. Fewer enterprises exist to weather-proof touristry, as suggested for touristry in Phuket, Thailand ( Raksakulthai, 2003 ) . Another of import measure towards implementing a nation-wide hazard direction scheme for touristry and clime alteration would be the function of all touristry substructure, every bit good as the hazard of assorted jeopardies in different locations. The Department of Environment in their clime alteration policy or the Ministry of Tourism in their hazard direction program are best advised to prosecute steps that offer win-win state of affairss, viz. for version, extenuation, wider environmental direction and development. Examples of such steps are re-afforestation, H2O preservation, and the usage of renewable energy beginnings. It is recommended that the synergisms between version, extenuation, and sustainable development be explored farther and that the effects be quantified where possible ; i.e. , how much C can be saved as a consequence of a peculiar step and what costs are involved. This is even more of import given the deficiency of resources in Fiji, which requires maximizing benefits from any enforced step ( Dang et al. , 2003 ) . Reducing the ingestion of hot H2O for wash and showers and cut downing the H2O temperature are salvaging steps. Other energy usage decreases steps in adjustment are illuming, including energy efficient visible radiation bulbs, detector lighting in the garden, solar panel visible radiations, and room keys used to run visible radiations inside the room. Although energy efficient bulbs are good option they are expensive and do non last long because of the fluctuating supply of power from generators. In the smaller islands the energy costs of transporting are higher, so the directors tend to increase the ship burden with riders on board with nutrient, waste or H2O. One manner of salvaging fuel is to minimise transportation trips. The addition in planetary average temperature to 2 grades above pre-industrial degrees is necessary to maintain the hazard of unsafe clime alteration at an acceptable degree and to restrict clime impacts. Temperatures increase certain degree of atmospheric concentration. The consequences indicate that in order to hold a good opportunity of restricting planetary mean temperature in the long tally to 2 grades atmospheric concentration of all nursery gases needs to be stabilised. Intergovernmental policy on clime alteration i.e. IPCC indicates that maintaining concentration in the scope of 445-490 ppm requires planetary emanations to top out by 2015, and to fall by between 50-85 % by 2050. Current tendencies would ensue in much higher concentrations and high hazards of ruinous clime alteration. The clean development mechanism means to do conformity with easier mark committednesss, the Kyoto Protocol allows utilizing offset credits from emanations decrease undertakings in developing states, under the Clean Development Mechanism ( CDM ) . Governments can suggest and implement emanations decreases on a project-by-project footing under CDM. The ensuing credits are bought by authoritiess that are under emanations decrease duties. Large undertakings classs are renewable energy chiefly utilizing hydropower alternatively of fossil fuels, decrease of methane emanations from landfills and coal mines, emanations from cement production, and devastation of powerful industrial gases. There were over three 1000s CDM undertakings underway in may 2008, which, is implemented and approved, would give expected emanations decreases of 2.5 billion dozenss of C dioxide. The Asian Pacific part histories for 80 per cent of the CDM credits that expected to be generated. The World Bank cites supply e stimations of 1.4 to 2.2 billion credits by 2012Decisions and recommendationsGlobal concern over clime alteration impacts and hazards has increased greatly in recent times, and clime alteration is recognised non merely an environmental challenge but besides an economic challenge. The Pacific part is home to the fast growth, big economic systems in the universe and the dominant beginning of growing in nursery gas emanations. To restrict and cut down emanations action is required in developing states. There is big figure of chances to cut down emanations but most of these are expensive and can non be implemented unless policy scenes change. More ambitious policies will be needed to turn emanation tendencies around in developing and developed states. The international kineticss are of the reciprocally reenforcing type: one country`s action depends on other states making their spot. The more states commit to important policies, the easier it will go to pull others in. In contrast, if so me states refuse to take portion in corporate action, others will besides decline to make so. An effectual response to planetary clime alteration will necessitate to affect bilateral trade or many-sided understanding. Large and medium sized economic systems will necessitate to be a portion of it. For an understanding to win, the door must be kept broad unfastened for developing states to prosecute to the full in policies, with the support of high income states. Climate alteration analysts predict that within the coming decennaries, sea degree will lift bit by bit. So the impacting state might hold begun placing the effects of clime alteration on touristry activities and overall people populating in that part. Small islands are at hazard to accommodate to the inauspicious affects of clime alteration because of high costs every bit good as benefits. Not merely merely people but alone human civilizations are besides at high hazard. Migration is another option for local people but once more the cost factor is important, as most of these people are illiterate and unemployed. They will hold to relocate unwillingly. Survival is the chief concern in this instance. It is besides extremely impossible for any recipient state to allow refuge to an full state. The larger impact of clime alteration will dispute the capacity of the state. The secondary impacts will be H2O scarceness, nutrient security, wellness services, land scarceness. At some point man y land countries will go incapable of prolonging life and people will be forced to migrate.

Development: Advantages and Disadvantages

Development is often defined in terms of progress, forwardness, and modernity. It is characterized by high-rise building, state-of-the-art gadgets, consumer goods, and an over all idea of a good life. However, according to Amartya Sen, development â€Å"is a process of expanding the real freedoms that people enjoy† and also a â€Å"process of removing unfreedoms and of expressing the substantive freedoms of different types that people have reason to value† (Gasper, 2000). This definition has to be further analyzed because the real effects of this so-called development are very much contested.There have been numerous debates whether development infused positive or negative consequences. Development is a very controversial term and much is to be known with regards to its effects, whether it is indeed beneficial to those under it or it is a curse that they are better off without. Development brings a more comfortable life but at the expense of the environment and the tradi tional culture of the people. The advantageous effects of development which is primarily focus in the idea of giving a better life for the people under it has also been discussed and taken into account.This is best described by the changes in the way of life of the Ladakhi people in the midst of development. One of the most important contributions of development is through health and the decrease in life mortality. It is through the progress of science and technology especially in the field of medicine that treatments for diseases which were incurable before are now given solutions. The traditional life of the Ladakhis is a good example. Previously, people in Ladakh die from diseases that western medicine has found a cure for but the introduction of development in this place has aided in solving this problem.Furthermore, infant mortality in Ladakh which is estimated to be as high as fifteen percent decreases due to improvement in health conditions (Norberg-Hodge, 1991). Development has also given the opportunity for people coming from one part of the world to be more accustomed and familiar with those living in the other parts of the globe. The idea of development has paved the way for better communication and interaction by means of the media, trade, and other methods of progress.Many Ladakhis are enjoying some benefits of development as the introduction of money and technology made their lives more comfortable than before. They enjoy the ability to travel to new places and buy various kinds of material goods outside like imported rice and sugar which have become parts of the everyday meal of the Ladakhis (Norberg-Hodge, 1991). Development has also answered one of the serious problems in Ladakh, which is illiteracy. It is through the idea of development that new opportunities for education are provided.Education gives those people who traditionally belongs to the socially disadvantaged the chance to acquire higher position. People do not have to be contented by simply being a blacksmith because they could apply for a better job by educating themselves. This opportunity is especially seductive to younger people because of the freedom and mobility that they associate in living the modern world (Norberg-Hodge, 1991). Moreover, education also opens new horizons for these people as they could learn different things coming from various places instead of being confined in their own environment.Development has brought real improvements to the traditional society of Ladakh. The introduction of money, technology, as well as improvement in the medical conditions entail with it significant benefits for the Ladakhi people. Using these aforementioned factors as a gauge, it can be said that their condition is better and far more comfortable as compared before (Norberg-Hodge, 1991). The effects of development is not always seen in an advantageous lens because there have been instance wherein it has bring more harm rather than good.This is greatly felt in third world countries or the so-called developing countries that are just recently undergoing the path of industrialization. The study of Ladakh before and after the influences of development came into their place is a good example in order to measure the negative outcomes of development. One of its adverse effects is in terms of the environment. The establishment of factories, buildings, and other form of modernity has taken its toll in the ecological condition of society. The western idea of development has forgotten to include the importance of sustainable development.A good example is Ladakh, a territory that is situated in the Indian region of Jammu and Kashmir. It is known for its breathtaking environmental beauty especially its mountain formations. For 500 years, the Ladakhis have been self-sufficient as they are only dependent upon their environment where they acquire their basic needs as well as their little luxuries in life. However, this kind of situation changes drast ically with the presence of westerners that insisted in changing Ladakh in a more progressive territory.The usual source of living of the citizens that is greatly through agricultural means is now changed with employment in factories at the center of town. Majority of Ladakhis have their own land but they have foregone tilling their own soil to acquire occupations that give them money in return rather than natural resources that they need. Such kind of thinking is highly influenced by tourists coming in Ladakh that are instilling the idea that the their form of life is backward and that through the aid of science they could even maximize the products that they get from the environment.This perception of development is producing discontentment and greed among the people that forces them to destroy the environment which have been a source of their livelihood for many years just so they could satisfy this new form of desire (Norberg-Hodge, 1991). The presence of new source of modernity in return is polluting their environment. The rivers that have been a source of life for these people could not even be drunk anymore. The fresh air that they once breathed is now polluted and even the land that play an important role in their traditional culture and local economy is being replaced by infrastructures.Being the case, it is just evident that the idea of sustainable of development is not given due importance but rather what is observable is the destruction of the environment that is inconsiderate of the succeeding generations’ welfare. Another important drawback that is brought about by development is its ability to destroy the traditional culture that has been the very roots of people’s identity. New ideas of what development is, of what is modern and what is not, and even the idea of what is civilized from what is not are threatening the values and traditions that local people uphold.In the case of the Ladakhs, as their way of life is infiltrated by mo dernity their value system is also being in changed. These people strongly believed in their strong relationship with nature and among themselves. This is rooted from the idea that each and every life form is dependent upon each other. Nature and everything in it as well as the people have an interdependent and intertwined interaction. One cannot survive without the other and vice versa. Unfortunately, this had changed dramatically.The old tradition wherein they acquire their fundamental means of living in the environment has its limit but this is not the case anymore as progress persist ecological boundaries are being transcended. This is even observably in the relationship of the Ladakhis, which is communal in nature. Before, to be able to sustain their everyday needs they work together characterized by cooperation and harmony amongst them with each individual taking equal responsibilities in the accomplishment of a particular task.They do such as a mutually beneficial practice be cause whatever they gained as a group would eventually be advantageous for them individually as well. As the idea of stiff competition enters the frame of mind of these individuals they started to take for granted their communal identity and instead focused on their personal gains. Such incident resulted in the break down of communities as less interaction among them exists due to the fact that they no longer work together in acquiring their needs but rather they compete against each other in order to acquire a job.The kind of work that allows them to be source of cheap labor, which is seen in the establishment of call centers in India (Can, 2004). This competition has been the cause of friction among citizens. The Indians and Muslims in Ladakh who has live side by side in harmony for many ears are now experiencing conflict due to the struggle for scarce resources, the unequal competition in the market, and the over all idea of greediness (Norberg-Hodge, 1991). Lastly, development c laims to bring security through employment, maximization of resources, and easier access to other parts of the world.Ironically, its outcomes brought more insecurity not only to the environment but most especially to the perception of the people towards themselves. By means of western tourists and the influence of media, the idea of comparison is produced wherein people like the Ladakhis evaluate their way of life based upon the lifestyle of those in the west. This intends creates an idea of inferiority to these people because they cannot measure up to the western idea of what a good life is. They feel ashamed of what they are as well as to the values and traditions that they once uphold.Their choices and actions changed in a way that they want to pattern it with the west. Ladakhi people lost their self-esteem and their very sense of self-identity (Norberg-Hodge, 1991). Such kind of mentality is exemplified even in their form leisure. If before they find pleasure by bonding among th emselves they now seek new ways of enjoyment. Children now play with toys like Barbie and Rambo and the adults want to watch movies and read magazines. Being the case, this resulted in less time for the family and even changed their perspective of how to view a man from a woman.A woman should give value to her aesthetic importance while a man should maintain a macho imaged which the media enforces. Even the idea of education has a polarized perspective as it is based upon the western curriculum. Traditional form of education is based upon ones’ experienced as how it would be useful in their environment unlike the western education that specializes on a particular field that limits a person capability. These aforementioned situations, heightens the insecurity of these people to see themselves as second class citizens and forced them to be prototypes of the westerners.There are two faces in the idea of development. One side of development has its positive or advantageous effect s. Using freedom as a lens could aid in seeing the beneficial outcomes of development. There are three important roles that development contributes in the attainment of freedom. First, its â€Å"direct importance† that enables people to decide for themselves without any constraints. Even the poorer section of the society could participate in the market place as they are given the chance to participate in the activities within the market. Second, development entails â€Å"instrumental importance†.This paves the way for people to achieve their desired results through the freedom that development gives them. Development provides the means or methodology that enables individuals to accomplish their objectives. Lastly, its â€Å"constructive role† that provides the venue for easier exchanged of information. This allows people to participate more in the formation of policy as they have the ability to express their opinions and suggestions. Development empowers them to participate more and enables them to highlight important issues that should be immediately addressed. However, development also has its negative side.It is seen in the adverse outcomes that it brought. This is mostly highlighted in the case of Ladakh wherein it has experienced drastic changes in its environment, its way of life, and its people’s perception of themselves. Development has affected the ecological state of Ladakh that diminishes the source of natural means for its people. The idea of environmental sustainability has been neglected in order for modernity to take place. The once beautiful place of Ladakh has very disturbing problems of pollution. Another adverse consequence of development is how it undermines the traditional culture of local people.They no longer adhere to their usual practice of communal activities. The people become more individualistic that resulted in the breakdown of communities. This affected their relationship that is previously grounded in the belief of the interconnectedness of their lives with nature and among each other but has changed due to development. Furthermore, even the way people look at themselves have changed as they lost their self-esteem and identity. They compared their way of life to that of the west, which resulted for them to feel a sense of inferiority.Ladakhis have to change their selves in order to measure up to their western counterparts. The advantages and disadvantages that development brings should be further studied. A deeper understanding of its effects could aid in finding the balance of how development could best be practiced in such a way that it could helped the people to live a life of comfort without undermining their local values and traditions. The lesson that can be learned from these outcomes is that the meaning of development should be re-assessed and re-evaluated.Development should not simply be taken as it is especially if the only basis of what development is comes from the p olarized definition of western standards. Another factor that also have to be taken into consideration is who really benefits from development. If its really after the good of all or just a few. A better understanding of development and a sense of awareness of how it takes place as well as its results are effective means by which development could be gauged whether it really has advantageous or disadvantageous effects. References Can, M. ed. (2004). Chains of Future: Linking Women Producers and Workers in the Global Markets. London: Commonwealth Secretaries. Gasper, D. (2000). â€Å"Development as Freedom: Taking Economics Beyond Commodities- The Cautious Boldness of Amartya Sen†. Journal of International Development. 12. 989-1001 Norberg-Hodge, H. (1991). â€Å"Nothing is Black, Nothing is White†. In Ancient Futures: Learning from Ladakh. London: Random. Norberg-Hodge, H. (1991). â€Å"The Development Hoax†. In Ancient Futures: Learning from Ladakh. London: Random. Sen, A. (1999). â€Å"The Perspective of Freedom†. In Sen, A Development as Freedom. Oxford.

Outline of Chinese Americans and Mexican Americans Assignment

Outline of Chinese Americans and Mexican Americans - Assignment Example A second wave of immigrants came during World War II in order to supply construction, farm and domestic labor under the â€Å"Bracero Program†. During the last quarter of the 20th century there was large scale immigration both legal and illegal from Mexico to the US due to Mexico’s severe economic problems. The first large scale Chinese immigration to America was in 1848 when the California Gold Rush led many to believe they could find their fortune and escape economic hardship especially in Canton province because of British dominance( Le 2012) They also came to Hawaii as contract workers in sugar plantations, and to continental US as merchants, gardeners, domestics, laundry workers, farmers and starting in 1865 as railroad workers. Public Policies In 1848 the Treaty of Guadalupe Hidalgo guaranteed Mexican Americans all the rights of citizens of the United States including free enjoyment of their liberty and property. However despite these promised protections they wer e largely dispossessed of their land by an Anglo run legal system that administered land holdings Kutty 2008) This caused a severe reduction in their economic status into the 20th century. In addition to this economic discrimination, Mexican Americans also suffered racial and legal prejudice with civic segregation similar to the blacks in various areas until the 1950s and 1970s. Even the US Congress expressed the view that Mexicans were racially inferior. During the Great Depression, because of welfare burdens the federal government pursued a policy of forced repatriation of Mexican Americans to Mexico. The public policies affecting Chinese Americans were the Naturalization Act of 1870 restricting all immigration into the US to white persons and persons of African descent and the Chinese Exclusion Act of 1882 ( thinkquest). The former act made Chinese ineligible for citizenship until 1943 and was the first significant bar on free immigration in America’s history. The latter a ct was to prevent an excess of cheap labor Ways Policies Affected Immigration Success Not only were Mexicans deprived of their property after the Guadalupe Hidalgo Treaty, but had to pay discriminating taxes as well( Kutty Policy) For example in California they were subject to the Foreign Miners License Tax which was enforced only against the non Europeans. This policy was a success in its’ unstated goal of forcing 2/3 of the Mexican miners to return home. Also the wealth of Mexican Americans in New Mexico was depleted by usury laws at excessive rates when Mexicans tried to buy back the land and in Texas violent sabotage of their business interests prevailed. Segregation into inferior housing, education, employment and civic services has contributed to stereotyping by white society. The repatriation policies during the Great Depression forced about 1/3 of Mexican Americans to leave the US mostly because of violence, harassment and diminished opportunities. The Naturalization Act of 1870 arose out of resentment against the frugal, hard working, low waged Chinese Americans

How Businesses Use Learning & Memory to Affect Consumers Essay

How Businesses Use Learning & Memory to Affect Consumers - Essay Example If we look at the apple, you can directly say that it means high quality and luxury of computers, if we say it is Mercedes luxury cars and sports cars BMW means (Natale 2007 45-52). Those traders have learned to their brand, successfully using touch system and stimulus. They may even offer its popular brand for rent to other companies that are not brand ill bred with the negative image. Discussion Businesses are usually relatively little power to use punishment or negative reinforcement. However, parking meters often used to prevent consumers from taking valuable parking space and manufacturers may void your warranty if consumers take their product to unauthorized repair facility (Watkins 2006 294-303). Several factors influence effectiveness of operant learning. In general, more time effects of behavior, especially. In other words, power companies will be more likely to encourage consumers to use less electricity at peak times, when consumers actually have to pay when they used elec tricity (e.g. - slot), but not at end of month. Learning is also more likely to occur when the person can between behavior and consequences (but learning can occur even if link is not aware). Another problem is that building programs and extinction. Extinction occurs when behavior continues to have the impact on behavior and then eventually stops happening. For example, if the passenger finds that scream at check -in staff did not receive its upgrade to first class, it is likely to stop this behavior. Sometimes the person is rewarded each time you run behavior (e.g., consumer receives the non- alcoholic beverage whenever coins were introduced machine). Nevertheless, it is not necessary for learning time occurs. Even rewarded only from time to time, behavior can be studied. Several building programs are available: Fixed interval, consumer gets the free dessert every Tuesday, when he or she eats at the certain restaurant. Fixed ratio: behavior is rewarded (or punished) for each nth ti me it is performed. (For example, every tenth loyalty presented supplied free). Variable ratio: Each time the action is performed, there is some chance that be given. For example, each time user enters store, he or she receives the lottery ticket. With each ticket, there is the 20% chance to get the free burger. Consumers can get the free burger twice, or he or she can go ten times without getting the hamburger once. Variable Reinforcement Is Least Vulnerable To Extinction Sometimes training may be necessary to teach consumer desired behavior. In other words, it may be possible to directly teach consumers to adopt desired behavior. For example, user can first get the good free product (product itself, if it is good, it is the reward), then buy with the large cents off coupon, and finally buy at high prices. Thus, we are strengthening approaches desired behavior. Instead of introducing Coca -Cola directly in Indonesia, fruit soft drinks were introduced because they were more like dri nks are consumed (Anderson & Farkas 2003 88-93). Consumer does not always have to go through learning process itself; sometimes it can be learned by observing consequences of others. For example, stores can make the big deal out of bullpen continued shop is not so much because they want to stop this behavior among those who were, and to discourage behaviors in others? In addition, viewers can identify with characters in advertising that

Twe assingments each one is one page Essay Example | Topics and Well Written Essays - 500 words

Twe assingments each one is one page - Essay Example However, the war has already crossed the 60 days limit and the Congress still remains silent. The March 21 notification of Obama was that U.S would engage in a ‘limited and well-defined mission’ in Libya. However, by May 20, as Huffpost Politics reports, Obama wrote to the Congress that U.S is ‘no longer in lead’ but the participation involves non-kinetic support like intelligence and refueling, and kinetic attacks on Libyan air defenses and NATO-led forces. Obama violated the law by not seeking the formal approval of the Congress for military operations in Libya. Secondly, he violated the law by not certifying in writing that there is ‘unavoidable military necessity’ in Libya so that the military operation continues after 60 days. Now, what Obama has to do is to seek and receive the permission from Congress before the completion of 90 days. In addition, he will have to withdraw all the forces and resources from Libya in 90 days. II Article II, Section I of the American Constitution states that Congress would decide the date of appointment of electors. So, in 1845, Congress enacted a law providing that Tuesday after the first Monday of November of the year in which the electors are to be appointed is the Election Day. There are many reasons behind the selection of Tuesday as the Election Day.

The History of Sugar and Its Influence Assignment - 1

The History of Sugar and Its Influence - Assignment Example In seeking to integrate with such an understanding and leverage a further realization for how current society ingests larger and larger amounts of sugar, as well as the ways in which societal stakeholders can seek to lessen the impacts of sugar consumption, the August 2013 issue of National Geographic features a cover story that is entitled â€Å"Sugar (A Not so Sweet Love Story)†. The following analysis will seek to derail the discussion and summary of the analysis which the author performs. It is the hope of this student that such a summary will be useful in helping not only to understand the key points of the authors argument but also with regards to utilizing these understandings and prescriptions for a better life and an overall decrease in the level of obesity and health impacts that the consumption of too much sugar has been tied to. Firstly, the author traces the history of how sugar came to be introduced to the West and subsequently the remainder of the world. As with so many inventions and development in human history, the spread of Empire was ultimately the vehicles through which most of the world came to integrate with the consumption of sugar. The author indicates that era conquerors were the first to spread an awareness and appreciation for sugar and the lands that they conquered. In comparing to the spread of sugar throwing paint at a fan, the author discusses the way through which an appreciation of refined sugar and the means through which it can be added to see dishes and ingredients as a means of making things tastier, the author points to how the spread of sugar into the West was first evidenced around 500 B.C.E. him from this point, sugar production spread into much of the Western world and was incorporated into the diet of individuals; albeit to a much lesser degree that it is within the current ti me.

ACC 202 MOD 5 CA Essay Example | Topics and Well Written Essays - 500 words

ACC 202 MOD 5 CA - Essay Example As the company has the capacity to produce 20,000 units in a year, Paul Peco should focus on utilizing the firm’s maximum production capacity, as there is a high level of demand for the product. Pecos has the capacity to manufacture 20,000 units per year without any increase in the fixed costs. The most profitable solution for Paul Peco would be to sell 20,000 units in a year, so that the company’s maximum capacity is utilized. From this volume, the contribution required from a single unit to cover the fixed costs can be computed (Weston and Copeland). The profit margin originally set by Paul Peco was a minimum of $ 10 per unit. In the revised plan, a minimum profit of $ 12.50 per unit is fixed. Hence the revised minimum selling price is at $ 280 per unit. It is evident that Paul Peco would have sold 1,925 units in the last month. Assuming a constant demand every month, Paul Peco will easily be able to sell 20,000 units in the first year. The last month’s contribution margin income statements for the two rules are presented below. From the revised plan, it is evident that Ms. Goodperson’s decision to accept the contract at $290 per unit was profitable. Ms.Goodperson should be hired again. Also, based on the revised decision rule, Paul Peco should instruct his sales staff to accept orders at any price above $ 280 per

Chapter 14 Study Questions Essay Example | Topics and Well Written Essays - 750 words

Chapter 14 Study Questions - Essay Example Financial markets could make the economy worse if they cease to exist and similarly could up the economic basis of any nation with their presence. A financial market takes care of both the market economy as well as the non-market economy at a single time. (Lyons, 2001) A financial intermediary, technically speaking, is an institution that provides indirect fund means from the people who want to save or lend towards the people who wish to invest or borrow at the same time. The institution acts as the middleman between the firms that raise funds and the investors in essence. The financial intermediary is basically a financial institution in the most basic sense. As an example, this institution could either be a bank or a credit union. At times, the financial intermediary is seen in the instance of an insurance company as well. The financial intermediary could also be an individual who has the basic role of intermediation under a financial context between two or more parties. The funds are channeled in an easy way through the financial intermediary and the lenders and borrowers have a direct basis with the financial intermediary while indirectly they deal with each other. Money markets are those markets which act as substitutes for money. Capital markets are the markets for the long run basis of securities where they take care of both debt and equity. Money markets look at overnight to short term funds while capital markets mobilize long term savings so that financing of long term investments could be made. Money markets have maturity of one year or less than a year while capital markets have long term measures within them. Money markets collect different markets under them for a number of different instruments while capital markets look to include both the lending and borrowing regimes. Within money markets, the credit worthiness of the participants is deemed as significant while

Quality and Safety Two Sides of the Same Coin Coursework

Quality and Safety Two Sides of the Same Coin - Coursework Example According to National Transportation Safety Board, civil aviation accidents in the United States for 2011 had slightly increased than the previous year (i.e. 2010). Numbers of civil accidents have increased from 1500 in the year 2010 to 1550 in the year 2011. Simultaneously, the fatalities have increased from 473 in the year 2010 to 485 in the year 2011. In the year 2011, 28 mishaps were accounted for the Part 121 air carriers and 4 accidents were reported for the part 135 commuter1. This statistic show that the differences between the accidents rates between the two years do not vary much, even though efforts have been made that are needed to ensure higher quality and considerable safety to the consumers. It has undoubtedly become one among the top most priorities of the aviation companies. Aviation industry is an important industry in the United Kingdom and influences the country’s overall economy and lives of the people. The British Airways based in the UK is one of the bes t airline companies operating in the world. Its brand revolves round the customer satisfaction and its primary goal is to provide utmost satisfaction to its customers through ensuring quality and safety. The aviation industry in the United Kingdom is booming at a tremendous rate every year, thereby creating the need to maintain best quality along with the highest safety measures to its customers2. The continuously changing environment in the aviation sector has posed certain significant challenges to improve the quality in service and the safety measures provided to the customers. Risk in Aviation Industry Risk in aviation industry emanates from a number of factors. Pilots usually have to work in complex circumstances and operate together with different technologies. In such circumstances, the risk is very high and threats come from variety of sources3. Icing during the winters causes dense layer of clouds cover that may cause difficulties to the pilot while flying. Icing and freezi ng are common in European countries including the United Kingdom. The other major reason that acts as a hurdle in the aviation is the wind factor. During taking off or at the time of landing, strong wind possesses considerable risk to the aircrafts. During thunderstorm, lightening is caused which discharges electric that may destroy the aircraft and may cause fire in the fuel tank which may result into explosion in the aircraft4. In addition to this, it can disrupt the communication system and the navigation tool present in the aircraft. When there is a large storm, it may accompany with the hail stone that may damage the skin of the aircraft. It is not only the natural factors that cause damages and create threats while flying in aircrafts but the most imperative factor that is responsible for the aircraft mishap is the technical errors. Today, alike other sector, the aviation sector is also highly dependent on technology. Despite technological advancements, there have been a numbe r of cases of technological failures such as engine failure and communication error during the course of a flight. Engine failure is probably the most dangerous situation that may lead to fatal accidents. Engine failure may be caused by the contamination of fuel or the pump failure. Another reason for the engine failure is the spark plugs that may not function properly during the flying stage5. Technological factors can be avoided, if proper inspection is conducted before the

Philosophy Meaning Essay Example for Free

Philosophy Meaning Essay PHILOSOPHY greek meaning â€Å"love of wisdom†, encompassed the love of all wisdom, but only in recent centuries came to refer to a special branch of enquiry, separate from other sciences, such as â€Å"natural philosophy†. * is universally defined as â€Å"the study of the wisdom or knowledge about the general problems, facts, and situations connected with human existence, values, reasons, and general reality. † It seeks reasons, answers, and general explanations to life and its factors. Thus, if we talk about philosophy, we talk about a school of thoughts. â€Å"philosophers† which makes a profession of studying things in their separation from human life and practice. The main branches of Philosophy are Logic, Epistemology, Metaphysics and Ethics. Western philosophy is referred to as the school of thought from Greek philosophy that influenced the greater part of Western civilization. * takes its roots from Rome and Christianity, specifically Judeo-Christianity. * Latin * Rational, Scientific, Logical schools. Western civilization is more individualistic, trying to find the meaning of life here and now with self at the center as it is already given and part of the divine. Eastern philosophy is based mainly in Asia, more specifically the Chinese philosophy. * Confucianism, Mahayana Buddhism, and Taoism. Chinese. Hinduism, Integral Yoga, Islam, Zen * Relationship with religion; Integration Search for absolute truth: * Systemic approach – all events in the universe are interconnected * Searching inside yourself – by becoming a part of the universe through meditation and right living. Eastern philosophy is drawn much more into groups or society or people’s actions and thoughts as one in order to find meaning in life as they try to get rid of the false â€Å"me† concept and find meaning in discovering the true â€Å"me† in relation to everything around them, or as part of a bigger scheme. Summary: * Western philosophy is mainly used in the Western parts of the world, such as in the European countries, while the Eastern philosophy is prevalent in Asian countries. * Both philosophies center on virtues. * West’s Individualism ( and the East’s Collectivism (A human being is an integral part of the universe and the society. People are fundamentally connected. Duty towards all others is a very important matter. Collectivism is stronger. ) * Eastern philosophy takes more of a spiritual approach while Western philosophy is more hands-on. The Ionian Philosophers * comes from Aristotle; first source to attempt systematic exposition of their doctrines. Thales * Prediction of the eclipse, and other astronomical activities. * Prediction of solstices * Mathematical discoveries (geometry ) * Cosmology * Natural phenomena including the heavens could be discussed as processes governed by natural laws. * Believed that the Earth was a large (? at) disk ? oating on an in? nite ocean of water, and that earthquakes resulted from disturbances in this ocean that shook and cracked the Earth. * concept of â€Å"unity underlying diversity† some fundamental principles tying together all the multitude of things we see on Earth * water was the fundamental element from which all things were derived. Anaximander * Zoogony and anthropogony * thought the Universe formed out of an in? nite chaos he called the â€Å"boundless† due to a â€Å"separating out† of opposites (such as hot and cold, wet and dry). * ? rst recorded attempt to model the Universe. (the Earth was a cylinder and that the Sun, Moon and stars were all located on concentric cylinders, or hoops, rotating about the Earth. ) Anaximenes * one ruling material principle is air; imperceptible. * Air was the fundamental material of all things. * ? rst attempt to explain the diversity of the world with qualitative differences in terms of quantitative differences. Babylonians and Egyptians were excellent at mathematics. Greeks began to move away from their mythical view of the world and started to seek explanations of natural phenomena; later called science. * All questioned the origin of the Universe, what was here in the beginning, and what things are made from. They all believed that material substance (rather than some spiritual or supernatural substance; thus the name materialists) made up the Universe. In other words, matter is the only substance, and reality is identical with the actually occurring states of energy and matter. * physicalism. to distance oneself from what seems a historically important but no longer scientifically relevant thesis of materialism.

Development and Importance of Solar Electricity

Development and Importance of Solar Electricity Noxious gasses, acrid fumes, scarred landscapes, a massive carbon footprint, and a warming atmosphere. These are the consequences of obtaining energy from nonrenewable resources such as coal, natural gas, and petroleum. These are the sources we use to produce electricity, endangering the very planet we live on through their harmful impacts on the environment. These destructive effects include, but are not limited to, the creation of a blanket of carbon dioxide which traps heat in the atmosphere and thus warms it, water and ground contamination from spills and other mishaps, and air pollution. There is a better answer to obtaining electricity, one which reduces greenhouse gas emissions and has a much, much smaller impact on the environment: the photovoltaic (PV) cell, also known as the solar cell. Because the solar cell has these incredible benefits, our nation should invest much more money into research and development of solar power to generate electricity. Thanks to considerable public investment in green energy that came from the US, Germany, and China during the Great Recession, recent American and European regulations that have de-incentivized coal power plants [,] competition among manufacturers, and technological know-how (R. Meyer How Solar and Wind Got So Cheap, So Fast 1), solar energy has become much cheaper, and thus, economically viable. While costs do vary between regions and types of solar panels, the average cost is around 60 cents per watt (R. Meyer How Solar and Wind Got So Cheap, So Fast 1). Solar cell technology has been around since 1839 when French physicist Alexandre Edmond Becquerellar first demonstrated the photovoltaic effect, or the ability of a solar cell to convert sunlight into electricity (R. Meyer History of Solar Power 1). Forty-four years later, in 1883, the American inventor Charles Fritts created the worlds first rooftop solar array in New York (R. Meyer History of Solar Power 1). Up to this point, however, the process behind the photovoltaic effect (also known as the photoelectric effect) was not understood. The process continued to elude scientists until 1905 when Albert Einstein wrote a paper explaining the photoelectric effect (R. Meyer History of Solar Power 1). Together, Becquerellar and Einstein paved the way for the development of photovoltaic technology. During the 1950s, the U.S. military funded research on PV technologys potential to power satellites (R. Meyer History of Solar Power 1), and in 1964 the National Aeronautics and Space Administra tion (NASA) launched its first satellite equipped with solar panels. However, it wasnt until the Arab oil embargo of 1973 and the ensuing energy crisis that the United States started to earnestly develop solar energy. The U.S. governments first step was passing the Solar Heating and Cooling Demonstration Act of 1974 (R. Meyer History of Solar Power 1), which created the Solar Energy Coordination and Management Project, an organization designed to direct agencies like NASA, the National Science Foundation, and the Department of Housing and Urban Development to improve solar energy technology (R. Meyer History of Solar Power 1). When Jimmy Carter became President in 1977, he labeled the energy crisis as the moral equivalent of war and made energy policy a top priority of his administration (R. Meyer History of Solar Power 2). That same year, he created the Department of Energy and pushed through Congress several acts relating to renewable energy use. The goal of Carters efforts and th ose of Congress was to make solar viable and affordable and market it to the public (R. Meyer History of Solar Power 2). In facilitating this goal, Congress created the commercial investment tax credit (ITC) and the residential energy credit (or residential ITC) to provide financial incentives for the public to purchase solar properties (R. Meyer History of Solar Power 2). Unfortunately, the tax credit failed to increase Americas use of solar power, as solar comprised a negligible amount of electricity generation (R. Meyer History of Solar Power 2). However, declining domestic oil production and rising oil imports throughout the early 2000s (R. Meyer History of Solar Power 2) led to the Energy Policy Act of 2005 (EPAct). This act raised the commercial ITC to a temporary 30 percent rate and reinstated the residential ITC [which had expired in 1985] (R. Meyer History of Solar Power 2). Today, in addition to tax credits and grants, the government continues to heavily subsidize the indu stry with research and development, commercialization, and regulatory support (R. Meyer History of Solar Power 3). In 1985, total renewable energy production and consumption amounted to 6084 trillion Btu. Out of that amount, less than half trillion Btu came from solar power, less than 0.0008 percent of total renewable energy. In comparison in 2015, total renewable energy production and consumption amounted to 9466 trillion Btu. Out of that amount, 427 trillion Btu came from solar power, about 4.5 percent of total renewable energy. This means from 1985 to 2015 total renewable energy production and consumption increased by 3382 trillion Btu, while in the same time period, solar energy consumption and production has increased by around. 426.5 trillion Btu (US EIA Monthly Energy Review January 2017 151). Electricity is an extremely important factor of our everyday lives, but we should obtain this essential resource much more responsibly through solar power. Solar power produces significantly less greenhouse gas emissions (more specifically carbon dioxide) and has a very high technical potential. According to the United States Environmental Protection Agency (EPA), greenhouse gases are gases that trap heat in the atmosphere (EPA 1). In 2014, 81% of all greenhouse gas emissions in the United States came from carbon dioxide, which amounted to 556,470,000 metric tons (EPA 1). This carbon dioxide enters the atmosphere through burning fossil fuels (such as coal, natural gas, and oil), as well as solid waste, trees and wood products, and also as a result of certain chemical reactions (EPA 1). According to the EPA, 37% of carbon dioxide produced comes from generation of electricity (EPA 1). If our nation used solar power to generate electricity, the amount of carbon dioxide we produce would drastically decrease, as the carbon footprint of the solar industry is much, much smaller than that of the oil or gas business (R. Meyers The Solar Industry Has Paid Off Its Carbon Debts 2). This is made possible because the energy put into making solar panels, such as quart and copper be[ing] mi ned. The raw materials be[ing] converted into wafers, then [being] encased in protective material Has the solar industry really saved any energy at all? (R. Meyers The Solar Industry Has Paid Off Its Carbon Debts). Researchers at the University of Utrecht and the University of Groningen have determined that the answer is yes, using a type of research called lifecycle analysis, which investigates the total environmental impact of a product over time (R. Meyer The Solar Industry Has Paid Off Its Carbon Debts 2). According to Meyers, this kind of research is tricky: researchers must find and calibrate years of economic and energy data, collected across 40 years, in many different countries, with different goals in mind (R. Meyers The Solar Energy Has Paid Off Its Carbon Debts 2). Scott Hershey, a professor of chemical and environmental engineering at Olin College, stated in an email that their [the researchers] methods are solid, but this type of analysis is fraught with assumptions (R . Meyer The Solar Energy Has Paid Off Its Carbon Debts 2). While exact numbers are not known relating to how much carbon dioxide solar power produces, it is known that it is much less than amounts from nonrenewable sources. However, this carbon dioxide can be removed from the atmosphere by being absorbed by plants as part of the biological carbon cycle. Unfortunately, all plants have a limit to how much carbon dioxide they can absorb, and all the plants in the world cannot absorb all the carbon dioxide just the U.S. produces (EPA 1). Solar power produces much less carbon dioxide than power plants burning fossil fuels, and there is very high technical potential. Technical potential refers to the achievable energy generation of a particular technology given system performance, topographical limitations, environmental, and land-use constraints (Lopez, Roberts, Heimiller, Blair, Porro 1). In other words, it is the amount of energy a technology can produce within strict parameters. The process for generating these technical potential estimates is very exact, requiring complex calculations and surveying of the land. However, there are three different types of solar technologies, and the technical potential for each drastically varies. The three different types of solar technologies are utility-scale PV, rooftop PV, and concentrating solar power (CSP). According to NREL, utility-scale PV is generation of electricity through large-scale PV (NREL 3). However, NREL has estimated that 3,212,324 km2 of land is available for utility-scale solar production in the U.S. (Anthony Lopez, Billy Roberts, Donna Heimiller, Nate Blair, and Gian Porro 10,11), out of 9,833,517 km2, which is the total land area of the United States (The World Factbook 1). This means 32.66% of U.S. land is suitable for production of electricity, which could produce up to 282,844,911 gigawatt hours (GWh) of electricity (Anthony Lopez, Billy Roberts, Donna Heimiller, Nate Blair, and Gian Porro 10, 11). In 2015, the United States produced 4.103 trillion (4,103,000,000) kilowatt hours (KWh) of electricity, which is equal to 4,103,000 gigawatt hours (GWh) of electricity (Philipp Beiter, and Tian Tian 7)[i]. In other words, using just utility-scale solar power plants, we could produce almost 68 percent of all the energy we consume using just solar power! However, many fossil fuel executives and politicians are opposed to solar power, among other reasons, because they say that it is costly and the construction of the solar panels still cause emissions. These critics are correct: solar power is still costly and the manufacture of solar power does create emissions. However, historically, prices today are much lower than those at the turn of the century. In an email from Jenny Chase, the head of the solar department at Bloomberg New Energy Financial, she stated that reductions in the cost of solar panels have to do with the experience curve. This means that the more of something we do, the better we get at it (Robinson Meyer How Solar and Wind Got So Cheap So Fast 2). Cost cutbacks also have to do with manufacturers improving their fabrication of materials in photovoltaic cells, including an essential material called polysilicon. Prices for polysilicon got as high as $400 per kilogram. That enticed more manufacturers to get into the indu stry, creating a supply glut and a price crash (Robinson Meyer How Solar and Wind Got So Cheap So Fast 2). As a result, current prices are much lower than prices from years ago. While solar panels themselves create very few greenhouse gas emissions, their production can, depending on where they are produced. According to Robinson Meyer, many solar panels are manufactured in Europe and China (Robinson Meyer The Solar Industry Has Paid Off Its Carbon Debts 2). However, the environmental situations in these two regions are drastically different, because China relies on coal burning for much of its electricity, and it has fairly lax environmental protections. The EU [European Union], on the other hand, already heavily relies on clean energy, and it has a large and entrenched environmental bureaucracy (Robinson Meyer The Solar Industry Has Paid Off Its Carbon Debts 2). This means that solar panels produced in China are more than likely produced in factories require a lot of energy and produce relatively dirty emissions (Robinson Meyer The Solar Industry Has Paid Off Its Carbon Debts 3. Meanwhile, in Europe, factories producing solar panels require relatively litt le energy and produce cleaner emissions (Robinson Meyer The Solar Industry Has Paid Off Its Carbon Debts 3). However, China has toughened its environmental protection laws, as they attempt to curb pollution. This means that in the future, China may produce solar panels with fewer emissions. If you dont believe solar power is the better choice for producing our electricity, there are other options to choose from that still protects our environment, including wind, geothermal, tidal, hydroelectric, and biomass. However, if none of those options suit you either, then think about the consequences of using nonrenewable sources. Pollution. Changes in global weather patterns. Flooding. Drought. Desertification. Health consequences. These consequences spell out the destruction of the planet we live on. It may take years, but with continuous reliance on fossil fuels, these effects are inevitable. We still have a chance to turn around, by using solar power, or other forms of renewable resources. Yes, this would require sacrifices and change. It would require courage to go against the status quo. It would require risk. But if we chose to use solar power to generate electricity, we could make the world a little bit better. For ourselves, our world, and our posterity. Works Cited Beiter, Philipp, and Tian Tian. 2015 Renewable Energy Data Book. 2015 Renewable Energy Data Book | Department of Energy. U.S. Department of Energy (U.S. DOE), Nov. 2016. Web. 04 Mar. 2017. . Bolinger, Mark, and Joachim Seel. Utility-Scale Solar 2015: An Empirical Analysis of Project Cost, Performance, and Pricing Trends in the United States. Electricity Markets and Policy Group. Lawrence Berkeley National Laboratory, Aug. 2016. Web. 04 Mar. 2017. . History of Solar Power. IER. U.S. Department of Energy (U.S. DOE), 18 Feb. 2016. Web. 04 Mar. 2017. . Lopez, Anthony, Billy Roberts, Donna Heimiller, Nate Blair, and Gian Porro. U.S. Renewable Energy Technical Potentials: A GIS-Based Analysis. National Renewable Energy Laboratory Documents Archive. U.S. Department of Energy (U.S. DOE), July 2012. Web. 04 Mar. 2017. . Meyer, Robinson. How Solar and Wind Got So Cheap, So Fast. The Atlantic. Atlantic Media Company, 02 Dec. 2015. Web. 04 Mar. 2017. . Meyer, Robinson. The Solar Industry Has Paid Off Its Carbon Debts Robinson Meyer. QOSHE. The Atlantic, 13 Dec. 2016. Web. 04 Mar. 2017. . Meyer, Robinson. The Solar Industry Has Paid Off Its Carbon Debts. The Atlantic. Atlantic Media Company, 13 Dec. 2016. Web. 04 Mar. 2017. . Overview of Greenhouse Gases. EPA. Environmental Protection Agency, 14 Feb. 2017. Web. 04 Mar. 2017. . Thetford, Kyle. Charting the Fall of Solar Prices. The Atlantic. Atlantic Media Company, 19 Aug. 2013. Web. 04 Mar. 2017. . The World Factbook: UNITED STATES. Central Intelligence Agency. Central Intelligence Agency, 12 Jan. 2017. Web. 04 Mar. 2017. . [i] The actual report gave the amount of energy in quadrillion Btu, but all my other sources gave it in terms of watts, so in this case, I converted Btu to watts.

Legal Issues Case Study For Nursing Essay -- essays research papers

Legal Issues Case Study for Nursing Case 2 Nursing Situation: Cindy Black (fictitious name), a four-year-old child with wheezing, was brought into the emergency room by her mother for treatment at XYZ (fictitious name) hospital at 9:12 p.m. on Friday, May 13. Initial triage assessment revealed that Cindy was suffering from a sore throat, wheezing bilaterally throughout all lung fields, seal-like cough, shortness of breath (SOB), bilateral ear pain. Vital signs on admission were pulse rate 160, respiratory rate 28, and a temperature of 101.6 Â °Fahrenheit (F) (rectal). Cindy Black was admitted to the emergency department for treatment. Notes written by the emergency department physician on initial examination read, "Croupy female; course breath sounds with wheezing; mild bilateral tympanic membrane hyperemia. Chest X-ray reveals bilateral infiltrates." Medication prescribed included Tylenol (acetaminophen) 325 mg orally for elevated temperature, Bronkephrine (ethylnorepinephrine hydrochloride) 0.1 millimeter subcutaneous, and monitor results. Nurse Slighta Hand, RN (fictitious name) administered the medication as ordered and the child was observed for thirty minutes. Miss Hand's charting was brief, almost illegible, and read, "Medicines given as prescribed. Cindy observed without positive results. Physician notified." The physician examined the child; notes read that the child had "minimal clearing" in response to the bronchodilator. The following medications were then prescribed: Elixir of turpenhydrate with codeine one milliliter by mouth, Gantrinsin (sulfisoxazole) 10 Case 3 milliliters, and Quibron (theophylline-glycerol guaiacolate) 10 milliliters. Nurse Slighta Hand, RN charted the medications were given as prescribed. Her note at 11:08 p.m. read, "Vomiting; unable to retain medicine. Respiration increased (54), temperature 101.4Â °F (rectal); wheezing with increased difficulty breathing." No further notes were made regarding Cindy's condition on the emergency department record by the nurse, except to state that at 12:04 am, "child released from emergency department." Thirty minutes after discharge from the emergency department, Cindy Black was brought back to the hospital. This time her vital signs were absent, her skin was warm without mottling, and the pupils of the eye were dilated but reacted slowl... ...30 minutes) Â · Pulse rate, rhythm, quality (every 15 minutes) Â · Respiratory rate, rhythm, character (every 15 minutes) Â · Patency of the airway (at least every 15 minutes, more if in distress) Â · Blood pressure (every 30 to 60 minutes) Â · Skin color and temperature (every 15 minutes) Â · Level of consciousness (every 15 minutes) Â · Emesis amount, character, and frequency Summary: Communication throughout the nursing process is crucial for the provision of safe patient care consistent with the prevailing professional standard. Spoken communication among all members of the health-care team, and especially between nurse and physician for clarifying orders, planning patient care, and reporting significant patient observations is vital to the nursing process. Equally important is written communication by the nurse in the form of prompt and accurate entries in the medical record. References Bernzweig, E. (1996). The nurse's liability for malpractice. (6th ed.). St. Louis: Mosby Creasia, J. and Parker, B. (1991). Conceptual foundations of professional nursing practice. St. Louis: Mosby Earnest, V. (1993). Clinical skills in nursing practice. (2nd ed.). Philadelphia: J. B. Lippincott

Victorian Architecture :: Architecture

Victorian Architecture During the Victorian period, there was a revival of classical (Greek and Roman), Gothic, Renaissance and Baroque architecture. Romantic architects replicated Greek and Roman buildings, which were revered as the ultimate examples of beauty (Sporre 487; Tansey 932). Increased nationalism in England also sparked a revival of Gothic architecture. After the Houses of Parliament burnt down in London (1834), the task of redesign the new building was assigned to Charles A. Barry and Augustus W. N. Pugin. Their Gothic design of the new Houses of Parliament make it a prime example of Victorian architecture today (Tansey 955). It is important to recognize that Romantic architecture was not only a return to the past. Modern technologies and materials, as well as non-European influences, also played a role. (Sporre 495-98; Tansey 956). One example is the Crystal Palace designed by Sir Joseph Paxton for the Great Exhibition in London (1851). Made of iron and glass, it was designed to be rapidly put together and taken apart. Another noted architectural example of this period was John Nash’s Royal Pavilion in Brighton (1815-18). The design of this palace was greatly influenced by Islamic and Eastern architecture (Flynn; Sporre 495-98; Tansey 956, 1014). Victorian architecture was both a rediscovery of the past and the precursor of Modern architecture. Some buildings embodied both of these characteristics. The Houses of Parliament and the Crystal Palace’s outside architecture had little to do with their functions and internal design. Their architects were revolutionizing the world of architecture and ushering in the

Media and Social Responsibility Essay

Do information media have social responsibility? If yes, in what ways? If no, why not? I, along with many other people will agree we are not sure what responsibilities are that information media has, but they do have some sort of responsibility. Media outlets need to remain unbiased, but we all know that none of them do. The certain news sites that I read, do usually seem one sided, but I keep an open mind when it comes to believing what the say. If I have any doubt what I am reading is far fetched, I turn to other sources to make sure I get the whole story. Over the years when I have done research on a particular topic, I have always used multiple sources. As far as the information media having any responsibilities, the have many. The main responsibility that they have is to report the most truthful news that they possibly can. They need to stay away from what their views are and report the truth, not just what they speculate. That is the biggest problem they have. If we can’t believe what they say, how are we going to find out what really happened. Their responsibilities need to stay focused on what is needed to be done, and that is to get a factual story out to the general public. Too many media outlets rush to get the â€Å"BIG† story out before the others and they leave out many key factors and the story usually doesn’t make much sense. The ones that are usually guilty of that are the local sites. They want to break the news when the have no information what so ever.

Plato Report

Does Plato Believe There can ever be a Just Society In answering this question I first need to describe what a just society would consist of. A perfect state can only be lead under perfect conditions. Civil Society would be a better name for this state. A just state would be made up of three parts. First, a state is a structure with parts that work together like an organism. If the parts do not work well together then the whole thing breaks down. It must have virtues, voices, it can be wise and brave. The state must have everyone performing there jobs to their best ability. For a state to be just the people within the state must also be just. A man is just when he has a well ordered soul because then you will do the right thing by performing good and just actions. A soul must be allowed to perform its proper function. In a state you cannot define justice by a man because a man can decay into ugliness. Instead you must define justice based on forms. Plato says that the forms are eternal and ever lasting. What constitutes an unjust society is a lack of knowledge. So ignored to create a just society we must educate people. The society must be well rounded in their education for if they are not they will have problems in society. A society must be fit, participation in athletics, they need to be sensitive to prose poetry, and have knowledge of mathematics and science. Education can not be on specialties, but everything mind, spirit, and body. Having a well rounded education will help people to communicate in all areas. The more you know in many different areas the better over all communication a society has. One of the reason there are inequalities in a society is due to lack of knowledge. Everyone in the society must to some extent be a philosopher because they seek education and knowledge. A just society must also have a just ruler. A just ruler would need to be a philosopher, he would have to offer honest leadership which reflects the will and knowledge of society. A perfect society must have temperance, knowledge, and wisdom. In justices occur because of a lack of knowledge resulting in greed. In order to get rid of injustice everyone in the society must be educated starting at birth. Women and men need to be equally educated in a well rounded fashion in order to promote a just society. In asking if this society could ever work the answer is no. The only way it could work is if all of society is willing to accept knowledge and work hard for education. Even though there is no such thing as a truly unjust society a totally just society will never happen until people are willing to work for it. Another reason there can never be a perfectly just society is because everyone†s perception of just is different. We know that the idea of justice is there, but to explain it to where everyone agrees to the idea would be hard to achieve. However, in trying to find true justice the society becomes stronger and more just. Expressing individuality that benefits or hurts a society however, reflects assertiveness, incentive, thought, and creativity, which strengthens the society. If a society ever got to the point of being just, the society would no longer have greed, drive for a better life, it would not have poverty or wealth. The society would just stop. There would be no more invention, growth, or change. The only change from Plato†s time to ours is technology. We are still searching for the perfect government, the question of who is better than who is still asked, and education is still a major principle to whether or not you are successful.

Stochastic Calculus Solution Manual

Stochastic Calculus for Finance, Volume I and II by Yan Zeng Last updated: August 20, 2007 This is a solution manual for the two-volume textbook Stochastic calculus for ? nance, by Steven Shreve. If you have any comments or ? nd any typos/errors, please email me at [email  protected] edu. The current version omits the following problems. Volume I: 1. 5, 3. 3, 3. 4, 5. 7; Volume II: 3. 9, 7. 1, 7. 2, 7. 5–7. 9, 10. 8, 10. 9, 10. 10. Acknowledgment I thank Hua Li (a graduate student at Brown University) for reading through this solution manual and communicating to me several mistakes/typos. 1. 1. Stochastic Calculus for Finance I: The Binomial Asset Pricing Model 1. The Binomial No-Arbitrage Pricing Model Proof. If we get the up sate, then X1 = X1 (H) = ? 0 uS0 + (1 + r)(X0 ? ?0 S0 ); if we get the down state, then X1 = X1 (T ) = ? 0 dS0 + (1 + r)(X0 ? ?0 S0 ). If X1 has a positive probability of being strictly positive, then we must either have X1 (H) > 0 or X1 (T ) > 0. (i) If X1 (H) > 0, then ? 0 uS0 + (1 + r)(X0 ? ?0 S0 ) > 0. Plug in X0 = 0, we get u? 0 > (1 + r)? 0 . By condition d < 1 + r < u, we conclude ? 0 > 0.In this case, X1 (T ) = ? 0 dS0 + (1 + r)(X0 ? ?0 S0 ) = ? 0 S0 [d ? (1 + r)] < 0. (ii) If X1 (T ) > 0, then we can similarly deduce ? 0 < 0 and hence X1 (H) < 0. So we cannot have X1 strictly positive with positive probability unless X1 is strictly negative with positive probability as well, regardless the choice of the number ? 0 . Remark: Here the condition X0 = 0 is not essential, as far as a property de? nition of arbitrage for arbitrary X0 can be given. Indeed, for the one-period binomial model, we can de? ne arbitrage as a trading strategy such that P (X1 ?X0 (1 + r)) = 1 and P (X1 > X0 (1 + r)) > 0. First, this is a generalization of the case X0 = 0; second, it is â€Å"proper† because it is comparing the result of an arbitrary investment involving money and stock markets with that of a safe investment involving only money market. This can also be seen by regarding X0 as borrowed from money market account. Then at time 1, we have to pay back X0 (1 + r) to the money market account. In summary, arbitrage is a trading strategy that beats â€Å"safe† investment. Accordingly, we revise the proof of Exercise 1. 1. as follows.If X1 has a positive probability of being strictly larger than X0 (1 + r), the either X1 (H) > X0 (1 + r) or X1 (T ) > X0 (1 + r). The ? rst case yields ? 0 S0 (u ? 1 ? r) > 0, i. e. ?0 > 0. So X1 (T ) = (1 + r)X0 + ? 0 S0 (d ? 1 ? r) < (1 + r)X0 . The second case can be similarly analyzed. Hence we cannot have X1 strictly greater than X0 (1 + r) with positive probability unless X1 is strictly smaller than X0 (1 + r) with positive probability as well. Finally, we comment that the above formulation of arbitrage is equivalent to the one in the textbook. For details, see Shreve [7], Exercise 5. . 1. 2. 1 5 Proof. X1 (u) = ? 0 ? 8 + ? 0 ? 3 ? 5 (4? 0 + 1. 20? 0 ) = 3? 0 + 1. 5? 0 , a nd X1 (d) = ? 0 ? 2 ? 4 (4? 0 + 1. 20? 0 ) = 4 ? 3? 0 ? 1. 5? 0 . That is, X1 (u) = ? X1 (d). So if there is a positive probability that X1 is positive, then there is a positive probability that X1 is negative. Remark: Note the above relation X1 (u) = ? X1 (d) is not a coincidence. In general, let V1 denote the ? ? payo? of the derivative security at time 1. Suppose X0 and ? 0 are chosen in such a way that V1 can be ? 0 ? ?0 S0 ) + ? 0 S1 = V1 . Using the notation of the problem, suppose an agent begins ? replicated: (1 + r)(X with 0 wealth and at time zero buys ? 0 shares of stock and ? 0 options. He then puts his cash position ? 0 S0 ? ?0 X0 in a money market account. At time one, the value of the agent’s portfolio of stock, option and money market assets is ? X1 = ? 0 S1 + ? 0 V1 ? (1 + r)(? 0 S0 + ? 0 X0 ). Plug in the expression of V1 and sort out terms, we have ? X1 = S0 (? 0 + ? 0 ? 0 )( S1 ? (1 + r)). S0 ? Since d < (1 + r) < u, X1 (u) and X1 (d) have opposite signs. So if the price of the option at time zero is X0 , then there will no arbitrage. 1. 3. S0 1 Proof. V0 = 1+r 1+r? d S1 (H) + u? ? r S1 (T ) = 1+r 1+r? d u + u? 1? r d = S0 . This is not surprising, since u? d u? d u? d u? d this is exactly the cost of replicating S1 . Remark: This illustrates an important point. The â€Å"fair price† of a stock cannot be determined by the risk-neutral pricing, as seen below. Suppose S1 (H) and S1 (T ) are given, we could have two current prices, S0 and S0 . Correspondingly, we can get u, d and u , d . Because they are determined by S0 and S0 , respectively, it’s not surprising that risk-neutral pricing formula always holds, in both cases. That is, 1+r? d u? d S1 (H) S0 = + u? 1? r u? d S1 (T ) 1+r S0 = 1+r? d u ? d S1 (H) + u ? 1? r u ? d S1 (T ) 1+r . Essentially, this is because risk-neutral pricing relies on fair price=replication cost. Stock as a replicating component cannot determine its own â€Å"fair† price via the risk-n eutral pricing formula. 1. 4. Proof. Xn+1 (T ) = = ? n dSn + (1 + r)(Xn ? ?n Sn ) ?n Sn (d ? 1 ? r) + (1 + r)Vn pVn+1 (H) + q Vn+1 (T ) ? ? Vn+1 (H) ? Vn+1 (T ) (d ? 1 ? r) + (1 + r) = u? d 1+r = p(Vn+1 (T ) ? Vn+1 (H)) + pVn+1 (H) + q Vn+1 (T ) ? ? ? = pVn+1 (T ) + q Vn+1 (T ) ? ? = Vn+1 (T ). 1. 6. 2 Proof. The bank’s trader should set up a replicating portfolio whose payo? s the opposite of the option’s payo?. More precisely, we solve the equation (1 + r)(X0 ? ?0 S0 ) + ? 0 S1 = ? (S1 ? K)+ . 1 Then X0 = ? 1. 20 and ? 0 = ? 2 . This means the trader should sell short 0. 5 share of stock, put the income 2 into a money market account, and then transfer 1. 20 into a separate money market account. At time one, the portfolio consisting of a short position in stock and 0. 8(1 + r) in money market account will cancel out with the option’s payo?. Therefore we end up with 1. 20(1 + r) in the separate money market account. Remark: This problem illustrates why we are in terested in hedging a long position.In case the stock price goes down at time one, the option will expire without any payo?. The initial money 1. 20 we paid at time zero will be wasted. By hedging, we convert the option back into liquid assets (cash and stock) which guarantees a sure payo? at time one. Also, cf. page 7, paragraph 2. As to why we hedge a short position (as a writer), see Wilmott [8], page 11-13. 1. 7. Proof. The idea is the same as Problem 1. 6. The bank’s trader only needs to set up the reverse of the replicating trading strategy described in Example 1. 2. 4. More precisely, he should short sell 0. 1733 share of stock, invest the income 0. 933 into money market account, and transfer 1. 376 into a separate money market account. The portfolio consisting a short position in stock and 0. 6933-1. 376 in money market account will replicate the opposite of the option’s payo?. After they cancel out, we end up with 1. 376(1 + r)3 in the separate money market ac count. 1. 8. (i) 2 s s Proof. vn (s, y) = 5 (vn+1 (2s, y + 2s) + vn+1 ( 2 , y + 2 )). (ii) Proof. 1. 696. (iii) Proof. ?n (s, y) = vn+1 (us, y + us) ? vn+1 (ds, y + ds) . (u ? d)s 1. 9. (i) Proof. Similar to Theorem 1. 2. 2, but replace r, u and d everywhere with rn , un and dn .More precisely, set pn = 1+rn ? dn and qn = 1 ? pn . Then un ? dn Vn = pn Vn+1 (H) + qn Vn+1 (T ) . 1 + rn (ii) Proof. ?n = (iii) 3 Vn+1 (H)? Vn+1 (T ) Sn+1 (H)? Sn+1 (T ) = Vn+1 (H)? Vn+1 (T ) . (un ? dn )Sn 10 10 Proof. un = Sn+1 (H) = Sn +10 = 1+ Sn and dn = Sn+1 (T ) = Sn ? 10 = 1? Sn . So the risk-neutral probabilities Sn Sn Sn Sn at time n are pn = u1? dnn = 1 and qn = 1 . Risk-neutral pricing implies the price of this call at time zero is ? ? 2 2 n ? d 9. 375. 2. Probability Theory on Coin Toss Space 2. 1. (i) Proof. P (Ac ) + P (A) = (ii) Proof. By induction, it su? ces to work on the case N = 2.When A1 and A2 are disjoint, P (A1 ? A2 ) = A1 ? A2 P (? ) = A1 P (? ) + A2 P (? ) = P (A1 ) + P (A2 ). When A1 and A2 are arbitrary, using the result when they are disjoint, we have P (A1 ? A2 ) = P ((A1 ? A2 ) ? A2 ) = P (A1 ? A2 ) + P (A2 ) ? P (A1 ) + P (A2 ). 2. 2. (i) 1 3 1 Proof. P (S3 = 32) = p3 = 8 , P (S3 = 8) = 3p2 q = 3 , P (S3 = 2) = 3pq 2 = 8 , and P (S3 = 0. 5) = q 3 = 8 . 8 Ac P (? ) + A P (? ) = P (? ) = 1. (ii) Proof. E[S1 ] = 8P (S1 = 8) + 2P (S1 = 2) = 8p + 2q = 5, E[S2 ] = 16p2 + 4  · 2pq + 1  · q 2 = 6. 25, and 3 1 E[S3 ] = 32  · 1 + 8  · 8 + 2  · 3 + 0.  · 8 = 7. 8125. So the average rates of growth of the stock price under P 8 8 5 are, respectively: r0 = 4 ? 1 = 0. 25, r1 = 6. 25 ? 1 = 0. 25 and r2 = 7. 8125 ? 1 = 0. 25. 5 6. 25 (iii) 8 1 Proof. P (S3 = 32) = ( 2 )3 = 27 , P (S3 = 8) = 3  · ( 2 )2  · 1 = 4 , P (S3 = 2) = 2  · 1 = 2 , and P (S3 = 0. 5) = 27 . 3 3 3 9 9 9 Accordingly, E[S1 ] = 6, E[S2 ] = 9 and E[S3 ] = 13. 5. So the average rates of growth of the stock price 9 6 under P are, respectively: r0 = 4 ? 1 = 0. 5, r1 = 6 ? 1 = 0. 5, and r2 = 13. 5 ? 1 = 0. 5. 9 2. 3. Proof. Apply conditional Jensen’s inequality. 2. 4. (i) Proof.En [Mn+1 ] = Mn + En [Xn+1 ] = Mn + E[Xn+1 ] = Mn . (ii) 2 n+1 Proof. En [ SSn ] = En [e? Xn+1 e? +e ] = 2 ? Xn+1 ] e? +e E[e = 1. 2. 5. (i) 2 2 Proof. 2In = 2 j=0 Mj (Mj+1 ? Mj ) = 2 j=0 Mj Mj+1 ? j=1 Mj ? j=1 Mj = 2 j=0 Mj Mj+1 + n? 1 n? 1 n? 1 n? 1 2 2 2 2 2 2 2 2 Mn ? j=0 Mj+1 ? j=0 Mj = Mn ? j=0 (Mj+1 ? Mj ) = Mn ? j=0 Xj+1 = Mn ? n. n? 1 n? 1 n? 1 n? 1 n? 1 (ii) Proof. En [f (In+1 )] = En [f (In + Mn (Mn+1 ? Mn ))] = En [f (In + Mn Xn+1 )] = 1 [f (In + Mn ) + f (In ? Mn )] = 2 v v v g(In ), where g(x) = 1 [f (x + 2x + n) + f (x ? 2x + n)], since 2In + n = |Mn |. 2 2. 6. 4 Proof. En [In+1 ?In ] = En [? n (Mn+1 ? Mn )] = ? n En [Mn+1 ? Mn ] = 0. 2. 7. Proof. We denote by Xn the result of n-th coin toss, where Head is represented by X = 1 and Tail is 1 represented by X = ? 1. We also suppose P (X = 1) = P (X = ? 1) = 2 . De? ne S1 = X1 and Sn+1 = n Sn +bn (X1 ,  ·  ·  · , Xn )Xn+1 , where bn ( ·) is a bounded function on {? 1, 1} , to be determined later on. Clearly (Sn )n? 1 is an adapted stochastic process, and we can show it is a martingale. Indeed, En [Sn+1 ? Sn ] = bn (X1 ,  ·  ·  · , Xn )En [Xn+1 ] = 0. For any arbitrary function f , En [f (Sn+1 )] = 1 [f (Sn + bn (X1 ,  ·  ·  · , Xn )) + f (Sn ? n (X1 ,  ·  ·  · , Xn ))]. Then 2 intuitively, En [f (Sn+1 ] cannot be solely dependent upon Sn when bn ’s are properly chosen. Therefore in general, (Sn )n? 1 cannot be a Markov process. Remark: If Xn is regarded as the gain/loss of n-th bet in a gambling game, then Sn would be the wealth at time n. bn is therefore the wager for the (n+1)-th bet and is devised according to past gambling results. 2. 8. (i) Proof. Note Mn = En [MN ] and Mn = En [MN ]. (ii) Proof. In the proof of Theorem 1. 2. 2, we proved by induction that Xn = Vn where Xn is de? ned by (1. 2. 14) of Chapter 1. In other words, the sequence (Vn )0? n?N can be realized as the value process of a portfolio, Xn which consists of stock and money market accounts. Since ( (1+r)n )0? n? N is a martingale under P (Theorem Vn 2. 4. 5), ( (1+r)n )0? n? N is a martingale under P . (iii) Proof. (iv) Proof. Combine (ii) and (iii), then use (i). 2. 9. (i) (H) S1 (H) 1 = 2, d0 = S1S0 = 2 , S0 (T and d1 (T ) = S21 (TT)) = 1. S 1 1 0 ? d So p0 = 1+r? d0 0 = 2 , q0 = 2 , p1 (H) u0 5 q1 (T ) = 6 . Therefore P (HH) = p0 p1 (H) = 1 , 4 5 q0 q1 (T ) = 12 . Vn (1+r)n = En VN (1+r)N , so V0 , V1 1+r ,  ·Ã‚ ·Ã‚ ·, VN ? 1 , VN (1+r)N ? 1 (1+r)N is a martingale under P . Proof. u0 = u1 (H) = =S2 (HH) S1 (H) = 1. 5, d1 (H) = S2 (HT ) S1 (H) = 1, u1 (T ) = S2 (T H) S1 (T ) =4 1+r1 (H)? d1 (H) u1 (H)? d1 (H) 1 = 1 , q1 (H) = 2 , p1 (T ) = 2 1 4, 1+r1 (T )? d1 (T ) u1 (T )? d1 (T ) 1 12 1 = 6 , and P (HT ) = p0 q1 (H) = P (T H) = q0 p1 (T ) = and P (T T ) = The proofs of Theorem 2. 4. 4, Theorem 2. 4. 5 and Theorem 2. 4. 7 still work for the random interest rate m odel, with proper modi? cations (i. e. P would be constructed according to conditional probabilities P (? n+1 = H|? 1 ,  ·  ·  · , ? n ) := pn and P (? n+1 = T |? 1 ,  ·  ·  · , ? n ) := qn . Cf. notes on page 39. ). So the time-zero value of an option that pays o?V2 at time two is given by the risk-neutral pricing formula V0 = E (1+r0V2 1 ) . )(1+r (ii) Proof. V2 (HH) = 5, V2 (HT ) = 1, V2 (T H) = 1 and V2 (T T ) = 0. So V1 (H) = 2. 4, V1 (T ) = p1 (T )V2 (T H)+q1 (T )V2 (T T ) 1+r1 (T ) p1 (H)V2 (HH)+q1 (H)V2 (HT ) 1+r1 (H) = = 1 9, and V0 = p0 V1 (H)+q0 V1 (T ) 1+r0 ? 1. 5 (iii) Proof. ?0 = (iv) Proof. ?1 (H) = 2. 10. (i) Xn+1 Proof. En [ (1+r)n+1 ] = En [ ? n Yn+1 Sn + (1+r)n+1 (1+r)(Xn n Sn ) ] (1+r)n+1 Xn (1+r)n . V2 (HH)? V2 (HT ) S2 (HH)? S2 (HT ) V1 (H)? V1 (T ) S1 (H)? S1 (T ) = 1 2. 4? 9 8? 2 = 0. 4 ? 1 54 ? 0. 3815. = 5? 1 12? 8 = 1. = ?n Sn (1+r)n+1 En [Yn+1 ] + Xn Sn (1+r)n = ?n Sn (1+r)n+1 (up + dq) + Xn n Sn (1+r)n = ?n Sn +Xn n Sn (1+r)n = (ii) Proof . From (2. 8. 2), we have ? n uSn + (1 + r)(Xn ? ?n Sn ) = Xn+1 (H) ? n dSn + (1 + r)(Xn ? ?n Sn ) = Xn+1 (T ). So ? n = Xn+1 (H)? Xn+1 (T ) uSn ? dSn and Xn = En [ Xn+1 ]. To make the portfolio replicate the payo? at time N , we 1+r VN X must have XN = VN . So Xn = En [ (1+r)N ? n ] = En [ (1+r)N ? n ]. Since (Xn )0? n? N is the value process of the N unique replicating portfolio (uniqueness is guaranteed by the uniqueness of the solution to the above linear VN equations), the no-arbitrage price of VN at time n is Vn = Xn = En [ (1+r)N ? ]. (iii) Proof. En [ Sn+1 ] (1 + r)n+1 = = < = 1 En [(1 ? An+1 )Yn+1 Sn ] (1 + r)n+1 Sn [p(1 ? An+1 (H))u + q(1 ? An+1 (T ))d] (1 + r)n+1 Sn [pu + qd] (1 + r)n+1 Sn . (1 + r)n Sn (1+r)n+1 (1? a)(pu+qd) Sn+1 If An+1 is a constant a, then En [ (1+r)n+1 ] = Sn (1+r)n (1? a)n . = Sn (1+r)n (1? a). Sn+1 So En [ (1+r)n+1 (1? a)n+1 ] = 2. 11. (i) Proof. FN + PN = SN ? K + (K ? SN )+ = (SN ? K)+ = CN . (ii) CN FN PN Proof. Cn = En [ (1+r)N ? n ] = En [ (1+ r)N ? n ] + En [ (1+r)N ? n ] = Fn + Pn . (iii) FN Proof. F0 = E[ (1+r)N ] = 1 (1+r)N E[SN ? K] = S0 ? K (1+r)N . (iv) 6 Proof.At time zero, the trader has F0 = S0 in money market account and one share of stock. At time N , the trader has a wealth of (F0 ? S0 )(1 + r)N + SN = ? K + SN = FN . (v) Proof. By (ii), C0 = F0 + P0 . Since F0 = S0 ? (vi) SN ? K Proof. By (ii), Cn = Pn if and only if Fn = 0. Note Fn = En [ (1+r)N ?n ] = Sn ? So Fn is not necessarily zero and Cn = Pn is not necessarily true for n ? 1. (1+r)N S0 (1+r)N ? n (1+r)N S0 (1+r)N = 0, C0 = P0 . = Sn ? S0 (1 + r)n . 2. 12. Proof. First, the no-arbitrage price of the chooser option at time m must be max(C, P ), where C=E (SN ? K)+ (K ? SN )+ , and P = E . (1 + r)N ? m (1 + r)N ? That is, C is the no-arbitrage price of a call option at time m and P is the no-arbitrage price of a put option at time m. Both of them have maturity date N and strike price K. Suppose the market is liquid, then the chooser option is equivalent to receiving a payo? of max(C, P ) at time m. Therefore, its current no-arbitrage price should be E[ max(C,P ) ]. (1+r)m K K By the put-call parity, C = Sm ? (1+r)N ? m + P . So max(C, P ) = P + (Sm ? (1+r)N ? m )+ . Therefore, the time-zero price of a chooser option is E K (Sm ? (1+r)N ? m )+ P +E (1 + r)m (1 + r)m =E K (Sm ? (1+r)N ? m )+ (K ? SN )+ . +E (1 + r)N (1 + r)mThe ? rst term stands for the time-zero price of a put, expiring at time N and having strike price K, and the K second term stands for the time-zero price of a call, expiring at time m and having strike price (1+r)N ? m . If we feel unconvinced by the above argument that the chooser option’s no-arbitrage price is E[ max(C,P ) ], (1+r)m due to the economical argument involved (like â€Å"the chooser option is equivalent to receiving a payo? of max(C, P ) at time m†), then we have the following mathematically rigorous argument. First, we can construct a portfolio ? 0 ,  ·  ·  · , ? m? 1 , whose payo? at time m is max(C, P ).Fix ? , if C(? ) > P (? ), we can construct a portfolio ? m ,  ·  ·  · , ? N ? 1 whose payo? at time N is (SN ? K)+ ; if C(? ) < P (? ), we can construct a portfolio ? m ,  ·  ·  · , ? N ? 1 whose payo? at time N is (K ? SN )+ . By de? ning (m ? k ? N ? 1) ? k (? ) = ? k (? ) ? k (? ) if C(? ) > P (? ) if C(? ) < P (? ), we get a portfolio (? n )0? n? N ? 1 whose payo? is the same as that of the chooser option. So the no-arbitrage price process of the chooser option must be equal to the value process of the replicating portfolio. In Xm particular, V0 = X0 = E[ (1+r)m ] = E[ max(C,P ) ]. (1+r)m 2. 13. (i) Proof.Note under both actual probability P and risk-neutral probability P , coin tosses ? n ’s are i. i. d.. So n+1 without loss of generality, we work on P . For any function g, En [g(Sn+1 , Yn+1 )] = En [g( SSn Sn , Yn + = pg(uSn , Yn + uSn ) + qg(dSn , Yn + dSn ), which is a function of (Sn , Yn ). So (Sn , Yn )0? n? N is Markov un der P . (ii) 7 Sn+1 Sn Sn )] Proof. Set vN (s, y) = f ( Ny ). Then vN (SN , YN ) = f ( +1 Vn = where En [ Vn+1 ] 1+r = n+1 En [ vn+1 (S1+r ,Yn+1 ) ] N n=0 Sn N +1 ) = VN . Suppose vn+1 is given, then = 1 1+r [pvn+1 (uSn , Yn + uSn ) + qvn+1 (dSn , Yn + dSn )] = vn (Sn , Yn ), vn (s, y) = n+1 (us, y + us) + vn+1 (ds, y + ds) . 1+r 2. 14. (i) Proof. For n ? M , (Sn , Yn ) = (Sn , 0). Since coin tosses ? n ’s are i. i. d. under P , (Sn , Yn )0? n? M is Markov under P . More precisely, for any function h, En [h(Sn+1 )] = ph(uSn ) + h(dSn ), for n = 0, 1,  ·  ·  · , M ? 1. For any function g of two variables, we have EM [g(SM +1 , YM +1 )] = EM [g(SM +1 , SM +1 )] = pg(uSM , uSM )+ n+1 n+1 qg(dSM , dSM ). And for n ? M +1, En [g(Sn+1 , Yn+1 )] = En [g( SSn Sn , Yn + SSn Sn )] = pg(uSn , Yn +uSn )+ qg(dSn , Yn + dSn ), so (Sn , Yn )0? n? N is Markov under P . (ii) y Proof. Set vN (s, y) = f ( N ? M ).Then vN (SN , YN ) = f ( N K=M +1 Sk N ? M ) = VN . Suppose vn+1 is already given. a) If n > M , then En [vn+1 (Sn+1 , Yn+1 )] = pvn+1 (uSn , Yn + uSn ) + qvn+1 (dSn , Yn + dSn ). So vn (s, y) = pvn+1 (us, y + us) + qvn+1 (ds, y + ds). b) If n = M , then EM [vM +1 (SM +1 , YM +1 )] = pvM +1 (uSM , uSM ) + vn+1 (dSM , dSM ). So vM (s) = pvM +1 (us, us) + qvM +1 (ds, ds). c) If n < M , then En [vn+1 (Sn+1 )] = pvn+1 (uSn ) + qvn+1 (dSn ). So vn (s) = pvn+1 (us) + qvn+1 (ds). 3. State Prices 3. 1. Proof. Note Z(? ) := P (? ) P (? ) = 1 Z(? ) . Apply Theorem 3. 1. 1 with P , P , Z replaced by P , P , Z, we get the nalogous of properties (i)-(iii) of Theorem 3. 1. 1. 3. 2. (i) Proof. P (? ) = (ii) Proof. E[Y ] = (iii) ? Proof. P (A) = (iv) Proof. If P (A) = A Z(? )P (? ) = 0, by P (Z > 0) = 1, we conclude P (? ) = 0 for any ? ? A. So P (A) = A P (? ) = 0. (v) Proof. P (A) = 1 P (Ac ) = 0 P (Ac ) = 0 P (A) = 1. (vi) A P (? ) = Z(? )P (? ) = E[Z] = 1. Y (? )P (? ) = Y (? )Z(? )P (? ) = E[Y Z]. Z(? )P (? ). Since P (A) = 0, P (? ) = 0 for any ? ? A. So P (A) = 0. 8 Proof. Pick ? 0 such that P (? 0 ) > 0, de? ne Z(? ) = 1 P (? 0 ) 0, 1 P (? 0 ) , if ? = ? 0 Then P (Z ? 0) = 1 and E[Z] = if ? = ? 0 .  · P (? 0 ) = 1. =? 0 Clearly P (? {? 0 }) = E[Z1? {? 0 } ] = Z(? )P (? ) = 0. But P (? {? 0 }) = 1 ? P (? 0 ) > 0 if P (? 0 ) < 1. Hence in the case 0 < P (? 0 ) < 1, P and P are not equivalent. If P (? 0 ) = 1, then E[Z] = 1 if and only if Z(? 0 ) = 1. In this case P (? 0 ) = Z(? 0 )P (? 0 ) = 1. And P and P have to be equivalent. In summary, if we can ? nd ? 0 such that 0 < P (? 0 ) < 1, then Z as constructed above would induce a probability P that is not equivalent to P . 3. 5. (i) Proof. Z(HH) = (ii) Proof. Z1 (H) = E1 [Z2 ](H) = Z2 (HH)P (? 2 = H|? 1 = H) + Z2 (HT )P (? 2 = T |? 1 = H) = 3 E1 [Z2 ](T ) = Z2 (T H)P (? 2 = H|? = T ) + Z2 (T T )P (? 2 = T |? 1 = T ) = 2 . (iii) Proof. V1 (H) = [Z2 (HH)V2 (HH)P (? 2 = H|? 1 = H) + Z2 (HT )V2 (HT )P (? 2 = T |? 1 = T )] = 2. 4, Z1 (H)(1 + r1 (H)) [Z2 (T H)V2 (T H)P (? 2 = H|? 1 = T ) + Z2 (T T )V2 (T T )P (? 2 = T |? 1 = T )] 1 = , Z1 (T )(1 + r1 (T )) 9 3 4. 9 16 , Z(HT ) = 9 , Z(T H) = 8 3 8 and Z(T T ) = 15 4 . Z1 (T ) = V1 (T ) = and V0 = Z2 (HH)V2 (HH) Z2 (HT )V2 (HT ) Z2 (T H)V2 (T H) P (HH) + P (T H) + 0 ? 1. 1 1 1 1 P (HT ) + 1 (1 + 4 )(1 + 4 ) (1 + 4 )(1 + 4 ) (1 + 4 )(1 + 1 ) 2 3. 6. Proof. U (x) = have XN = 1 x, (1+r)N ? Z so I(x) = = 1 Z] 1 x. Z (3. 3. 26) gives E[ (1+r)N 1 X0 (1 + r)n Zn En [Z  ·X0 N Z (1 + r) . 0 = Xn , where ? Hence Xn = (1+r)N ? Z X En [ (1+r)N ? n ] N ] = X0 . So ? = = En [ X0 (1+r) Z n 1 X0 . By (3. 3. 25), we 1 ] = X0 (1 + r)n En [ Z ] = the second to last â€Å"=† comes from Lemma 3. 2. 6. 3. 7. Z ? Z Proof. U (x) = xp? 1 and so I(x) = x p? 1 . By (3. 3. 26), we have E[ (1+r)N ( (1+r)N ) p? 1 ] = X0 . Solve it for ? , we get ? ?p? 1 1 1 ? ? =? ? X0 p E 1 Z p? 1 Np ? ? ? = p? 1 X0 (1 + r)N p (E[Z p? 1 ])p? 1 1 p . (1+r) p? 1 ? Z So by (3. 3. 25), XN = ( (1+r)N ) p? 1 = 1 1 Np ? p? 1 Z p? 1 N (1+r) p? 1 = X0 (1+r) p? 1 E[Z p p? 1 Z p? 1 N (1+r) p? 1 = (1+r)N X0 Z p? 1 E[Z p p? 1 1 . ] ] 3. 8. (i) 9 d d Proof. x (U (x) ? yx) = U (x) ? y. So x = I(y) is an extreme point of U (x) ? yx. Because dx2 (U (x) ? yx) = U (x) ? 0 (U is concave), x = I(y) is a maximum point. Therefore U (x) ? y(x) ? U (I(y)) ? yI(y) for every x. 2 (ii) Proof. Following the hint of the problem, we have E[U (XN )] ? E[XN ? Z ? Z ? Z ? Z ] ? E[U (I( ))] ? E[ I( )], N N N (1 + r) (1 + r) (1 + r) (1 + r)N ? ? ? ? ? i. e. E[U (XN )] ? ?X0 ? E[U (XN )] ? E[ (1+r)N XN ] = E[U (XN )] ? ?X0 . So E[U (XN )] ? E[U (XN )]. 3. 9. (i) X Proof. Xn = En [ (1+r)N ? n ]. So if XN ? 0, then Xn ? 0 for all n. N (ii) 1 Proof. a) If 0 ? x < ? and 0 < y ? ? , then U (x) ? yx = ? yx ? and U (I(y)) ? yI(y) = U (? ) ? y? = 1 ? y? ? 0. So U (x) ? yx ? U (I(y)) ? yI(y). 1 b) If 0 ? x < ? and y > ? , then U (x) ? yx = ? yx ? 0 and U (I(y)) ? yI(y) = U (0) ? y  · 0 = 0. So U (x) ? yx ? U (I(y)) ? yI(y). 1 c) If x ? ? and 0 < y ? ? , then U (x) ? yx = 1 ? yx and U (I(y)) ? yI(y) = U (? ) ? y? = 1 ? y? ? 1 ? yx. So U (x) ? yx ? U (I(y)) ? yI(y). 1 d) If x ? ? and y > ? , then U (x) ? yx = 1 ? yx < 0 and U (I(y)) ? yI(y) = U (0) ? y  · 0 = 0. So U (x) ? yx ? U (I(y)) ? yI(y). (iii) XN ? Z Proof. Using (ii) and set x = XN , y = (1+r)N , where XN is a random variable satisfying E[ (1+r)N ] = X0 , we have ?Z ? Z ? E[U (XN )] ? E[ XN ] ? E[U (XN )] ? E[ X ? ]. (1 + r)N (1 + r)N N ? ? That is, E[U (XN )] ? ?X0 ? E[U (XN )] ? ?X0 . So E[U (XN )] ? E[U (XN )]. (iv) Proof. Plug pm and ? m into (3. 6. 4), we have 2N 2N X0 = m=1 pm ? m I( m ) = m=1 1 pm ? m ? 1{ m ? ? } . So X0 ? X0 ? {m : = we are looking for positive solution ? > 0). Conversely, suppose there exists some K so that ? K < ? K+1 and K X0 1 m=1 ? m pm = ? . Then we can ? nd ? > 0, such that ? K < < ? K+1 . For such ? , we have Z ? Z 1 E[ I( )] = pm ? m 1{ m ? ? } ? = pm ? m ? = X0 . N (1 + r) (1 + r)N m=1 m=1 Hence (3. 6. 4) has a solution. 0 2N K 2N X0 1 m=1 pm ? m 1{ m ? ? } . Suppose there is a solution ? to (3. 6. 4), note ? > 0, we then can conclude 1 1 1 m ? ? } = ?. Let K = max{m : m ? ? }, then K ? ? < K+1 . So ? K < ? K+1 and K N m=1 pm ? m (Note, however, that K could be 2 . In this case, ? K+1 is interpreted as ?. Also, note = (v) ? 1 Proof. XN (? m ) = I( m ) = ? 1{ m ? ? } = ?, if m ? K . 0, if m ? K + 1 4. American Derivative Securities Before proceeding to the exercise problems, we ? rst give a brief summary of pricing American derivative securities as presented in the textbook. We shall use the notation of the book.From the buyer’s perspective: At time n, if the derivative security has not been exercised, then the buyer can choose a policy ? with ? ? Sn . The valuation formula for cash ? ow (Theorem 2. 4. 8) gives a fair price for the derivative security exercised according to ? : N Vn (? ) = k=n En 1{? =k} 1 1 Gk = En 1{? ?N } G? . (1 + r)k? n (1 + r)? ?n The buyer wants to consider all the possible ? ’s, so that he c an ? nd the least upper bound of security value, which will be the maximum price of the derivative security acceptable to him. This is the price given by 1 De? nition 4. 4. 1: Vn = max? ?Sn En [1{? ?N } (1+r)? n G? ]. From the seller’s perspective: A price process (Vn )0? n? N is acceptable to him if and only if at time n, he can construct a portfolio at cost Vn so that (i) Vn ? Gn and (ii) he needs no further investing into the portfolio as time goes by. Formally, the seller can ? nd (? n )0? n? N and (Cn )0? n? N so that Cn ? 0 and Sn Vn+1 = ? n Sn+1 + (1 + r)(Vn ? Cn ? ?n Sn ). Since ( (1+r)n )0? n? N is a martingale under the risk-neutral measure P , we conclude En Cn Vn+1 Vn =? ? 0, ? n+1 n (1 + r) (1 + r) (1 + r)n Vn i. e. ( (1+r)n )0? n? N is a supermartingale. This inspired us to check if the converse is also true.This is exactly the content of Theorem 4. 4. 4. So (Vn )0? n? N is the value process of a portfolio that needs no further investing if and only if Vn (1+r)n Vn (1+r)n is a supermartingale under P (note this is independent of the requirement 0? n? N Vn ? Gn ). In summary, a price process (Vn )0? n? N is acceptable to the seller if and only if (i) Vn ? Gn ; (ii) is a supermartingale under P . 0? n? N Theorem 4. 4. 2 shows the buyer’s upper bound is the seller’s lower bound. So it gives the price acceptable to both. Theorem 4. 4. 3 gives a speci? c algorithm for calculating the price, Theorem 4. 4. establishes the one-to-one correspondence between super-replication and supermartingale property, and ? nally, Theorem 4. 4. 5 shows how to decide on the optimal exercise policy. 4. 1. (i) Proof. V2P (HH) = 0, V2P (HT ) = V2P (T H) = 0. 8, V2P (T T ) = 3, V1P (H) = 0. 32, V1P (T ) = 2, V0P = 9. 28. (ii) Proof. V0C = 5. (iii) Proof. gS (s) = |4 ? s|. We apply Theorem 4. 4. 3 and have V2S (HH) = 12. 8, V2S (HT ) = V2S (T H) = 2. 4, V2S (T T ) = 3, V1S (H) = 6. 08, V1S (T ) = 2. 16 and V0S = 3. 296. (iv) 11 Proof. First, we note the simple inequality max(a1 , b1 ) + max(a2 , b2 ) ? max(a1 + a2 , b1 + b2 ). >† holds if and only if b1 > a1 , b2 < a2 or b1 < a1 , b2 > a2 . By induction, we can show S Vn = max gS (Sn ), S S pVn+1 + Vn+1 1+r C P P pV C + Vn+1 pVn+1 + Vn+1 + n+1 1+r 1+r C C pVn+1 + Vn+1 1+r ? max gP (Sn ) + gC (Sn ), ? max gP (Sn ), P C = Vn + Vn . P P pVn+1 + Vn+1 1+r + max gC (Sn ), S P C As to when â€Å" C C pVn+1 +qVn+1 1+r or gP (Sn ) > P P pVn+1 +qVn+1 1+r and gC (Sn ) < C C pVn+1 +qVn+1 }. 1+r 4. 2. Proof. For this problem, we need Figure 4. 2. 1, Figure 4. 4. 1 and Figure 4. 4. 2. Then ? 1 (H) = and ? 0 = V2 (HH) ? V2 (HT ) 1 V2 (T H) ? V2 (T T ) = ? , ? 1 (T ) = = ? 1, S2 (HH) ? S2 (HT ) 12 S2 (T H) ?S2 (T T ) V1 (H) ? V1 (T ) ? ?0. 433. S1 (H) ? S1 (T ) The optimal exercise time is ? = inf{n : Vn = Gn }. So ? (HH) = ? , ? (HT ) = 2, ? (T H) = ? (T T ) = 1. Therefore, the agent borrows 1. 36 at time zero and buys the put. At the same time, to hedge the long position, he needs to borr ow again and buy 0. 433 shares of stock at time zero. At time one, if the result of coin toss is tail and the stock price goes down to 2, the value of the portfolio 1 is X1 (T ) = (1 + r)(? 1. 36 ? 0. 433S0 ) + 0. 433S1 (T ) = (1 + 4 )(? 1. 36 ? 0. 433 ? 4) + 0. 433 ? 2 = ? 3. The agent should exercise the put at time one and get 3 to pay o? is debt. At time one, if the result of coin toss is head and the stock price goes up to 8, the value of the portfolio 1 is X1 (H) = (1 + r)(? 1. 36 ? 0. 433S0 ) + 0. 433S1 (H) = ? 0. 4. The agent should borrow to buy 12 shares of stock. At time two, if the result of coin toss is head and the stock price goes up to 16, the value of the 1 1 portfolio is X2 (HH) = (1 + r)(X1 (H) ? 12 S1 (H)) + 12 S2 (HH) = 0, and the agent should let the put expire. If at time two, the result of coin toss is tail and the stock price goes down to 4, the value of the portfolio is 1 1 X2 (HT ) = (1 + r)(X1 (H) ? 12 S1 (H)) + 12 S2 (HT ) = ? 1.The agent should exercise the put to get 1. This will pay o? his debt. 4. 3. Proof. We need Figure 1. 2. 2 for this problem, and calculate the intrinsic value process and price process of the put as follows. 2 For the intrinsic value process, G0 = 0, G1 (T ) = 1, G2 (T H) = 3 , G2 (T T ) = 5 , G3 (T HT ) = 1, 3 G3 (T T H) = 1. 75, G3 (T T T ) = 2. 125. All the other outcomes of G is negative. 12 2 5 For the price process, V0 = 0. 4, V1 (T ) = 1, V1 (T H) = 3 , V1 (T T ) = 3 , V3 (T HT ) = 1, V3 (T T H) = 1. 75, V3 (T T T ) = 2. 125. All the other outcomes of V is zero. Therefore the time-zero price of the derivative security is 0. and the optimal exercise time satis? es ? (? ) = ? if ? 1 = H, 1 if ? 1 = T . 4. 4. Proof. 1. 36 is the cost of super-replicating the American derivative security. It enables us to construct a portfolio su? cient to pay o? the derivative security, no matter when the derivative security is exercised. So to hedge our short position after selling the put, there is no need to charge t he insider more than 1. 36. 4. 5. Proof. The stopping times in S0 are (1) ? ? 0; (2) ? ? 1; (3) ? (HT ) = ? (HH) = 1, ? (T H), ? (T T ) ? {2, ? } (4 di? erent ones); (4) ? (HT ), ? (HH) ? {2, ? }, ? (T H) = ? (T T ) = 1 (4 di? rent ones); (5) ? (HT ), ? (HH), ? (T H), ? (T T ) ? {2, ? } (16 di? erent ones). When the option is out of money, the following stopping times do not exercise (i) ? ? 0; (ii) ? (HT ) ? {2, ? }, ? (HH) = ? , ? (T H), ? (T T ) ? {2, ? } (8 di? erent ones); (iii) ? (HT ) ? {2, ? }, ? (HH) = ? , ? (T H) = ? (T T ) = 1 (2 di? erent ones). ? 4 For (i), E[1{? ?2} ( 4 )? G? ] = G0 = 1. For (ii), E[1{? ?2} ( 5 )? G? ] ? E[1{? ? ? 2} ( 4 )? G? ? ], where ? ? (HT ) = 5 5 1 4 4 ? 2, ? ? (HH) = ? , ? ? (T H) = ? ? (T T ) = 2. So E[1{? ? ? 2} ( 5 )? G? ? ] = 4 [( 4 )2  · 1 + ( 5 )2 (1 + 4)] = 0. 96. For 5 (iii), E[1{? ?2} ( 4 )? G? has the biggest value when ? satis? es ? (HT ) = 2, ? (HH) = ? , ? (T H) = ? (T T ) = 1. 5 This value is 1. 36. 4. 6. (i) Proof. The value of the put at time N , if it is not exercised at previous times, is K ? SN . Hence VN ? 1 = VN K max{K ? SN ? 1 , EN ? 1 [ 1+r ]} = max{K ? SN ? 1 , 1+r ? SN ? 1 } = K ? SN ? 1 . The second equality comes from the fact that discounted stock price process is a martingale under risk-neutral probability. By induction, we can show Vn = K ? Sn (0 ? n ? N ). So by Theorem 4. 4. 5, the optimal exercise policy is to sell the stock at time zero and the value of this derivative security is K ?S0 . Remark: We cheated a little bit by using American algorithm and Theorem 4. 4. 5, since they are developed for the case where ? is allowed to be ?. But intuitively, results in this chapter should still hold for the case ? ? N , provided we replace â€Å"max{Gn , 0}† with â€Å"Gn †. (ii) Proof. This is because at time N , if we have to exercise the put and K ? SN < 0, we can exercise the European call to set o? the negative payo?. In e? ect, throughout the portfolio’s lifetime, the portfolio has intrinsic values greater than that of an American put stuck at K with expiration time N . So, we must have V0AP ? V0 + V0EC ? K ?S0 + V0EC . (iii) 13 Proof. Let V0EP denote the time-zero value of a European put with strike K and expiration time N . Then V0AP ? V0EP = V0EC ? E[ K SN ? K ] = V0EC ? S0 + . (1 + r)N (1 + r)N 4. 7. VN K K Proof. VN = SN ? K, VN ? 1 = max{SN ? 1 ? K, EN ? 1 [ 1+r ]} = max{SN ? 1 ? K, SN ? 1 ? 1+r } = SN ? 1 ? 1+r . K By induction, we can prove Vn = Sn ? (1+r)N ? n (0 ? n ? N ) and Vn > Gn for 0 ? n ? N ? 1. So the K time-zero value is S0 ? (1+r)N and the optimal exercise time is N . 5. Random Walk 5. 1. (i) Proof. E[ 2 ] = E[? (? 2 1 )+? 1 ] = E[? (? 2 1 ) ]E[ 1 ] = E[ 1 ]2 . (ii) Proof. If we de? ne Mn = Mn+? ? M? m (m = 1, 2,  ·  ·  · ), then (M · )m as random functions are i. i. d. with (m) distributions the same as that of M . So ? m+1 ? ?m = inf{n : Mn = 1} are i. i. d. with distributions the same as that of ? 1 . Therefore E [ m ] = E[? (? m m? 1 )+(? m? 1 m? 2 )+ ·Ã‚ ·Ã‚ ·+? 1 ] = E[ 1 ]m . (m) (m) (iii) Proof. Yes, since the argument of (ii) still works for asymmetric random walk. 5. 2. (i) Proof. f (? ) = pe? ? qe , so f (? ) > 0 if and only if ? > f (? ) > f (0) = 1 for all ? > 0. (ii) 1 1 1 n+1 Proof. En [ SSn ] = En [e? Xn+1 f (? ) ] = pe? f (? ) + qe f (? ) = 1. 1 2 (ln q ? ln p). Since 1 2 (ln q ln p) < 0, (iii) 1 Proof. By optional stopping theorem, E[Sn 1 ] = E[S0 ] = 1. Note Sn 1 = e? Mn 1 ( f (? ) )n 1 ? e?  ·1 , by bounded convergence theorem, E[1{? 1 1 for all ? > ? 0 . v (ii) 1 1 Proof. As in Exercise 5. 2, Sn = e? Mn ( f (? ) )n is a martingale, and 1 = E[S0 ] = E[Sn 1 ] = E[e? Mn 1 ( f (? ) )? 1 ? n ]. Suppose ? > ? 0 , then by bounded convergence theorem, 1 = E[ lim e? Mn 1 ( n>? 1 n 1 1 ? 1 ) ] = E[1{? 1 K} ] = P (ST > K). Moreover, by Girsanov’s Theorem, Wt = Wt + in Theorem 5. 4. 1. ) (iii) Proof. ST = xe? WT +(r? 2 ? 1 2 1 2 t ( )du 0 = Wt ? ?t is a P -Brownian motion (set ? )T = xe? WT +(r+ 2 ? 1 2 1 2 )T . So WT v > ? d+ (T, x) T = N (d+ (T, x)). P (ST > K) = P (xe? WT +(r+ 2 ? )T > K) = P 46 5. 4. First, a few typos. In the SDE for S, â€Å"? (t)dW (t)† > â€Å"? (t)S(t)dW (t)†. In the ? rst equation for c(0, S(0)), E > E. In the second equation for c(0, S(0)), the variable for BSM should be ? ? 1 T 2 1 T r(t)dt, ? (t)dt? . BSM ? T, S(0); K, T 0 T 0 (i) Proof. d ln St = X = ? is a Gaussian with X ? N ( (ii) Proof. For the standard BSM model with constant volatility ? and interest rate R, under the risk-neutral measure, we have ST = S0 eY , where Y = (R? 1 ? 2 )T +? WT ? N ((R? 1 ? )T, ? 2 T ), and E[(S0 eY ? K)+ ] = 2 2 eRT BSM (T, S0 ; K, R, ? ). Note R = 1 T (rt 0 T T dSt 1 2 1 1 2 2 St ? 2St d S t = rt dt + ? t dWt ? 2 ? t dt. So ST = S0 exp{ 0 (rt ? 2 ? t )dt + 0 T 1 2 2 ? t )dt + 0 ? t dWt . The ? rst term in the expression of X is a number and the T 2 random variable N (0, 0 ? t dt), since both r and ? ar deterministic. Th erefore, T T 2 2 (rt ? 1 ? t )dt, 0 ? t dt),. 2 0 ?t dWt }. Let second term ST = S0 eX , 1 T (E[Y ] + 1 V ar(Y )) and ? = 2 T, S0 ; K, 1 T 1 T V ar(Y ), we can get 1 V ar(Y ) . T E[(S0 eY ? K)+ ] = eE[Y ]+ 2 V ar(Y ) BSM So for the model in this problem, c(0, S0 ) = = e? ? T 0 1 E[Y ] + V ar(Y ) , 2 rt dt E[(S0 eX ? K)+ ] e BSM T, S0 ; K, 1 T T 0 T 0 1 rt dt E[X]+ 2 V ar(X) 1 T ? 1 E[X] + V ar(X) , 2 1 V ar(X) T ? = 1 BSM ? T, S0 ; K, T 0 T rt dt, 2 ? t dt? . 5. 5. (i) 1 1 Proof. Let f (x) = x , then f (x) = ? x2 and f (x) = 2 x3 . Note dZt = ? Zt ? t dWt , so d 1 Zt 1 1 1 2 2 2 ? t ? 2 t = f (Zt )dZt + f (Zt )dZt dZt = ? 2 (? Zt )? t dWt + 3 Zt ? t dt = Z dWt + Z dt. 2 Zt 2 Zt t t (ii) Proof. By Lemma 5. 2. 2. , for s, t ? 0 with s < t, Ms = E[Mt |Fs ] = E Zs Ms . So M = Z M is a P -martingale. (iii) Zt Mt Zs |Fs . That is, E[Zt Mt |Fs ] = 47 Proof. dMt = d Mt  · 1 Zt = 1 1 1 ? M t ? t M t ? 2 ? t ? t t dMt + Mt d + dMt d = dWt + dWt + dt + dt. Zt Zt Zt Zt Zt Zt Zt (iv) Proof. In part (iii), we have dMt = Let ? t = 5. 6. Proof. By Theorem 4. 6. 5, it su? ces to show Wi (t) is an Ft -martingale under P and [Wi , Wj ](t) = t? ij (i, j = 1, 2). Indeed, for i = 1, 2, Wi (t) is an Ft -martingale under P if and only if Wi (t)Zt is an Ft -martingale under P , since Wi (t)Zt E[Wi (t)|Fs ] = E |Fs . Zs By It? ’s product formula, we have o d(Wi (t)Zt ) = Wi (t)dZt + Zt dWi (t) + dZt dWi (t) = Wi (t)(? Zt )? (t)  · dWt + Zt (dWi (t) + ? i (t)dt) + (? Zt ? t  · dWt )(dWi (t) + ? i (t)dt) d t M t ? t M t ? 2 ? t ? t ? t M t ? t t dWt + dWt + dt + dt = (dWt + ? t dt) + (dWt + ? t dt). Zt Zt Zt Zt Zt Zt then dMt = ? t dWt . This proves Corollary 5. 3. 2. ?t +Mt ? t , Zt = Wi (t)(? Zt ) j=1 d ?j (t)dWj (t) + Zt (dWi (t) + ? i (t)dt) ? Zt ? i (t)dt = Wi (t)(? Zt ) j=1 ?j (t)dWj (t) + Zt dWi (t) This shows Wi (t)Zt is an Ft -martingale under P . So Wi (t) is an Ft -martingale under P . Moreover,  ·  · [Wi , Wj ](t) = Wi + 0 ?i (s)ds, Wj + 0 ?j (s)ds (t) = [Wi , Wj ](t) = t? ij . Combined, this proves the two-dimensional Girsanov’s Theorem. 5. 7. (i) Proof. Let a be any strictly positive number. We de? e X2 (t) = (a + X1 (t))D(t)? 1 . Then P X2 (T ) ? X2 (0) D(T ) = P (a + X1 (T ) ? a) = P (X1 (T ) ? 0) = 1, and P X2 (T ) > X2 (0) = P (X1 (T ) > 0) > 0, since a is arbitrary, we have proved the claim of this problem. D(T ) Remark: The intuition is that we invest the positive starting fund a into the money market account, and construct portfolio X1 from zero cost. Their sum should be able to beat the return of money market account. (ii) 48 Proof. We de? ne X1 (t) = X2 (t)D(t) ? X2 (0). Then X1 (0) = 0, P (X1 (T ) ? 0) = P X2 (T ) ? X2 (0) D(T ) = 1, P (X1 (T ) > 0) = P X2 (T ) > X2 (0) D(T ) > 0. 5. 8.The basic idea is that for any positive P -martingale M , dMt = Mt  · sentation Theorem, dMt = ? t dWt for some adapted process ? t . So martingale must be the exponential of an integral w. r. t. Brownian motion. Taking into account d iscounting factor and apply It? ’s product rule, we can show every strictly positive asset is a generalized geometric o Brownian motion. (i) Proof. Vt Dt = E[e? 0 Ru du VT |Ft ] = E[DT VT |Ft ]. So (Dt Vt )t? 0 is a P -martingale. By Martingale Represent tation Theorem, there exists an adapted process ? t , 0 ? t ? T , such that Dt Vt = 0 ? s dWs , or equivalently, ? 1 t ? 1 t ? 1 Vt = Dt 0 ? dWs . Di? erentiate both sides of the equation, we get dVt = Rt Dt 0 ? s dWs dt + Dt ? t dWt , i. e. dVt = Rt Vt dt + (ii) Proof. We prove the following more general lemma. Lemma 1. Let X be an almost surely positive random variable (i. e. X > 0 a. s. ) de? ned on the probability space (? , G, P ). Let F be a sub ? -algebra of G, then Y = E[X|F] > 0 a. s. Proof. By the property of conditional expectation Yt ? 0 a. s. Let A = {Y = 0}, we shall show P (A) = 0. In? 1 1 deed, note A ? F, 0 = E[Y IA ] = E[E[X|F]IA ] = E[XIA ] = E[X1A? {X? 1} ] + n=1 E[X1A? { n >X? n+1 } ] ? 1 1 1 1 1 P (A? {X ? 1})+ n=1 n+1 P (A? n > X ? n+1 }). So P (A? {X ? 1}) = 0 and P (A? { n > X ? n+1 }) = 0, ? 1 1 ? n ? 1. This in turn implies P (A) = P (A ? {X > 0}) = P (A ? {X ? 1}) + n=1 P (A ? { n > X ? n+1 }) = 0. ? ? t Dt dWt . T 1 Mt dMt . By Martingale Repre? dMt = Mt ( Mtt )dWt , i. e. any positive By the above lemma, it is clear that for each t ? [0, T ], Vt = E[e? t Ru du VT |Ft ] > 0 a. s.. Moreover, by a classical result of martingale theory (Revuz and Yor [4], Chapter II, Proposition (3. 4)), we have the following stronger result: for a. s. ?, Vt (? ) > 0 for any t ? [0, T ]. (iii) 1 1 Proof. By (ii), V > 0 a. s. so dVt = Vt Vt dVt = Vt Vt Rt Vt dt + ? t Dt dWt ? t = Vt Rt dt + Vt Vt Dt dWt = Rt Vt dt + T ?t Vt dWt , where ? t = 5. 9. ?t Vt Dt . This shows V follows a generalized geometric Brownian motion. Proof. c(0, T, x, K) = xN (d+ ) ? Ke? rT N (d? ) with d ± = then f (y) = ? yf (y), cK (0, T, x, K) = xf (d+ ) 1 v ? T x (ln K + (r  ± 1 ? 2 )T ). Let f (y) = 2 y v1 e? 2 2? 2 , ?d+ ? d? ? e? rT N (d? ) ? Ke? rT f (d? ) ? y ? y ? 1 1 = xf (d+ ) v ? e? rT N (d? ) + e? rT f (d? ) v , ? TK ? T 49 and cKK (0, T, x, K) x ? d? e? rT 1 ? d+ d? ? v ? e? rT f (d? ) + v (? d? )f (d? ) xf (d+ ) v f (d+ )(? d+ ) 2 ? y ? y ? y ? TK ? TK ?T x xd+ ? 1 ? 1 e? rT d? ?1 v v ? e? rT f (d? ) v ? v f (d? ) v f (d+ ) + v f (d+ ) ? T K2 ? TK K? T K? T ? T K? T x d+ e? rT f (d? ) d? v [1 ? v ] + v f (d+ ) [1 + v ] 2? T K ? T K? T ? T e? rT x f (d? )d+ ? 2 2 f (d+ )d? . K? 2 T K ? T = = = = 5. 10. (i) Proof. At time t0 , the value of the chooser option is V (t0 ) = max{C(t0 ), P (t0 )} = max{C(t0 ), C(t0 ) ? F (t0 )} = C(t0 ) + max{0, ? F (t0 )} = C(t0 ) + (e? r(T ? t0 ) K ? S(t0 ))+ . (ii) Proof. By the risk-neutral pricing formula, V (0) = E[e? rt0 V (t0 )] = E[e? rt0 C(t0 )+(e? rT K ? e? rt0 S(t0 )+ ] = C(0) + E[e? rt0 (e? r(T ? t0 ) K ? S(t0 ))+ ]. The ? st term is the value of a call expiring at time T with strike price K and the second term is the value of a put expiring at time t0 with strike price e? r(T ? t0 ) K. 5. 11. Proof. We ? rst make an analysis which leads to the hint, then we give a formal proof. (Analysis) If we want to construct a portfolio X that exactly replicates the cash ? ow, we must ? nd a solution to the backward SDE dXt = ? t dSt + Rt (Xt ? ?t St )dt ? Ct dt XT = 0. Multiply Dt on both sides of the ? rst equation and apply It? ’s product rule, we get d(Dt Xt ) = ? t d(Dt St ) ? o T T Ct Dt dt. Integrate from 0 to T , we have DT XT ? D0 X0 = 0 ? d(Dt St ) ? 0 Ct Dt dt. By the terminal T T ? 1 condition, we get X0 = D0 ( 0 Ct Dt dt ? 0 ? t d(Dt St )). X0 is the theoretical, no-arbitrage price of the cash ? ow, provided we can ? nd a trading strategy ? that solves the BSDE. Note the SDE for S ? R gives d(Dt St ) = (Dt St )? t (? t dt + dWt ), where ? t = ? t? t t . Take the proper change of measure so that Wt = t ? ds 0 s + Wt is a Brownian motion under the new measure P , we get T T T Ct Dt dt = D0 X0 + 0 T 0 ?t d(Dt St ) = D0 X0 + 0 ?t (Dt St )? t dWt . T This says the random variable 0 Ct Dt dt has a stochastic integral representation D0 X0 + 0 ? t Dt St ? dWt . T This inspires us to consider the martingale generated by 0 Ct Dt dt, so that we can apply Martingale Representation Theorem and get a formula for ? by comparison of the integrands. 50 (Formal proof) Let MT = Xt = ?1 Dt (D0 X0 T 0 Ct Dt dt, and Mt = E[MT |Ft ]. Then by Martingale Representation Theot 0 rem, we can ? nd an adapted process ? t , so that Mt = M0 + + t 0 ?t dWt . If we set ? t = T 0 ?u d(Du Su ) ? t 0 ?t Dt St ? t , we can check Cu Du du), with X0 = M0 = E[ Ct Dt dt] solves the SDE dXt = ? t dSt + Rt (Xt ? ?t St )dt ? Ct dt XT = 0. Indeed, it is easy to see that X satis? es the ? rst equation.To check the terminal condition, we note T T T XT DT = D0 X0 + 0 ? t Dt St ? t dWt ? 0 Ct Dt dt = M0 + 0 ? t dWt ? MT = 0. So XT = 0. Thus, we have found a trading strategy ? , so that the corresponding portfolio X replicates the cash ? ow and has zero T terminal value. So X0 = E[ 0 Ct Dt dt] is the no-arbitrage price of the cash ? ow at time zero. Remark: As shown in the analysis, d(Dt Xt ) = ? t d(Dt St ) ? Ct Dt dt. Integrate from t to T , we get T T 0 ? Dt Xt = t ? u d(Du Su ) ? t Cu Du du. Take conditional expectation w. r. t. Ft on both sides, we get T T ? 1 ? Dt Xt = ? E[ t Cu Du du|Ft ]. So Xt = Dt E[ t Cu Du du|Ft ].This is the no-arbitrage price of the cash ? ow at time t, and we have justi? ed formula (5. 6. 10) in the textbook. 5. 12. (i) Proof. dBi (t) = dBi (t) + ? i (t)dt = martingale. Since dBi (t)dBi (t) = P. (ii) Proof. dSi (t) = = = R(t)Si (t)dt + ? i (t)Si (t)dBi (t) + (? i (t) ? R(t))Si (t)dt ? ?i (t)Si (t)? i (t)dt d d ? ij (t) ? ij (t) d d j=1 ? i (t) ? j (t)dt = j=1 ? i (t) dWj (t) + ? ij (t)2 d e j=1 ? i (t)2 dt = dt, by L? vy’s Theorem, Bi ? ij (t) d j=1 ? i (t) dWj (t). So Bi is a is a Brownian motion under R(t)Si (t)dt + ? i (t)Si (t)dBi (t) + j=1 ?ij (t)? j (t)Si (t)dt ? Si (t) j=1 ?ij (t)? j (t)dt R(t)Si (t)dt + ? (t)Si (t)dBi (t). (iii) Proof. dBi (t)dBk (t) = (dBi (t) + ? i (t)dt)(dBj (t) + ? j (t)dt) = dBi (t)dBj (t) = ? ik (t)dt. (iv) Proof. By It? ’s product rule and martingale property, o t t t E[Bi (t)Bk (t)] = E[ 0 t Bi (s)dBk (s)] + E[ 0 t Bk (s)dBi (s)] + E[ 0 dBi (s)dBk (s)] = E[ 0 ?ik (s)ds] = 0 ?ik (s)ds. t 0 Similarly, by part (iii), we can show E[Bi (t)Bk (t)] = (v) ?ik (s)ds. 51 Proof. By It? ’s product formula, o t t E[B1 (t)B2 (t)] = E[ 0 sign(W1 (u))du] = 0 [P (W1 (u) ? 0) ? P (W1 (u) < 0)]du = 0. Meanwhile, t E[B1 (t)B2 (t)] = E[ 0 t sign(W1 (u))du [P (W1 (u) ? 0) ? P (W1 (u) < 0)]du = 0 t = 0 t [P (W1 (u) ? ) ? P (W1 (u) < u)]du 2 0 = < 0, 1 ? P (W1 (u) < u) du 2 for any t > 0. So E[B1 (t)B2 (t)] = E[B1 (t)B2 (t)] for all t > 0. 5. 13. (i) Proof. E[W1 (t)] = E[W1 (t)] = 0 and E[W2 (t)] = E[W2 (t) ? (ii) Proof. Cov[W1 (T ), W2 (T )] = E[W1 (T )W2 (T )] T T t 0 W1 (u)du] = 0, for all t ? [0, T ]. = E 0 T W1 (t)dW2 (t) + 0 W2 ( t)dW1 (t) T = E 0 W1 (t)(dW2 (t) ? W1 (t)dt) + E 0 T W2 (t)dW1 (t) = ? E 0 T W1 (t)2 dt tdt = ? 0 1 = ? T 2. 2 5. 14. Equation (5. 9. 6) can be transformed into d(e? rt Xt ) = ? t [d(e? rt St ) ? ae? rt dt] = ? t e? rt [dSt ? rSt dt ? adt]. So, to make the discounted portfolio value e? t Xt a martingale, we are motivated to change the measure t in such a way that St ? r 0 Su du? at is a martingale under the new measure. To do this, we note the SDE for S is dSt = ? t St dt+? St dWt . Hence dSt ? rSt dt? adt = [(? t ? r)St ? a]dt+? St dWt = ? St Set ? t = (? t ? r)St ? a ? St (? t ? r)St ? a dt ? St + dWt . and Wt = t ? ds 0 s + Wt , we can ? nd an equivalent probability measure P , under which S satis? es the SDE dSt = rSt dt + ? St dWt + adt and Wt is a BM. This is the rational for formula (5. 9. 7). This is a good place to pause and think about the meaning of â€Å"martingale measure. † What is to be a martingale?The new measure P should be such that the discounted value pro cess of the replicating 52 portfolio is a martingale, not the discounted price process of the underlying. First, we want Dt Xt to be a martingale under P because we suppose that X is able to replicate the derivative payo? at terminal time, XT = VT . In order to avoid arbitrage, we must have Xt = Vt for any t ? [0, T ]. The di? culty is how to calculate Xt and the magic is brought by the martingale measure in the following line of reasoning: ? 1 ? 1 Vt = Xt = Dt E[DT XT |Ft ] = Dt E[DT VT |Ft ]. You can think of martingale measure as a calculational convenience.That is all about martingale measure! Risk neutral is a just perception, referring to the actual e? ect of constructing a hedging portfolio! Second, we note when the portfolio is self-? nancing, the discounted price process of the underlying is a martingale under P , as in the classical Black-Scholes-Merton model without dividends or cost of carry. This is not a coincidence. Indeed, we have in this case the relation d(Dt Xt ) = ? t d(Dt St ). So Dt Xt being a martingale under P is more or less equivalent to Dt St being a martingale under P . However, when the underlying pays dividends, or there is cost of carry, d(Dt Xt ) = ? d(Dt St ) no longer holds, as shown in formula (5. 9. 6). The portfolio is no longer self-? nancing, but self-? nancing with consumption. What we still want to retain is the martingale property of Dt Xt , not that of Dt St . This is how we choose martingale measure in the above paragraph. Let VT be a payo? at time T , then for the martingale Mt = E[e? rT VT |Ft ], by Martingale Representation rt t Theorem, we can ? nd an adapted process ? t , so that Mt = M0 + 0 ? s dWs . If we let ? t = ? t e t , then the ? S value of the corresponding portfolio X satis? es d(e? rt Xt ) = ? t dWt . So by setting X0 = M0 = E[e? T VT ], we must have e? rt Xt = Mt , for all t ? [0, T ]. In particular, XT = VT . Thus the portfolio perfectly hedges VT . This justi? es the risk-neutral pricing of Europea n-type contingent claims in the model where cost of carry exists. Also note the risk-neutral measure is di? erent from the one in case of no cost of carry. Another perspective for perfect replication is the following. We need to solve the backward SDE dXt = ? t dSt ? a? t dt + r(Xt ? ?t St )dt XT = VT for two unknowns, X and ?. To do so, we ? nd a probability measure P , under which e? rt Xt is a martingale, t then e? rt Xt = E[e? T VT |Ft ] := Mt . Martingale Representation Theorem gives Mt = M0 + 0 ? u dWu for some adapted process ?. This would give us a theoretical representation of ? by comparison of integrands, hence a perfect replication of VT . (i) Proof. As indicated in the above analysis, if we have (5. 9. 7) under P , then d(e? rt Xt ) = ? t [d(e? rt St ) ? ae? rt dt] = ? t e? rt ? St dWt . So (e? rt Xt )t? 0 , where X is given by (5. 9. 6), is a P -martingale. (ii) 1 1 Proof. By It? ’s formula, dYt = Yt [? dWt + (r ? 2 ? 2 )dt] + 2 Yt ? 2 dt = Yt (? dWt + rdt). So d(e? rt Yt ) = o t a ? e? rt Yt dWt and e? rt Yt is a P -martingale.Moreover, if St = S0 Yt + Yt 0 Ys ds, then t dSt = S0 dYt + 0 a dsdYt + adt = Ys t S0 + 0 a ds Yt (? dWt + rdt) + adt = St (? dWt + rdt) + adt. Ys This shows S satis? es (5. 9. 7). Remark: To obtain this formula for S, we ? rst set Ut = e? rt St to remove the rSt dt term. The SDE for U is dUt = ? Ut dWt + ae? rt dt. Just like solving linear ODE, to remove U in the dWt term, we consider Vt = Ut e Wt . It? ’s product formula yields o dVt = = e Wt dUt + Ut e Wt 1 ( )dWt + ? 2 dt + dUt  · e Wt 2 1 ( )dWt + ? 2 dt 2 1 e Wt ae? rt dt ? ? 2 Vt dt. 2 53 Note V appears only in the dt term, so multiply the integration factor e 2 ? e get 1 2 1 2 d(e 2 ? t Vt ) = ae? rt Wt + 2 ? t dt. Set Yt = e? Wt +(r? 2 ? (iii) Proof. t 1 2 1 2 t on both sides of the equation, )t , we have d(St /Yt ) = adt/Yt . So St = Yt (S0 + t ads ). 0 Ys E[ST |Ft ] = S0 E[YT |Ft ] + E YT 0 t a ds + YT Ys T t T a ds|Ft Ys E YT |Ft ds Ys E[YT ? s ]ds t = S0 E[YT |Ft ] + 0 a dsE[YT |Ft ] + a Ys t t T = S0 Yt E[YT ? t ] + 0 t a dsYt E[YT ? t ] + a Ys T t = = S0 + 0 t a ds Yt er(T ? t) + a Ys ads Ys er(T ? s) ds S0 + 0 a Yt er(T ? t) ? (1 ? er(T ? t) ). r In particular, E[ST ] = S0 erT ? a (1 ? erT ). r (iv) Proof. t dE[ST |Ft ] = aer(T ? t) dt + S0 + 0 t ads Ys a (er(T ? ) dYt ? rYt er(T ? t) dt) + er(T ? t) (? r)dt r = S0 + 0 ads Ys er(T ? t) ? Yt dWt . So E[ST |Ft ] is a P -martingale. As we have argued at the beginning of the solution, risk-neutral pricing is valid even in the presence of cost of carry. So by an argument similar to that of  §5. 6. 2, the process E[ST |Ft ] is the futures price process for the commodity. (v) Proof. We solve the equation E[e? r(T ? t) (ST ? K)|Ft ] = 0 for K, and get K = E[ST |Ft ]. So F orS (t, T ) = F utS (t, T ). (vi) Proof. We follow the hint. First, we solve the SDE dXt = dSt ? adt + r(Xt ? St )dt X0 = 0. By our analysis in part (i), d(e? t Xt ) = d(e? rt St ) ? ae? rt dt. Integrate fr om 0 to t on both sides, we get Xt = St ? S0 ert + a (1 ? ert ) = St ? S0 ert ? a (ert ? 1). In particular, XT = ST ? S0 erT ? a (erT ? 1). r r r Meanwhile, F orS (t, T ) = F uts (t, T ) = E[ST |Ft ] = S0 + t ads 0 Ys Yt er(T ? t) ? a (1? er(T ? t) ). So F orS (0, T ) = r S0 erT ? a (1 ? erT ) and hence XT = ST ? F orS (0, T ). After the agent delivers the commodity, whose value r is ST , and receives the forward price F orS (0, T ), the portfolio has exactly zero value. 54 6. Connections with Partial Di? erential Equations 6. 1. (i) Proof. Zt = 1 is obvious.Note the form of Z is similar to that of a geometric Brownian motion. So by It? ’s o formula, it is easy to obtain dZu = bu Zu du + ? u Zu dWu , u ? t. (ii) Proof. If Xu = Yu Zu (u ? t), then Xt = Yt Zt = x  · 1 = x and dXu = = = = Yu dZu + Zu dYu + dYu Zu au ? ?u ? u ? u du + dWu Zu Zu [Yu bu Zu + (au ? ?u ? u ) + ? u ? u ]du + (? u Zu Yu + ? u )dWu Yu (bu Zu du + ? u Zu dWu ) + Zu (bu Xu + au )du + (? u Xu + ? u )dWu . + ? u Z u ? u du Zu Remark: To see how to ? nd the above solution, we manipulate the equation (6. 2. 4) as follows. First, to u remove the term bu Xu du, we multiply on both sides of (6. 2. 4) the integrating factor e? bv dv . Then d(Xu e? ? Let Xu = e? u t u t bv dv ) = e? u t bv dv (au du + (? u + ? u Xu )dWu ). u t bv dv Xu , au = e? ? u t bv dv au and ? u = e? ? bv dv ? ? u , then X satis? es the SDE ? ? ? dXu = au du + (? u + ? u Xu )dWu = (? u du + ? u dWu ) + ? u Xu dWu . ? ? a ? ? ? ? To deal with the term ? u Xu dWu , we consider Xu = Xu e? ? dXu = e? u t u t ?v dWv . Then ?v dWv ?v dWv ? ? [(? u du + ? u dWu ) + ? u Xu dWu ] + Xu e? a ? u t u t 1 ( u )dWu + e? 2 u t ?v dWv 2 ? u du ? +(? u + ? u Xu )( u )e? ? ?v dWv du 1 ? 2 ? ? ? = au du + ? u dWu + ? u Xu dWu ? ?u Xu dWu + Xu ? u du ? ?u (? u + ? u Xu )du ? ? ? 1 ? 2 = (? u ? ?u ? u ? Xu ? u )du + ? u dWu , a ? ? 2 where au = au e? ? ? ? 1 d Xu e 2 u t ?v dWv 2 ? v dv and ? u = ? u e? ? ? = e2 1 u t 2 ? v dv u t ?v dWv . Finally, use the integrating factor e u t 2 ? v dv u 1 2 ? dv t 2 v , we have u t 1 ? ? 1 2 (dXu + Xu  · ? u du) = e 2 2 [(? u ? ?u ? u )du + ? u dWu ]. a ? ? Write everything back into the original X, a and ? , we get d Xu e? i. e. d u t bv dv? u t 1 ? v dWv + 2 u t 2 ? v dv = e2 1 u t 2 ? v dv? u t ?v dWv ? u t bv dv [(au ? ?u ? u )du + ? u dWu ], Xu Zu = 1 [(au ? ?u ? u )du + ? u dWu ] = dYu . Zu This inspired us to try Xu = Yu Zu . 6. 2. (i) 55 Proof.The portfolio is self-? nancing, so for any t ? T1 , we have dXt = ? 1 (t)df (t, Rt , T1 ) + ? 2 (t)df (t, Rt , T2 ) + Rt (Xt ? ?1 (t)f (t, Rt , T1 ) ? ?2 (t)f (t, Rt , T2 ))dt, and d(Dt Xt ) = ? Rt Dt Xt dt + Dt dXt = Dt [? 1 (t)df (t, Rt , T1 ) + ? 2 (t)df (t, Rt , T2 ) ? Rt (? 1 (t)f (t, Rt , T1 ) + ? 2 (t)f (t, Rt , T2 ))dt] 1 = Dt [? 1 (t) ft (t, Rt , T1 )dt + fr (t, Rt , T1 )dRt + frr (t, Rt , T1 )? 2 (t, Rt )dt 2 1 +? 2 (t) ft (t, Rt , T2 )dt + fr (t, Rt , T2 )dRt + frr (t, Rt , T2 )? 2 (t, Rt )dt 2 ? Rt (? 1 (t)f (t, Rt , T1 ) + ? 2 (t)f (t, Rt , T2 ))dt] 1 = ? 1 (t)Dt [? Rt f (t, Rt , T1 ) + ft (t, Rt , T1 ) + ? t, Rt )fr (t, Rt , T1 ) + ? 2 (t, Rt )frr (t, Rt , T1 )]dt 2 1 +? 2 (t)Dt [? Rt f (t, Rt , T2 ) + ft (t, Rt , T2 ) + ? (t, Rt )fr (t, Rt , T2 ) + ? 2 (t, Rt )frr (t, Rt , T2 )]dt 2 +Dt ? (t, Rt )[Dt ? (t, Rt )[? 1 (t)fr (t, Rt , T1 ) + ? 2 (t)fr (t, Rt , T2 )]]dWt = ? 1 (t)Dt [? (t, Rt ) ? ?(t, Rt , T1 )]fr (t, Rt , T1 )dt + ? 2 (t)Dt [? (t, Rt ) ? ?(t, Rt , T2 )]fr (t, Rt , T2 )dt +Dt ? (t, Rt )[? 1 (t)fr (t, Rt , T1 ) + ? 2 (t)fr (t, Rt , T2 )]dWt . (ii) Proof. Let ? 1 (t) = St fr (t, Rt , T2 ) and ? 2 (t) = ? St fr (t, Rt , T1 ), then d(Dt Xt ) = Dt St [? (t, Rt , T2 ) ? ?(t, Rt , T1 )]fr (t, Rt , T1 )fr (t, Rt , T2 )dt = Dt |[? t, Rt , T1 ) ? ?(t, Rt , T2 )]fr (t, Rt , T1 )fr (t, Rt , T2 )|dt. Integrate from 0 to T on both sides of the above equation, we get T DT XT ? D0 X0 = 0 Dt |[? (t, Rt , T1 ) ? ?(t, Rt , T2 )]fr (t, Rt , T1 )fr (t, Rt , T2 )|dt. If ? (t, Rt , T1 ) = ? (t, Rt , T 2 ) for some t ? [0, T ], under the assumption that fr (t, r, T ) = 0 for all values of r and 0 ? t ? T , DT XT ? D0 X0 > 0. To avoid arbitrage (see, for example, Exercise 5. 7), we must have for a. s. ?, ? (t, Rt , T1 ) = ? (t, Rt , T2 ), ? t ? [0, T ]. This implies ? (t, r, T ) does not depend on T . (iii) Proof. In (6. 9. 4), let ? 1 (t) = ? (t), T1 = T and ? (t) = 0, we get d(Dt Xt ) = 1 ? (t)Dt ? Rt f (t, Rt , T ) + ft (t, Rt , T ) + ? (t, Rt )fr (t, Rt , T ) + ? 2 (t, Rt )frr (t, Rt , T ) dt 2 +Dt ? (t, Rt )? (t)fr (t, Rt , T )dWt . This is formula (6. 9. 5). 1 If fr (t, r, T ) = 0, then d(Dt Xt ) = ? (t)Dt ? Rt f (t, Rt , T ) + ft (t, Rt , T ) + 2 ? 2 (t, Rt )frr (t, Rt , T ) dt. We 1 2 choose ? (t) = sign ? Rt f (t, Rt , T ) + ft (t, Rt , T ) + 2 ? (t, Rt )frr (t, Rt , T ) . To avoid arbitrage in this case, we must have ft (t, Rt , T ) + 1 ? 2 (t, Rt )frr (t, Rt , T ) = Rt f (t, Rt , T ), or equivalently, for any r in the 2 range of Rt , ft (t, r, T ) + 1 ? (t, r)frr (t, r, T ) = rf (t, r, T ). 2 56 6. 3. Proof. We note d ? e ds s 0 bv dv C(s, T ) = e? s 0 bv dv [C(s, T )(? bs ) + bs C(s, T ) ? 1] = ? e? s 0 bv dv . So integrate on both sides of the equation from t to T, we obtain e? T 0 bv dv C(T, T ) ? e? t 0 t 0 T bv dv C(t, T ) = ? t s 0 e? T t s 0 bv dv ds. Since C(T, T ) = 0, we have C(t, T ) = e 1 ? a(s)C(s, T ) + 2 ? 2 (s)C 2 (s, T ), we get A(T, T ) ? A(t, T ) = ? bv dv T t e? bv dv ds = T e t s bv dv ds. Finally, by A (s, T ) = T a(s)C(s, T )ds + t 1 2 ? 2 (s)C 2 (s, T )ds. t