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INTRODUCTORY MATHEMATICS AND STATISTICS FOR ISLAMIC FINANCE – Book Sample
Preface – INTRODUCTORY MATHEMATICS AND STATISTICS FOR ISLAMIC FINANCE
The objective of this book is to provide an introductory and uniﬁed training in mathematics and statistics for students in Islamic ﬁnance. Students enrolled in Islamic ﬁnance programs may have had different training in mathematical and statistical methods.
Some students may have advanced training in some mathematical or statistical topics; however, they may not have been sufﬁciently exposed to some topics that are highly relevant in Islamic ﬁnance or to the applications of quantitative methods in this ﬁeld. Other students may have had less advanced quantitative training.
It will be therefore necessary to provide a homogenous quantitative training in mathematics and statistics for students, with a view to enhancing their command of the theory and practice of Islamic ﬁnance.
In view of the nature of Islamic ﬁnance, students or professionals should acquire adequate skills in computational mathematics and statistics in order to accomplish their duties in any ﬁnancial or nonﬁnancial institutions where they might be employed.
Without computational skills, students or professionals may not be able to manipulate economic and ﬁnancial data; they may not meet the challenges of their ﬁnancial career. In fact, the ﬁnance industry has reached an extremely advanced stage in terms of the quantitative methods, computerization, product innovations, and arbitrage and trading programs that are used.
Many institutions such as hedge funds, pension funds, investment corporations, insurance companies, and asset management companies require advanced knowledge in actuaries, and models of investment and risk management. Professionals have to satisfy the standards required by these institutions and be able to use software, such as Microsoft Excel, EViews, Mathe- matica, MATLAB, and Maple, to process data and carry out computational tasks.
The Internet is rich in the use of computational tools. A student can plug in data and get instantaneous answers; however, it is important for a student to understand the theory underlying the computational procedures.
While existing books on ﬁnance cover the topics of mathematics or statistics only, this book covers fundamental topics in both mathematics and statistics that are essential for Islamic ﬁnance. The book is also a diversiﬁed and up-to-date statistical text and prepares students for more advanced concepts in mathematics, statistics, and ﬁnance.
Although most of the mathematical and statistical books concentrate on traditional mathematics or statistics, this book uses examples and sample problems drawn from ﬁnance theory to illustrate applications in Islamic ﬁnance. For instance, a student will be exposed to ﬁnancial products, asset pricing, portfolio selection theory, duration and convexity of assets, stock valuation, exchange rate pricing, and efﬁcient market hypothesis. Examples are provided for illustrating these important topics.
A special feature of the book is that it starts from elementary notions in mathematics and statistics before advancing to more complex concepts. As an introductory text, no prerequisite in mathematics or statistics is required. In mathematics, this book starts from elementary notions such as numbers, vectors, and matrices, before it advances to topics in calculus and linear algebra.
The same approach is applied in statistics; the book covers basic concepts in probability theory, such as events, probabilities, and distributions, and advances progressively to econometrics, time series analysis, and continuous time ﬁnance. Each chapter is aimed at an introductory level and does not go into detailed proofs or advanced concepts.
The questions at the end of each chapter repeat examples discussed in the chapter and students should be able to carry out computations using widely available software, such as Excel, Matlab, and Mathematica, online formulas, and other calculators. Internet presentations that illustrate many procedures in the book are also available. The successful resolution of these questions means that a student has a good understanding of the contents of the chapter.
Sampling Hypothesis Testing Theory
Sampling theory is a study of the relationship existing between a population and samples drawn from the population. It is useful in estimating unknown population parameters such as population mean μ and variance σ2 from knowledge of corresponding sample mean and variance, often called sample statistics. Sampling theory is also useful in determining whether the observed differences between two samples are due to chance variation or they are really signiﬁcant.
Such questions arise for testing differences in returns among assets. For instance, is the difference between returns on S&P 500 and the Nikkei 225 signiﬁcant? Likewise, is the volatility of one index different from the volatility of another index?
The analysis of differences in samples involves the formulation of hypotheses and applications of tests of signiﬁcance that are important in the theory of decisions. In order for the conclusions of sampling theory and statistical inference to be valid, samples must be chosen so as to be representative of the population. One way in which a representative sample may be obtained is by the process called random sampling, according to which each member of a population has an equal chance of being included in the sample.
This section covers the sampling distribution of the mean, the sampling distribution of proportions, and the sampling distribution of differences.
Consider all possible samples of size n that can be drawn from a given population. For each sample we can compute a statistic (such as the mean x and the standard deviation s) that will vary from sample to sample. In this manner we obtain a distribution of the statistic that is called its sampling distribution. If, for example, the particular statistic used is the sample mean x, then the distribution is called the sampling distribution of the mean. Similarly, we could have sampling distributions of variance, standard deviation, median, and proportion. For each sampling distribu- tion, we can compute the mean and standard deviation. Thus, we speak of the mean and the standard deviation of the sampling distribution of the mean or the sampling distribution of the variance.
Brownian Motion, Risk-Neutral Processes, and the Black- Scholes Model
In this chapter we cover some basic elements and results of continuous time ﬁnance. In fact, considerable advances in ﬁnance theory have been made in continuous time. Many asset pricing, risk analysis, and rate of return models have been developed in continuous time.
There is a close relationship between discrete time and continuous time analysis. Time series are stochastic processes deﬁned on discrete time intervals. More speciﬁcally, the observations were made at ﬁxed points in time such as at the close of the market, or end of the month, or every hour.
However, modern ﬁnance has also used continuous-time stochastic processes with inﬁnitesimal time intervals. The random variable is assumed to be continuous in time. Continuous-time stochastic processes are widely applied in ﬁnance theory. Many pricing models such as the Black- Scholes option pricing formula were developed in continuous time. In this chapter, we introduce some basic concepts of continuous-time stochastic models and show their applications in asset pricing theory.
We study a continuous stochastic process in the same way as a time series. We try to characterize the probability law of the random variable, that is, the data generating process; then, we determine the mean and variance of the process. We use the process either to predict the future values of the variable, or to study risk, and price an asset. We examine how to transform a process into a risk-neutral, or equivalently a martingale, process in order to be able to use it for pricing.
Brownian motion is a main element of continuous-time ﬁnance. We need to under- stand this concept in order to appreciate the many advances in ﬁnance theory. Brownian motion was originally described by the botanist R. Brown (1828) who described the irregular and random motion of a pollen particle suspended in a ﬂuid. Hence, this motion is called Brownian motion. The theory is far from complete without a mathematical model developed later by Wiener (1931). The stochastic model used for Brownian motion is also called the Wiener process. We see that the mathematical model has many desirable properties. Since 1931, Brownian motion has been used in mathematical theory for stock prices. It is nowadays a fashion to use Wiener process to study ﬁnancial markets
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