**Applied Science in Financial Mathematics and Industrial Statistics**

**MMT401XS3: Mathematics for Finance**

(45 Hours of lectures and tutorials)

**Objectives:**

- Survey mathematical models of fixed income securities.
- Discuss arbitrage free pricing and other derivatives.
- Formulate elementary models of asset allocation and their performance evaluation.

** ****Syllabus:**

- Introduction: Financial markets, market indices, Zero coupon bonds, coupon bonds, bond pricing, yield, yield curve, forward rates, Annual rate of return, notions and assumptions on market dynamics, No-arbitrage principle, Forwards and Futures contracts, equity, call and put options.
- Valuation: Bonds and their valuations, sensitivity of prices to the yield, Forward contracts and futures, arbitrage free pricing, Capital Asset Pricing Model(CAPM) and applications to the equity markets, Binomial Option Pricing Model(BOPM) and application applications to European and American options.
- Basics of Risk: Notion of Risk, Financial Risk, Credit Risk and Operational Risk, risk exposures in the fixed income securities and mitigation, concept of bond duration an applications, equity markets and equity portfolio construction, risk measures of equity portfolio, applications of options in risk management.

**Evaluation:**

- In-course Assessments 30%
- End-of-course Examination 70%

** ****Recommended Readings:**

- Frank J Fabozzi, “Fixed income Securities (2nd Edition)”, 2001.
- John C Hull, “Options, Futures and Other Derivatives (6th Edition)”, Prentice Hall of India, 2005.
- Mark S. Joshi, “The Concepts and Practice of Mathematical Finance (Mathematics, Finance and Risk)”. Cambridge University Press, 2008

**MMT402XS3: Actuarial Mathematics**

(45 Hours of lectures and tutorials)

**Objectives:**

- Explain basics actuarial valuation method
- Identify actuarial as general valuation techniques
- Value the different random cash flows
- Identify different life insurance models and their features

**Syllabus:**

Introduction to Actuarial concepts, valuation and actuarial valuation, importance of actuarial concepts and their applications in various finance fields, Introduction to insurance, Survival distribution, mortality rate, life expectancy, life table, Insurance and related models, Applications of insurance models, Life annuity and related models, Applications of life annuity models, Loss random variable and its applications in Insurance models, Premiums determination methods and related problems.

** ****Evaluation:**

- In-course Assessments 30%
- End-of-course Examination 70%

** ****Recommended Readings:**

- Bowers, NL., Gerber, HU., Jones, DA., Nesbitt CJ., Actuarial Mathematics,The Society of Actuaries, 1997.
- David Promislow, “Fundamentals of Actuarial Mathematics”, 2nd Edition, Wiley, 2010

**STA403XS3: Applied Multivariate Analysis for Real World Data**

(45 Hours of lectures and tutorials)

**Objectives:**

- Summarise complicated multivariate techniques using real world data sets
- Analyse datasets using visualization tools.
- Apply multivariate dataset analysis to Medicine, Biology, Sociology and Industry.

**Syllabus:**

- Introduction and theory necessary for Multivariate Analysis: Examples, descriptive Statistics and graphical techniques, application of multivariate techniques, matrix algebra and random vectors. The multivariate normal distribution, maximum likelihood estimation, sampling distribution of and s, large sample behavior of and s, detecting outliers and cleaning the data, transformation to approximate normality. Inferences about a mean vector, comparison of several multivariate means, multivariate linear regression models. Principal component: summarizing sample variations by Principal components, graphing the Principal components. Factor analysis: the orthogonal factor model, methods of estimation, factor rotation, factor scores. Canonical correlation analysis: canonical variance, canonical correlation, interpreting canonical variations. Discrimination and classification: separation and classification for two population, classification with two multivariate normal populations, evaluating classification functions, Fishers discriminant function, classification in several populations. Clustering, distance methods and ordination: similarity measures, Hierarchical clustering methods, non-hierarchical clustering methods, Multi dimensional scaling, graphical methods. Matlab/R code for multivariate analysis: This should consist code for doing all the analysis specified above, exercises containing data sets to illustrate all the methods explained above to be analyzed.

** ****Evaluation:**

- In-course Assessments 30%
- End-of-course Examination 70%

**Recommended Readings:**

- Richard A. Johnson and Dean W. Wichern, Applied Multivariate Statistical Analysis, (6th Edition), Pearson PLC, 2007.
- Alvin C. Rencher., Methods of Multivariate Analysis (Second Edition, Wiley Series in Probability and Statistics, 2002.
- Francois Husson, Sebastien Le, Jerome Pages, Exploratory Multivariate Analysis by Example Using R , Chapman & Hall/CRC Press, 2011.

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**STA404XS2: Advanced Statistical Computing**

(15 Hours of lectures and 30 Hours of practical)

**Objectives:**

- Develop statistical computing skills using an appropriate statistical software
- Apply statistical simulation techniques
- Demonstrate MCMC simulation techniques

**Syllabus:**

Writing simple functions to automate regular tasks.Introduction and overview of simulation analysis. Random number generation: The Box-Muller algorithm, The Inversion method, The Rejection method, The Polar algorithm. Variance reduction techniques.Data resampling methods, The Bootstrap method. Likelihood Methods: posterior modes, asymptotic confidence intervals, The Delta method, Bayesian calculations, The E-M algorithm. Markov Chain Monte Carlo Methods: The Gibbs Sampler, The Metropolis- Hastings algorithm. Data partitioning: Cross validation, Jackknife.

** ****Evaluation:**

- In-course Assessments 30%
- End-of-course Examination 70%

**Recommended Readings:**

- Wendy L. Martinez and Angel R. Marinez, Computational Statistics Handbook with MATLAB (2nd edition); Chapman & Hall/CRC Press,
- Venables, W.N. and Ripley B.D. , Modern Applied Statistics with S (4th edition), Springer Series, 2002.
- Steve Brooks, Andrew Gelman, Galin Jones and Xiao-Li Meng, Hand book of Markov chain Monte Carlo (1st Edition); Chapman & Hall/CRC Press, 2011.

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