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Department:

MATH

Course Number:

6783

Hours - Lecture:

3

Hours - Lab:

0

Hours - Recitation:

0

Hours - Total Credit:

3

Typical Scheduling:

Usually every spring semester

Fundamentals of statistical inference are presented and developed for models used in the modern analysis of financial data. Techniques are motivated by examples and developed in the context of applications. Crosslisted with ISYE 6783.

Prerequisites:

MATH 3215 (or equivalent), some knowledge of programming, and MS QCF standing or some previous exposure to the topics of stocks, bonds and options

Course Text:

No text

Topic Outline:

The following probability topics are covered in the models that are presented:

- Distributions such as the normal (Gaussian), lognormal, geometric, binomial, Poisson, Student's t, F, chi-square, gamma, and Pareto
- Characteristic functions, sums of independent random variables, a-stable random variables
- Limit Theorems for sums
- Order statistics
- Limit Theorems for extremes
- Elementary stochastic processes such as Markov chains
- Dynamic linear models
- Time series models

The following topics in statistical inference are covered in the models that are presented:

- Likelihood functions
- Estimation
- Testing Hypotheses via Neyman-Pearson tests, likelihood ratio tests, and Wald tests
- Tests of fit
- Markov chain and time series inference
- Regression
- Principal components analysis
- Non-parametric analyses

Applications to financial data are made throughout and include the topics such as the following:

- Testing hypotheses of independence, normality, homoscedascticity, and symmetry for returns, and the Bachelier and Mandelbrot models
- Efficient frontier in portfolio analysis under short selling and riskless borrowing and lending, optimal portfolio under single index and multi-index models, principal components analysis, stability tests of betas from auxiliary data
- Simulation and Monte-Carlo, estimation and assessment of accuracy of path integrals arising in option pricing
- Hill's estimator of the Pareto index, application to solvency analysis and ruin probabilities, connections with a-stability
- Analysis of ar, ma, arma, arima, arch, garch, and stochastic volatility time series models applied to exchange rates, indexes, interest rates, and returns.