Pricing of derivatives on mean-reverting assets. Are stock prices and returns are mean reverting or not - Essay Example

?Mondale 2022 XXXXXX XXXXXXX XXXXXX XXXXXX XXXXXX Table of contents CHARPTER 6 1. INTRODUCTION …………………………………………………………. 6 1.1 BACKGROUND …………………………………………………………. 6 1.2 PROBLEM STATEMENT ……………………………………………….. 6 1.3 OBJECTIVE ………………………………………………………………. 6 1.4 SCOPE OF THE STUDY …………………………………………………. 7 1.5 LIMITATIONS OF THE STUDY ………………………………………… 7 1.6 RATIONALE/JUSTIFICATION OF THE STUDY ……………………… 7 CHAPTER 2 …………………………………………………………………………… 9 2. LITERATURE REVIEW …………………………………………………… 9 2.1 Absolute mean reversion …………………………………………………… 9 2.2 Relative mean reversion ………...…………………………………………. 10 2.3 Mean reversion in stock returns …………………………………………… 11 CHAPTER 3 …………………………………………………………………………… 12 METHODOLOGY …………………………………………………………….. 12 3.1Introduction ………………………………………………………………… 12 3.2Research Design ……………………………………………………………. 12 3.3Population and Sampling …………………………………………………… 12 3.3.1 Population ………………………………………………………… 13 3.3.2 Sampling Design ………………………………………………….. 13 3.3.2.1 Sampling Frame ………………………………………………….. 13 3.3.2.2 Sampling Technique ………………………………………………. 13 3.3.2.3 Sample Size ………………………………………………………. 14 3.4 Data Collection Methods ………………………………………………...… 14 3.5 Research Procedures ………………………………………………….……. 15 3.6 Data Analysis Methods ……………………………………………………. 16 3.7 Summary …………………………………………………………………… 16 CHAPTER 4 ………………………………………………………………………….……. 17 FIGURES ………………………………...………………………………………… 17 4.1 Definition of mean reversion stock prices ……………………………….…….. 17 4.2 Permanent and transitory price components …………………………………… 17 4.3 Properties of multi-period returns ……………………………………….……... 18 4.4 One-period returns ……………………………………………………....…..…. 19 4.5 Multi-period returns ……………………………………………………………. 20 4.6 Higher-order AR model ……………………………………………………….... 22 4.7 The relation between mean reversion and covariance stationary …...………….. 23 CHAPTER 5 …………………………………………………………………………………… 25 5. Mean reversion and mean-variance efficient portfolios …………………………. 25 5.1 Outline ………………………………………………………………………….. 25 5.1.1 Expected returns and volatilities of stocks and bonds ………………... 25 5.1.2 Correlation between stock and bond returns …………………………. 26 5.1.3 Risk-free yield curve …………………………………………………. 26 5.1.4 Variance ratio of permanent and transitory returns …………….…….. 26 5.2 The role of the variance ratio …………………………………………… 26 5.6 Dealing with uncertainty about the variance ratio ……………………… 27 5.7 Trading strategies …………………………………………………….…. 28 . CHAPTER 6 ………………………………………………………………………... 30 CONCLUSSION …………………………………………………………… 30 RECOMMENDATIONS …………………………………………………… 30 REFERENCES ............................................................................................................. 31 PUBLIC RELATIONS APPROACH TO UTILITIES TO WIN MONDALE 2022 ABSTRACT A widely held belief in the world of economics is that what goes up must come down eventually. In terms of stock prices, this belief translates into the concept of mean reversion. The belief states that a decline in stock prices is most likely to remain followed by an upward price movement, and vice versa. The presence or absence of mean reversion, with regard to the issue of public relations, has important economic implications on the overall mean reversion in stock prices, with regard to Mondale as expected in 2022 induces negative autocorrelation in stock returns. The relatively low long-term volatility increases the attractiveness of stocks as a long-term investment. Furthermore if stock prices are mean reverting in the long run, low stock prices are followed by relatively high expected future returns, which could encourage long term investors such as pension funds to invest in equity after a stock market downturn . CHARPTER 1 1. INTRODUCTION 1.1 BACKGROUND Genital integrity is one of the many issues that have hit not only the contemporary society, but also the olden society in a manner that dismays many. Genital integrity is a branch of education that seeks to express the fact that each individual has a right to oppose any surgeries involving their genitals. The spread of this notion and education has however come to rise in the contemporary society even more provided the high amount of civilization that this generation has brought in. Groups on human rights have revealed the fact that surgery, mostly circumcision, happen to the males and thus the importance of going through this issue critically in an effort to bring out the reason as to why males need as much protection as females. 1.2 PROBLEM STATEMENT 1.3 OBJECTIVE The goal of this paper is not to provide a final answer to the question whether stock prices and returns are mean reverting or not. Instead, the research aims at making investors aware of the economic consequences of mean reverting behavior of stocks. This lies in regard with the Mondale as of 2022. The remainder of this paper therefore focuses on describing relevant properties of mean reverting stock prices and the resulting implications for long-term investors (Bekaert, 1999, 110). The challenges we faced during our research were difficulties in getting required data from the three companies, failure to secure appointments with some company managers and high costs. 1.6 RESEARCH FINDINGS AND DISCUSSION Although the mean-reverting price process defined later by Equations (1), (2), and (3) may seem restrictive, they are more general than is apparent at first sight. This generality explains the model’s suitability as a tool for describing the mean reverting behavior of stock prices. To allow the price process to be consistent with the efficient market hypothesis, the random walk should lie nested in the specification for the stock price (Frankel, 1995, 140). This explains why the permanent price component in the first equation lies chosen to be a random walk. The price process in Equation (1) follows a random walk for o= one, but deviates from the efficient market hypothesis for zero < o < 1. Hence, only the choice for a first-order autoregressive (AR) process could possibly be restrictive (Bekaert, 1999, 129). A seemingly more general specification defines zt with regard to the first equation as a covariance-stationary, mean-reverting process with mean 0.But every covariance stationary series can stand written as a moving average (MA) process of infinite order. If the MA process is invertible, it can be written as an AR process of infinite order, which brings us one-step closer to our AR (1) process. The only restrictive aspect of the first-order AR process is its order. We may therefore want to consider a generalization of the previously considered mean reverting price process by relaxing the assumption of a first order AR process for the transitory price component. Instead of an AR (1) process we could assume an AR (p) process as approximation of an AR process of infinite order, for any p= one, 2… (Bekaert, 1999, 65). The ability of the stock price model to allow for mean reversion in stock prices with regard to Mondale as expected in 2022, in combination with the simplicity of the specification based on a first order autoregressive transitory price component. It explains why most studies confine the analysis of the transitory price process to a first order autoregressive model. CHAPTER 2 2. LITERATURE REVIEW Since the seminal studies by researchers, an ongoing debate has emerged in the literature as to whether stock prices are mean reverting or not. Testing for mean reversion is difficult because of the limited supply of historical data on stock prices. Precise estimation of the degree of long-run mean reversion requires very long data series, which are not available. Moreover, structural breaks in the behavior of stock prices are likely to occur during long sample periods, and as a result complicate the statistical analysis of mean reversion. These issues explain why even after twenty years of mean reversion research, it is still difficult quantify the degree of mean reversion in stock prices with regard to Mondale as expected in 2022 (Poterba, 1987, 102). Two different methods have lied used in the literature to test for mean reversion. The first approach tests for mean reversion. This is in a way that does not require estimation of the fundamental value process pt* with regard to the first equation. This method lies known as the approach of absolute mean reversion. The second method however, proceeds in a relatively different way. It starts with the estimation of the fundamental value process pt*. This method stands as relative mean reversion, as it has stock prices reverting relative to a specified mean value. We will explain both methods in more detail in this section. We will also review some recent studies analyzing mean reversion in stock returns (Weron, 2007, 154). 2.1 Absolute mean reversion Prior research has helped derive a regression model to test whether the autocorrelation pattern in stock returns is consistent with the model defined by Equations (1), (2) and (3).For values of o close to unity, the relative negative-autocorrelation in stock returns is stronger for long-horizon than for short-horizon returns. Particularly, researchers therefore examine several investment horizons between one and ten years. This approach establishes significant mean reversion, explaining 25% – 40% of the variation in the 3 – 5 year stock returns (Frankel, 1995, 201). In prior research, there exists use of a specific property of the random walk to test for mean reversion. Mean reversion of stock prices implies that the variance of stock returns grows less than proportionally with time. The m-year variance ratio lies defined as the ratio of the m-year return variance to the one-year return variance divided by m. When this ratio lies equal to one, the random walk hypothesis cannot stand rejected. Researchers find mean reversion over long investment horizons in the United States. Similar results have remained established for several developed countries. The lack of importance in their results lies attributed to the absence of more powerful tests to reject the null hypothesis. 2.2 Relative mean reversion The lack of evidence for mean reversion often lies attributed to small sample sizes in combination with statistical tests for mean reversion that lack power. A significant improvement in estimation accuracy may stand achieved by explicitly specifying the fundamental value process around which the mean reversion occurs. The question here is how to proxy the fundamental value process, which is intrinsically unobserved. According to the Gordon growth model, the relative value of a stock equals the discounted future cash flows generated by the stock. In practice, these cash flows are the dividends to stand paid out to the owners. The authors conclude that financial ratios revert to their long-term average value. In earlier work, researchers link the dividend yield to the expected returns on a stock and find that the latter have a mean-reverting tendency (Frankel, 1995, 243). 2.3 Mean reversion in stock returns Predictability of stock returns from dividends or other fundamental factors may give rise to negative autocorrelation in stock returns. Negative autocorrelation, mainly in stock returns generally stands referred to in the literature as mean reversion in stock returns. In order to understand the issue in a rather concise manner, it is important to understand the various issues that girls have. In the traditional society, there was the widespread issue of Female Genital Mutilation. The society hailed this from the fact that it was illegal for a young woman to copulate before her marriage day. They thus sought to circumcise the girls in an effort to reduce their sexual urges. Many human rights groups of the contemporary society have come up with strict criticism on the issue stating that this carried out without the consent of the girl and thus leads to their not enjoying copulation. It is important to note that when this advocacy is going on, that none or very unsuccessful human rights groups protest against the circumcision of young boys. According to the concept of genital integrity, it is not right to induce surgery on a person’s body without their consent. However, circumcision is a surgery induced on young boys whose decisions are by their parents. Many are the times that people state that this has been a tradition well accepted by society and thus the necessity for its continuity. However, this is completely opposite from the values integrated by concepts of genital integrity. Te young boys have a right to decide as to whether they want to go through the process or not. Human Rights groups should also intensify campaigns aimed at reducing circumcision of boys. CHAPTER 3 METHODOLOGY 3.1 Introduction The research methodology comprises of research design to lie in use, description of the population and the sampling design, data collection methods, research procedures and data analysis methods. 3.2 Research Design It refers to the practical way in which the research lay conducted according to a systematic attempt to generate evidence to answer the research question. This case study involves a descriptive research on mean reversion in stock prices with regard to Mondale as expected in 2022, as well as implications for long term investors. Descriptive research is concerned with finding out what, who, when, where or how much. 3.3 Population and Sampling 3.3.2 Population From the Kenyan market, the study shall focus on Energy and Petroleum industry. In the study, researchers analyze the share prices for the following companies KenGen Co. Ltd, KenolKobil Ltd, and Kenya Power & Lighting Co. The limitation of the scope is that most companies could not provide us with required data thus a small sample. 3.3.3 Sampling Design The sampling design comprises of sampling frame, the sampling technique and the sampling size as described below for this proposal. 3.3.3.1 Sampling Frame Sampling is the process of selecting units from a population of interest so that by studying the sample we may fairly generalize our results back to the population from which they lay chosen. A sampling frame is the source material or device from which a sample is drawn. There however exists the stress for the importance of the sampling frame. 3.3.3.2 Sampling Technique Sampling may lie done either a probability or a non-probability basis. This lies as an important research design decision, and one that will depend on such factors as whether the theory behind the research is idealist or positivist, whether qualitative or quantitative methods lie used. A probability sampling method is any method of sampling that utilizes some form of random selection. These days, we tend to use computers as the mechanism for generating random numbers as the basis for random selection (Poterba, 1987, 118). A non-probability sampling does not involve random selection. That however, does not mean that non-probability samples are not representative of the population. However, it does mean that non-probability samples cannot depend upon the rationale of probability theory. 3.3.3.3 Sample Size In purposive sampling, this lies determined by judgment. Mainly in other more random types of sample, it remains calculated as a proportion of the sampling frame, the major criterion being to ensure that it is representative of the whole. While using stratified sampling, one may need to adjust your strata and collapse into smaller strata if you find that some of your sample sizes are too small. 3.4 Data Collection Methods Data collection means gathering information to address those critical evaluation questions that you have identified earlier in the evaluation process. There lie many methods, available to gather information, as well as a wide variety of information sources. The information you collect is the evidence you will have available to answer the evaluation questions. Poor evidence is information that is scant, cannot be trusted, or simply is not relevant to the questions asked. Superior evidence is information that comes from reliable sources and through trustworthy methods that address important questions. With reference to the topic in question, there exists two general types of information. These are descriptive and judgmental. The descriptive information can include the following examples: Characteristics of the project, reports of the accomplishments related to the project, current knowledge levels of project personnel and the target audience, amount of the relative participation by the target audience, policies concerning cost share, types of participants and demographic data. Judgmental information in the same regard may include varying examples such as, Opinions from experts or consultants, consumer preferences, target audience’s beliefs and values, technical agency personnel’s interpretation of laws, stakeholder’s perceived priorities, and farmers’ interpretation of guidelines. In Pilot Testing, and with regard to Wong, (2009, 287), One will need to test the process he or she designs or the information collection instrument, no matter the form or structure. As a result, one ought to plan time for this step and for any revisions that may result from this testing. With regard to the issue of bias, and the manner in which it is likely to apply in the case, Bias means to be prejudiced in opinion or judgment. Bias can enter the evaluation process in a variety of ways. For example, if you use a self-selected sample in cases where a person decides to participate in a study, rather than lying picked randomly by the researcher, how might these respondents be different from the people that chose not to participate? In matters pertaining reliability, the question at hand is whether the evaluation process you have designed consistently measure what you want it to measure. If you use multiple settings, interviews, or observers, will they consistently measure the same thing each time? If one designs an instrument, will people interpret the questions the same way each time? With reference to the issue of validity, the has to be the opinion on whether the information collection methods in designed, produce information that measures what lies to be measured. One ought to be sure that the information you collect is relevant to the evaluation questions you are intending to answer. 3.5 Research Procedures According to a researcher by the name Poterba, the underlying principle behind the guidelines suggested below is to ensure the validity of your instrument by making sure that your questions relate to the objectives of your study. Step 1 is to clearly define and individually list all the specific objectives or research questions for your study. Step 2 is for each research objective or questions, list all the associated questions that you want to answer through your study Step 3 is to take each research question listed in step two and list the information required to answer it and step four is to formulate questions to obtain this information. Having formulated the research problem, developed a study design, and selected a sample, constructed a research instrument, you then collect the data from which you will draw inferences and conclusions for your study. Depending upon ones plans, one might commence interviews, conduct experiments, mail out a questionnaire, or make observations. Data collection through any of the methods may involve some ethical issues in relation to the participants and the researcher. 3.6 Data Analysis Methods Analysis lie carried out to describing relevant properties of mean reverting stock prices and the resulting implications for long-term investors 3.7 Summary The research methodology chapter describes five key areas of a project proposal, which are the research design to lie in use, description of the population and the sampling design, data collection methods, research procedures and data analysis methods. CHAPTER 5 5. Mean reversion and mean-variance efficient portfolios This section assesses the economic implications of mean reversion in stock prices with regard to Mondale as expected in 2022 for long-term investors. The starting point exists as a mean variance efficient investor that determines its optimal portfolio weights solely based on the mean and covariance structure of the asset returns in its investment set. An investment mean variance efficient investor will hold the portfolio with the lowest volatility for a given level of the expected portfolio return. Similarly, it would also expect the same for the portfolio with the highest expected return for a given level of the portfolio volatility. 5.1 Outline We consider a long-term investor that wants to divide its wealth between stocks and bonds in mean variance optimal proportions. To construct a relevant example, we base the parameter values on historical data. 5.1.1 Expected returns and volatilities of stocks and bonds We assume the following values, corresponding to monthly stock returns µ=0.9% (expected return of stocks), o= 0.975 (reflecting a speed of mean reversion of 2.3 years), ?=3.2% (volatility of the error term in the transitory price component), and T=3.2% (volatility of the error term of the permanent price component). For the bond index, we assume a monthly-expected return of 0.7% and a volatility of 1.4%, the latter index is investable through various Exchange Traded Funds and Exchange Traded Notes. To detach the effect of mean reversion in stock prices with regard to Mondale as expected in 2022 on the optimal portfolio weights, we exclude a priority for the presence of mean reversion in the bond returns. 5.1.2 Correlation between stock and bond returns The monthly contemporaneous correlation between the stock and bond indices lies assumed 0.2, which equals the historical correlation between the aforementioned stock and bond indices during the sample period. The multi-period correlations lie assumed 0.18 for one year, 0.17 for 5 years, 0.17 for 10 years, and 0.17 for 20 years. 5.1.3 Risk-free yield curve The risk-free rate lies based on the nominal interest rate term structure. 5.1.4 Variance ratio of permanent and transitory returns Findings from research assume that the variance of the permanent and transitory price components contribute equally to the total stock return variance, with regard to a choice that has been motivated by the results of prior research. In the subsequent analysis, we define the variance ratio as the return variance of the permanent price component divided by the return variance of the transitory component (Right, 2001, 119). 5.2 The role of the variance ratio Our sensitivity analysis shows that the choice of the variance ratio may have substantial impact on investment decisions. If at all, the variance ratio proves to be high meaning that stock prices are strongly mean reverting stocks, become relatively less risky in the end, making it optimal to invest a relatively large share of wealth in stocks. However, if the true variance ratio is lower than the assumed value, the supposed risk exposure is lower than the actual risk exposure. Hence, too much wealth lies allocated to stocks, resulting in a non-optimal overexposure to risk. As noted by previously carried out research, estimation of the variance ratio is subject to estimation is only possible in an indirect way and requires very long sample periods owing to the slowly moving nature of the transitory price component. Available stock return series are relatively short estimates, which suffer from large parameter uncertainty. Precise assessment of the parameter uncertainty lies complicated by the use of overlapping returns (Weron, 2007, 197). 5.6 Dealing with uncertainty about the variance ratio Given the uncertainty involved with the variance ratio, it is prudent for a risk-averse investor to use a conservative estimate of this ratio. To see this, suppose that an investor is uncertain about the degree of mean reversion in stock prices with regard to Mondale as expected in 2022. If it overestimates the mean reversion in stock prices with regard to Mondale as expected in 2022, this will result in an overexposure to risk. In an adverse scenario with falling stock prices, this would end result in negative investment returns. Conversely, if the investor underestimates stock price mean reversion, it will cause less exposure of risk. This is amid rising stock prices, this would end result to too low investment returns. Because the investor is risk averse, it will always exhibit a preference for the risk of earning too low profits to the risk of incurring high losses. The strength of the relative investor’s preference was observed to depend on its success (Robbins, 2007, 87). With regard to the degree of risk averseness, for every risk adverse investor it is optimal to base the portfolio weights on an assumption of no or little stock price mean reversion. For an investor that is only little risk averse, by contrast, it is most favorable to assume a higher degree of mean reversion. In total, it is an optimal policy for any risk averse investor to make conservative assumptions about the degree of stock price mean reversion (Right, 2001, 142). 5.7 Trading strategies Several studies propose trading strategies based on mean reversion in stock prices with regard to Mondale as expected in 2022 and show that these strategies yield excess returns. The latter study establishes a half-life of 3and a half years, in a model of relative mean reversion. The authors consider a contrarian trading strategy, which lies composed of loosely speaking, of buying past losers en selling past winners (Wong, 2009, 143); based on rolling window estimates of the underlying mean reversion model. They show that such a strategy is able to generate risk-adjusted excess returns. Clearly, if stock prices were to follow a random walk, it would not be probable to earn excess returns. Hence, not only does the optimal asset allocation affect the degree of stock price mean reversion, instead, the profitability of trading strategies crucially depends on it as well. As highlighted in the previous chapter, in today’s financially tyrannical conditions, investors are seeking out higher returns. In fixed income, one relative option is that to move along the relative yield curve. On the contrary, this involves maturity risk. Bekaert, & Grenadier view another strategy as looking beyond safe-haven supreme bonds, at emerging markets, distressed sovereigns, and corporate and high yield bonds, but this involves credit risk as narrated by Bekaert (1999, 110). Investors can also look at real assets, but again these are relatively risky investments. Where there are risks, there exist frequent rewards as well. It was clear in the last chapter that the equity premium is large. A straightforward way of enhancing projected returns is to increase equity weightings. In the short run, the risks are proportionally large. Nevertheless, there exists a seductive argument that says equity risk falls the longer the investment horizon, a supposed corollary to the advice which investors ought to take a long-term view. CHAPTER 6 CONCLUSSION There has been an ongoing debate as to whether stock prices and stock returns are mean reverting in the end. In the research, we have discussed the implications of mean-reverting behavior in stock prices for long-term investors. We showed that the variance of stock returns increases less than proportionally with the investment horizon if stock prices are mean reverting. If stock prices are mean reverting, stocks are relatively less risky for longer investment horizons, so that a larger share of wealth may lie allocated to stocks. The same case scenario applies if stock returns show negative autocorrelation, which is often referred to in the literature as mean reversion in stock returns, in which case stock prices are not necessarily mean-reverting (Wong, 2009, 287). RECOMMENDATIONS Given the impact of mean-reverting behavior of stocks on asset allocation decisions and the profitability of trading strategies, it is imperative for investors to know whether stock prices and stock returns exhibit mean reversion. Until now, the literature has not yet found strong evidence in favor of mean-reverting behavior, but this may also be due to the difficulties involved in the empirical assessment of mean reversion. Hence, it is unclear whether stock prices and stock returns are mean reverting or not. This is with regard to the fact that the investor will underestimate the risk exposure of stocks if he or she overestimates the degree of mean reversion. It seems practical for a risk-averse investor to base investment decisions on conservative assumptions regarding the mean-reverting behavior of stocks. References Bekaert, G., & Grenadier, S. R. 1999. Stock and bond pricing in an affine economy. Cambridge, MA: National Bureau of Economic Research. Frankel, J. A., & Rose, A. 1995. A panel project on purchasing power parity: mean reversion within and between countries. Cambridge, MA: National Bureau of Economic Research. Lutz, B. 2010. Pricing of derivatives on mean-reverting assets. Heidelberg: Springer. Poterba, J, & Summers, L. H. 1987. Mean reversion in stock prices: evidence and implications. Cambridge (1050 Massachusetts Avenue, Cambridge, Mass. 02138): National Bureau of Economic Research. Right, S. 2001. Shuttle: Mondale's SST Flies. Nature, 232(5306), 82-82. Robbins, P., & Publications, i. 2007. Encyclopedia of environment and society. Thousand Oaks: Sage Publications. Weron, R. 2007. Modeling and Forecasting Electricity Loads and Prices a Statistical Approach.. Chichester: John Wiley & Sons. Wong, H. Y., & Lo, Y. W. 2009. Option Pricing With Mean Reversion And Stochastic Volatility. European Journal of Operational Research, 197(1), 179-187. Read More