One basic assumption of Black-Scholes model Essay Example Essay Example

One basic assumption of Black-Scholes model Essay Example Paper One basic assumption of Black-Scholes model Essay Introduction One basic assumption of Black-Scholes model is that the stock price is log-normally distributed with constant volatility. However, in option market, does this assumption hold? In our paper, we try to show how wrong Black-Scholes is by challenging this assumption and illustrate the difference between Black-Scholes and real world. Method used to exam mispricing problem of Black-Scholes model About Mixlognormal: The probability distribution of the stock price might be made up of a mixture of two lognormal distributions, one for the possibility of an increase in share price and the other one of a decrease. In this way, we can capture the empirical distribution of stock price; its shape must be more accurate and accordingly more likely to be the same as the distribution of the company’s share price in the real world. One basic assumption of Black-Scholes model Essay Body Paragraphs (Precisely, if we can employ more lognormal distributions to obtain the possibility for the movement of share price, we would get better distribution to describe real world.) Therefore, we could simply test the accuracy of Black-Scholes model by comparison. About Data: We chose Six Continents as our target company to do our analysis and made a comparison between its mixlognormal distribution and Black-Scholes lognormal distribution. We chose Feb 19th as the big event date for the company, because there was a takeover bid from the management of Pizza Express on that day. The share price of Six Continents jumped more than 10% on Feb 19th and the trading volume increased more than 200%. The dates before and after the big event date are Feb 18th and Feb 20th respectively. Following this way, the mispricing drawback of Black-Scholes might be detected more easily due to the noise of big event. Interpretation of the results Using the Excel VBA programme, we got the Black- Scholes lognormal distribution and mixlognormal distribution for these two particular days. The lognormal distribution of Black-Scholes is shown in blue and the mixlognormal distribution is shown in pink. Figure 1 Mixlognormal distribution and lognormal distribution for Six Continent options on 18th, Feb 2003. Figure 2 (Figure 1 and 2 are consistent with each other. Both illustrate the option is mispriced by Black-Scholes.) Comparing these two distributions (i.e. figure 1) of the date of Feb 18th, we find that: 1, the mixlognormal distribution has a fatter left tail and thinner right tail than the lognormal distribution (This phenomenon is also supported by empirical evidence when some people analysed the S;P 500 index), and 2. Both sides of the tails in mixlognormal distribution are longer as well. 3. The right tail is longer than the left tail. For the first finding, it means that in the more accurate scenario, the volatility used to price a low strike price option (i.e. a deep-out-of-the-money put or a deep-in-the-money call) is higher than that used to price a high-strike-price-option (i.e. a deep-in -the-money put or a deep-out-of-the-money call). The second finding means that the Black-Scholes lognormal distribution chops some small probability events, but in fact, these small probability events do exist in real world. The explanation of the third finding is that the bid proposal for Six Continent leaked before the announcement date. Investors have taken the probability of takeover into consideration and predict the price will go up. The probability of the share price going up is bigger than the probability of its going down, the right tail is longer. Figure 3 Mixlognormal distribution and lognormal distribution for Six Continent options on 18th, Feb 2003. Figure 4 Comparing the mixlognormal figure with the lognormal one (i.e. figure 3) of the date of Feb 20th, we find that: 1. This relationship of the two distributions is similar to those of the date of Feb 18th. 2. The left tail is still a little bit fatter and the right side a little thinner in the mixlognormal than that of lognormal. 3. The right tail is longer than of the left tail. The interpretations of this figure are the same to the above ones. From the above comparison and analysis, we can find the reason that Black-Scholes model is often shown to misprice out-of -the money and in-the-money options relative to their at-the-money counterparts: the assumption, which states that future stock price have a lognormal distribution and equivalently future returns have a normal distribution (which both results from the assumption of stock prices following a geometric Brownian motion.), is invalid. If this assumption is neglected, the risk neutral probability density does not necessarily fall into the family of lognormal distributions but can be shaped differently. This implies, that lognormal parametric estimation of risk-neutral probabilities leads to wrong result. When we move the two mixlognorma l distributions (i.e. figure 6) together and also the two Black-Scholes distributions (i.e. figure 5) together, it is obvious to find that beside the mean of the share price moving to high level, the volatility shape changes a little. The left tail of figure before the big event dates is much fatter than the left tail of the figure after the big event date. Figure 5 Lognormal distributions of Six Continent options on 18th and 20th of Feb 2003. Figure 6 Mixlognormal distribution of Six Continent options on 18th and 20th of Feb 2003. This is understandable if we accept the explanation for the volatility smile when concerns leverage. It says that as a company’s share price declines in the value, the company’s leverage increases. Thus the equity of the company becomes more risky and its volatility increases. When the company’s share price increases, leverage decreases, then the equity become less risky and its volatility decreases. Since the bid for the Six Continen ts was accepted as favourable news for the shareholders, the share price had a big jump. According to the above explanation, its shares become less risky and thus lower volatility, so the left tail becomes thinner. Conclusion We estimated risk neutral density of equity option prices and compared mixlognormal distribution and lognormal distribution before and after big event. We found that the assumption (stock prices are log-normally distributed) of Black-Scholes does not hold in the real world and this hole can make the options mispriced. Appendix 1: The raw data of lognormal distribution for Six Continent options on 18th, Feb 2003. Trade 18-Feb-03 Maturity 16-Apr-03 r 0 T 0.1561644 F 554.5 sigma 0.4091905 mkt call strikes BS theory Sq Error moneyness weigted sq error 196 360 194.58 2.01 194.5000 0.0103 166.5 390 164.89 2.59 164.5000 0.0157 138 420 135.86 4.57 134.5000 0.034 101.5 460 99.5 3.98 94.5000 0.0421 69 500 67.93 1.14 54.5000 0.0211 38 550 37.88 0.01 4.5000 0.0032 17.5 600 18.77 1.62 45.5000 0.0356 7 650 8.34 1.8 95.5000 0.0188 2.5 700 3.36 0.75 145.5000 0.0051 1 750 1.25 0.06 195.5000 0.0003 0.5 800 0.43 0 245.5000 0 18.53 0.1862 Appendix 2: The raw data of lognormal distribution for Six Continent options on 20th, Feb, 2003. Trade 37672 Maturity 37727 r 0 T 0.15068493 F 615.5 sigma 0.36813434 mkt call strikes BS theory Sq Error moneyness weigted sq error 256.5 360 255.5013903 0.997221354 255.5 0.003903019 227 390 225.5132704 2.210364876 225.5 0.009802062 197 420 195.5835263 2.006397745 195.5 0.010262904 157.5 460 156.0819138 2.010968563 155.5 0.012932274 119.5 500 118.0715084 2.040588233 115.5 0.017667431 76.5 550 75.70383997 0.633870793 65.5 0.009677417 42.5 600 42.91727918 0.174121912 15.5 0.011233672 21 650 21.38061239 0.144865789 34.5 0.004199008 9 700 9.401761762 0.161412513 84.5 0.001910207 3.5 750 3.688784898 0.035639738 134.5 0.000264979 1 800 1.308782808 0.095346823 184.5 0.000516785 0.5 850 0.425705755 0.005519635 234.5 2.35379E-05 10.5163 1797 0.082393295 Appendix 3: The raw data of mixlognormal distribution for Six Continent options on 18th, Feb, 2003. Trade 18-Feb-03 Maturity 16-Apr-03 r 0 T 0.1561644 F 554.5 F1 613.42684 sigm1 0.0050015 F2 545.13413 sigma2 0.4331614 p 0.1371431 Cmarket X Cimplied (Cmarket-Cimplied)^2 Implied sigma by BS 196 360 194.6650299 1.782145105 0.605651935 166.5 390 165.1475618 1.82908915 0.53766516 138 420 136.4577704 2.378472184 0.502173454 101.5 460 100.7421585 0.574323782 0.454991983 69 500 69.60810609 0.369793012 0.424942163 38 550 38.86792249 0.753289445 0.410555737 17.5 600 17.00376787 0.246246324 0.393289282 7 650 6.928636732 0.005092716 0.38547187 2.5 700 2.914632833 0.171920386 0.382378979 1 750 1.14301875 0.020454363 0.394114927 0.5 800 0.422795028 0.005960608 0.417111769 8.136787075 Appendix 4: The raw data of mixlognormal distribution for Six Continent options on 20th, Feb 2003 Trade 20-Feb-03 Maturity 16-Apr-03 r 0 T 0.1506849 F 615.5 F1 631.64945 sigm1 0.005 F2 613.47651 sigm a2 0.3942533 p 0.1113462 Cmarket X Cimplied (Cmarket-Cimplied)^2 Implied sigma by BS 256.5 360 255.5039835 0.992048772 0.684121728 227 390 255.5291768 2.163320768 0.634591317 197 420 195.6487533 1.825867624 0.544725925 157.5 460 156.3364319 1.353890697 0.457835271 119.5 500 118.6654525 0.696469529 0.411053745 76.5 550 76.50974266 9.49E-05 0.380098628 42.5 600 42.7445391 0.059799374 0.363618501 21 650 20.42471433 0.330953606 0.363939585 9 700 9.572591543 0.327861075 0.362150688 3.5 750 4.077689599 0.333725272 0.363395708 1 800 1.597153827 0.356592693 0.351554917 0.5 850 0.581926491 0.00671195 0.376206304 8.447336279 We will write a custom essay sample on One basic assumption of Black-Scholes model Essay Example specifically for you for only $16.38 $13.9/page Order now We will write a custom essay sample on One basic assumption of Black-Scholes model Essay Example specifically for you FOR ONLY $16.38 $13.9/page Hire Writer We will write a custom essay sample on One basic assumption of Black-Scholes model Essay Example specifically for you FOR ONLY $16.38 $13.9/page Hire Writer

The Process of Environmental Scanning Essay Example

The Process of Environmental Scanning Essay Example The Process of Environmental Scanning Paper The Process of Environmental Scanning Paper Environmental scanning is a key activity that begins a companys strategy planning process. The scanning must include both external forces like economic, technological, political-legal, and coloratura as well as internal competencies such as value, rareness, immutability, and organization in order to be effective (Wheeled Hunger, 2010). This thorough analysis will provide a foundation for a company to build their plan upon. Environmental scanning provides a company their strengths to build upon, their weaknesses to mitigate, their opportunities to seize, and their threats to be careful of. Large companies like Microsoft and Amazon are still operating and successful today because they take the time and resources to conduct frequent environmental scans to keep their strategic plans relevant to the ever evolving competitive market. Internal and External Environments Environmental scanning shows that Microsoft Corporation would have typical internal environments such employee turnover and retention, employee morale, and standard financial planning. Specific to Microsoft would be hiring and retaining employees that will help keep Microsoft at the cutting edge of technology. Bill Gates (2014) vision statement states Our vision is to create innovative technology that is accessible to everyone and that adapts to each persons needs. Accessible technology eliminates barriers for people with disabilities and it enables individuals to take full advantage of their capabilities. This statement has a significant impact on internal environments and strategic planning. External Environments for Microsoft Corporation include technology advancement within the industry, competition such as Apple and Google, and in recent years the government and anti-trust lawsuits. Microsoft has invested and pursued a significant amount of vertical growth and has been criticized for creating a monopoly as well. Internal environments for Amazon are similar to those of Microsoft. Amazons mission statement is a very high reaching goal, to be Earths most customer-centric company, where customers can find and discover anything they might want to buy online, and endeavors to offer its customers the lowest possible prices, (Amazon, 2014). This is the main Internal Environment specific to Amazon as this drives many of the strategic planning as it relates to internal operations. External Environments for Amazon are customer satisfaction, new vendors, new products, and competition. Competition is probably the largest External Environment due to Amazons broad offering of products; this reaches into almost every industry worldwide and as Amazon increases its offerings it increases the competition as well. Competitive Advantages and Strategies Like many other companies that achieved success in the business world, Microsoft rose to the top of the software and computer business via the right strategy for ensuring a competitive advantage. The Microsoft Corporations competitive advantages include size and stability. The single most important strategy that the Microsoft Corporation went after that was pivotal for quick profits is research and development (RD), which helped the Microsoft Corporation to grow. Growth is a popular strategy because larger businesses tend to survive longer than smaller companies due to the greater availability of financial resources, organizational routines, and external ties (Wheeled Hunger, 201 0, Chapter 7). The strategies that the Microsoft Corporation now uses include a strong brand name, comprehensive product portfolio, inclusion, and global diversity as a strategy for success (Wheeled Hunger, Chapter 8, 2010). Amazons competitive advantages include investing and innovation. The strategies that Amazon uses include a marketing strategy that concerns sales, distributions, and pricing of goods. Johnson (2014), Amazon at its roots is built to transform. When it finds opportunities to serve new customers, or existing customers in new ways, it conceives and builds new business models to exploit them. Amazon has the unique ability to launch and run entirely new types of businesses while simultaneously extracting value from existing businesses (Para. 7). Amazon innovates, changing the companys business model to maintain a competitive advantage. Creating Value Amazon sustains competitive advantage by offering the convenience of shopping online and having products delivered right to the customers door. They also offer a wide range of products from multiple sellers that can offer lower prices. Amazon also specializes in fast shipping and customer satisfaction. Burrs (2003), In order to have a lasting competitive advantage, it is important to develop a competitive strategy that includes a wide spectrum of techniques to gain advantage. The internet allows customers to search for a product and where they can buy it at the lowest price. If it is levered within two business days, that is more appealing. This service will keep customers returning for future purchases. Amazon has something for everyone, from books to clothing, to housewives and grocery items. Someone who is busy and does not have time to shop can do so while at work or on the go without having to go to the store. According to Microsoft Customer and Partner Experience (2014), Microsoft strives to continually create innovative technology products that transform the way people work, learn, play, and communicate. Microsoft has been one of the leaders in the computer software industry. They offer a wide variety of products from software to tablets and cell phones. Their variety of products can be used for personal and business use. They sustain competitive strategy and create value by creating new products and services that can be used at school, work, or for play. Customers are always looking for the newest technology, the fastest computer, and the best prices. Microsoft continually competes with other companies in the technology industry by making their prices reasonable for the company but affordable for the customer. Measurement Guidelines The measurement guidelines that the Microsoft Corporation is using to refry its strategic effectiveness are balanced scorecards. The balanced scorecard is a proven performance measurement system. It is a complete long-term performance management system and methodology. It is a fact that defines, refines and shares strategy, for expressing strategy to operational terms, and for sizing the success of strategy implementation. The balanced scorecard initiative is very successful in the noticeable guidelines for the Microsoft Corporation. They meet the standards that represent the period of events in its organizations gradual development even though it is challenging and relating to the process of cultural change. Microsofts success is defined, with many facets approaches that support education, communication, scorecard development, and ongoing execution. (Bloomfield, c. , 2002) The measurement guidelines that the Amazon is using to verify its strategic effectiveness is its feasible and original business model that values its customers, and builds around the changing market. It is a plan of action and a thorough formula for profit, which turned around the respectable industry of books. Amazon continued to expand beyond just books and they now include all sorts of goods and services increasing from its central into near contiguity. Years later Amazon took hold of its empty space when it invented a new valued plan of action offering a charge service for brokers, buyers, and used books sellers. Once again it moved into its empty space by evolving a system to render assistance to distinct customers, and third-party sellers. Amazon has transformed its business from a store-front to precise types of service or goods to several different on-site departments as well as to outside beings which forms into a class of many sellers under one network, and it receives commissions sales from other companies. (Johnson, M. W. , 2014) Microsoft and Amazon are successful and profitable because of their ability to scan the environment, create value and sustain competitive advantages, and verify their strategies effectiveness through measurements. They incorporate all of these elements into their strategic planning process. They are able to exploit their strengths and keep pace with the changing technology market and stay one step ahead of the competition. New products and services are regularly developed and tested before brought to market. Environmental scanning is part of the reason these companies have flourished where others have failed.

Jet Blue - Strategic Management Assignment Example | Topics and Well Written Essays - 1000 words

Jet Blue - Strategic Management - Assignment Example Jet Blue embarked on an ambitious expansion plan, adding new planes and routes much faster than it added passengers. With the added capacity for more passengers, but customer base falling due to competition, Jet Blue started to face its most crucial problems. The profits started declining until in 2006 Jet Blue reported its first ever loss. Jet Blue could not utilize its planes to full capacity to meet the increasing fuel charges. Till now, Jet Blue hasn't been able to recover fully although the company is profitable again. (Press Release - Jet Blue, 2006) (Kahn, 2004) The shareholders of a company want higher returns on the shares in the form of dividends and value of the share. On the other hand, a community does care what returns the shareholders get. They want the company to conduct a business that either gives a positive return to the community or does not hinder the community in either way. To balance between these two, a company must realize that most of the shareholders are a part of the community; hence they would want the well being of the community as well as their own interests in the company. Therefore a company should concentrate more on the needs of the community which eventually would be in the best interests of the shareholders as well. 2 - Getting loyal customers to put out a phone "request for proposal" soliciting competitors' bids - This is ethical since you don't your the customers actually don't need the bids, therefore, this practice is not encouraged. 5 - Buying competitors' products and taking them apart - after buying something, it becomes your property (unless stated otherwise) therefore taking it apart to analyze is okay. 2 - Hiring management consultants who have worked for competitors - consultants are not your employees thus this practice is not appropriate since it would be unethical if consultants give some information about the competition