# 35364 Statistics for Quantitative Finance

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particular session, location and mode of offering is the authoritative source
of all information about the subject for that offering. Required texts, recommended texts and references in particular are likely to change. Students will be provided with a subject outline once they enrol in the subject.

Subject handbook information prior to 2020 is available in the Archives.

*UTS: Science: Mathematical and Physical Sciences*

*Credit points:*6 cp

*Result type*: Grade and marks

*There are course requisites for this subject. See access conditions.*

### Description

This subject provides a foundation in probability and statistics, and introduces the basic concepts of stochastic processes and time series. Topics include random variables, expectations, law of large numbers, central limit theorem, estimation of parameters, testing hypothesis, linear regression, Gaussian and Markov stochastic processes, and basic time series analysis.

### Subject learning objectives (SLOs)

Upon successful completion of this subject students should be able to:

1. | understand the theoretical results presented in this subject |
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2. | be able to apply the ideas and results of probability theory and statistics presented in this subject |

3. | understand the concept of statistical techniques |

4. | further develop your ability to recognize problems amenable to the methods of estimation and modeling in finance |

5. | be able to develop estimation procedures for various applications |

6. | have an experience in using modern software for statistical analysis. |

### Contribution to the development of graduate attributes

The material presented in this subject and the method of presentation are linked to the following Science and UTS graduate attributes:

- An understanding of the nature, practice and application of the chosen discipline.
- An understanding of the scientific method of knowledge acquisition, including problem solving, critical thinking, and the ability to discover new understandings.
- The capacity to learn in, and from, new disciplines to enhance the application of scientific knowledge and skills in professional contexts.
- An awareness of the role of statistics and probability within a global culture.
- An ability to think and work creatively and the ability to apply skills to unfamiliar applications.
- The ability to develop computing skills.

### Teaching and learning strategies

Each week, this subject involves twelve classes combining lectures and tutorials. These three hours of classes are supposed to be complemented by regular individual work comprised of studying the material presented in the lectures, solving tutorial problems, working on assignments and practicing with software.

This subject outline, tutorial questions, and other supporting material will be distributed in class. Some material including the subject outline will be available at UTSOnline. Although the subject will make use of UTSOnline, this is intended to supplement the face-to-face teaching not replace it. Bound copies of lecture notes will also be made available.

As a student in this subject you are expected to attend all lectures and tutorials. As preparation for a tutorial, you are expected to attempt all questions distributed prior to this tutorial. As preparation for a lecture, you are expected to learn all material presented in the previous lectures.

If you have further questions or need further help with understanding the subject, please ask the lecturer during the consultation hours. If this is not possible, you can e-mail or phone the lecturer to negotiate a time to meet.

UTSOnline is located at: http://online.uts.edu.au

### Content (topics)

**Probability Space:**

- Sample space (discrete, geometric, general), events, probability axioms
- Conditional probability, independent events
- Total probability and Bayes’s formulae

**Random Variables:**

- Cumulative distribution function (cdf)
- Discrete and continuous random variables
- Probability density function (pdf)
- Frequently used distributions
- Quantiles, expected value (mean), variance
- Jensen’s inequality

**Random Variables:**

- Cumulative distribution function (cdf)
- Discrete and continuous random variables
- Probability density function (pdf)
- Frequently used distributions
- Quantiles, expected value (mean), variance
- Jensen’s inequality

**Multivariate Random Variables:**

- Joint distribution function
- Independent random variables
- Conditional distribution
- Covariance and correlation, copulas
- Gaussian vectors
- Simulation of random variables

**Markov Chains:**

- Transition probability matrix
- Stationary distribution of the Markov chain
- Fundamental matrix
- Reaching times – the “ruin” problem

**Sampling:**

- Estimation of parameters:
- Method of moments
- Maximum likelihood method
- Empirical cdf, sample mean and variance
- Special distributions, confidence intervals

**Goodness-of-fit tests:**

- Chi-square test
- Kolmogorov-Smirnov test
- Probability plots

**Nonparametric tests:**

- Sign tests
- Ranked sign (Wilcoxon) test
- Mann-Whitney test
- Tests based on runs

**Regression Analysis:**

- Univariate linear regression
- Multivariate linear regression
- Multivariate linear regression – matrix notation

**Elementary Stochastic Processes:**

- Poisson process
- Brownian motion (Wiener process)
- Brownian motion hitting times

### Assessment

#### Assessment task 1: Assignment

Objective(s): | This assessment task contributes to the development of course intended learning outcome(s): .0 |
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Weight: | 40% |

#### Assessment task 2: Exam

Objective(s): | This assessment task contributes to the development of course intended learning outcome(s): .0 |
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Weight: | 60% |

### Minimum requirements

In order to pass this subject, a student must achieve a final result of 50% or more and achieve 40% or more on the

final examination. The final result is simply the sum of all the marks gained in each piece of assessment. Students

who obtain 50 marks or more but fail to score 40% or more on the final examination will be given an X grade (fail).

### References

- Miller, M. & Miller, M. (1999) Mathematical Statistics, Sixth Edition, Prentice Hall
- Platen, E. & Heath, D. (2006) A Benchmark Approach to Quantitative Finance, Springer
- Rice, J. (1995) Mathematical Statistics and Data Analysis, 2nd edition, Duxbury Press
- Ross, S. (2000) Introduction to Probability Models, 7th edition, Harcourt Academic Press, Inc
- Shao, J: Mathematical Statistics. Springer 2003
- P. Gatti: Probability Theory and Mathematical Statistics for Engineers. Spon Press 2005.
- G. Ivchenko Yy. Medvedev A. Chistyakov: Problems in Mathematical Statistics. Mir publishers Moscow