week 1 quantitative analysis of financial markets overvie · introduction probability &...

Introduction Probability & Statistics Regression Analysis Time Series Modeling

Week 1Quantitative Analysis of Financial Markets

Overview

Christopher Ting

Christopher Ting

http://www.mysmu.edu/faculty/christophert/

k: [email protected]: 6828 0364

ÿ: LKCSB 5036

October 14, 2017

Christopher Ting QF 603 October 14, 2017 1/25

http://www.mysmu.edu/faculty/christophert/

mailto:[email protected]


Table of Contents

1 Introduction

2 Probability & Statistics

3 Regression Analysis

4 Time Series Modeling



Quantitative Analysis (20% of FRM)

The broad areas of knowledge covered in readings related toQuantitative Analysis include the following:

1 Discrete and continuous probability distributions2 Estimating the parameters of distributions3 Population and sample statistics4 Bayesian analysis5 Statistical inference and hypothesis testing6 Estimating correlation and volatility using EWMA and GARCH

models7 Volatility term structures8 Correlations and copulas9 Linear regression with single and multiple regressors

10 Time series analysis and forecasting11 Simulation methods



QA Modules1 Probabilities2 Basic Statistics3 Distributions4 Bayesian Analysis5 Hypothesis Testing and Confidence Intervals6 Linear Regression with One Regressor7 Regression with a Single Regressor8 Regression with Multiple Regressors9 Hypothesis Tests and Confidence Intervals in Multiple Regression

10 Modeling and Forecasting Trend11 Modeling and Forecasting Seasonality12 Characterizing Cycles13 Modeling Cycles: MA, AR, and ARMA Models14 Volatiliy15 Correlations and Copulas16 Simulation Modeling



QA-1 ProbabilitiesChapter 2.Michael Miller, Mathematics and Statistics for Financial Risk Management, 2nd Edition(Hoboken, NJ: John Wiley & Sons, 2013).

À Describe and distinguish between continuous and discreterandom variables.

À Define and distinguish between the probability density function,the cumulative distribution function, and the inverse cumulativedistribution function.

À Calculate the probability of an event given a discrete probabilityfunction.

À Distinguish between independent and mutually exclusive events.À Define joint probability, describe a probability matrix, and calculate

joint probabilities using probability matrices.À Define and calculate a conditional probability, and distinguish

between conditional and unconditional probabilities.Christopher Ting QF 603 October 14, 2017 5/25


QA-2 Basic StatisticsChapter 3.Michael Miller, Mathematics and Statistics for Financial Risk Management, 2nd Edition(Hoboken, NJ: John Wiley & Sons, 2013).

Á Interpret and apply the mean, standard deviation, and variance ofa random variable.

Á Calculate the mean, standard deviation, and variance of a discreterandom variable

Á Calculate and interpret the covariance and correlation betweentwo random variables.

Á Calculate the mean and variance of sums of variables.Á Describe the four central moments of a statistical variable or

distribution: mean, variance, skewness and kurtosis.Á Interpret the skewness and kurtosis of a statistical distribution,

and interpret the concepts of coskewness and cokurtosis.Á Describe and interpret the best linear unbiased estimator.



QA-3 Distributions

Chapter 4.Michael Miller, Mathematics and Statistics for Financial Risk Management, 2nd Edition(Hoboken, NJ: John Wiley & Sons, 2013).

Â Distinguish the key properties among the following distributions:uniform distribution, Bernoulli distribution, Binomial distribution,Poisson distribution, normal distribution, lognormal distribution,Chi-squared distribution, Student’s t, and F -distributions, andidentify common occurrences of each distribution.

Â Apply the Central Limit Theorem. item Describe the properties ofindependent and identically distributed (i.i.d.) random variables.

Â Describe a mixture distribution and explain the creation andcharacteristics of mixture distributions.



QA-4 Bayesian Analysis

Chapter 6. (pp. 113-124 only)Michael Miller, Mathematics and Statistics for Financial Risk Management, 2nd Edition(Hoboken, NJ: John Wiley & Sons, 2013).

Ã Describe Bayes’ theorem and apply this theorem in the calculationof conditional probabilities.

Ã Compare the Bayesian approach to the frequentist approach.

Ã Apply Bayes’ theorem to scenarios with more than two possibleoutcomes.



QA-5 Hypothesis Testing and Confidence Intervals

Chapter 7.Michael Miller, Mathematics and Statistics for Financial Risk Management, 2nd Edition(Hoboken, NJ: John Wiley & Sons, 2013).

Ê Calculate and interpret the sample mean and sample variance.Ê Construct and interpret a confidence interval.Ê Construct an appropriate null and alternative hypothesis. and

calculate an appropriate test statistic.Ê Differentiate between a one-tailed and a two-tailed test and

identify when to use each test.Ê Interpret the results of hypothesis tests with a specific level of

confidence.Ê Demonstrate the process of backtesting VaR by calculating the

number of exceedances.



QA-6 Linear Regression with One Regressor

Chapter 4.James Stock and Mark Watson, Introduction to Econometrics, Brief Edition (Boston:Pearson education, 2008).

Å Explain how regression analysis in econometrics measures therelationship between dependent and independent variables.

Å Interpret a population regression function, regression coefficients,parameters, slope, intercept, and the error term.

Å Interpret a sample regression function, regression coefficients,parameters, slope, intercept, and the error term.

Å Describe the key properties of a linear regression.Å Define an ordinary least squares (OLS) regression and calculate

the intercept and slope of the regression.



QA-6 Linear Regression with One Regressor (cont’d)

Å Describe the method and three key assumptions of OLS forestimation of parameters.

Å Summarize the benefits of using OLS estimators.Å Describe the properties of OLS estimators and their sampling

distributions, and explain the properties of consistent estimators ingeneral.

Å Interpret the explained sum of squares, the total sum of squares,the residual sum of squares, the standard error of the regression,and the regression R2.

Å Interpret the results of an OLS regression.



QA-7 Regression with a Single Regressor


Æ Calculate, and interpret confidence intervals for regressioncoefficients.

Æ Interpret the p-value.Æ Interpret hypothesis tests about regression coefficients.Æ Evaluate the implications of homoskedasticity and

heteroskedasticity.Æ Determine the conditions under which the OLS is the best linear

conditionally unbiased estimator.Æ Explain the Gauss-Markov Theorem and its limitations, and

alternatives to the OLS.Æ Apply and interpret the t-statistic when the sample size is small.



QA-8 Regression with Multiple RegressorsChapter 6.James Stock and Mark Watson, Introduction to Econometrics, Brief Edition (Boston:Pearson education, 2008).

Ç Define and interpret omitted variable bias, and describe themethods for addressing this bias.

Ç Distinguish between single and multiple regression.Ç Interpret the slope coefficient in a multiple regression.Ç Describe homoskedasticity and heterosckedasticity in a multiple

regression.Ç Describe the OLS estimator in a multiple regression.Ç Calculate and interpret measures of fit in multiple regression.Ç Explain the assumptions of the multiple linear regression model.Ç Explain the concept of imperfect and perfect multicollinearity and

their implications.Christopher Ting QF 603 October 14, 2017 13/25


QA-9 Hypothesis Tests and Confidence Intervalsin Multiple Regression


É Construct, apply, and interpret hypothesis tests and confidenceintervals for a single coefficient in a multiple regression.

É Construct, apply, and interpret hypothesis tests and confidenceintervals for multiple coefficients in a multiple regression.

É Interpret the F -statistic.É Interpret tests of single restrictions involving multiple coefficients.É Interpret confidence sets for multiple coefficients.É Identify examples of omitted variable bias in multiple regressions.É Interpret the R2 and adjusted R2 in a multiple regression.



QA-10 Modeling and Forecasting Trend

Chapter 5.Francis X. Diebold, Elements of Forecasting, 4th Edition (Mason, Ohio: CengageLearning, 2006).

æ Describe linear and nonlinear trends.

æ Describe trend models to estimate and forecast trends

æ Compare and evaluate model selection criteria, including meansquared error (MSE), s2, the Akaike information criterion (AIC),and the Schwarz information criterion (SIC).

æ Explain the necessary conditions for a model selection criterion todemonstrate consistency.



QA-11 Modeling and Forecasting Seasonality

Chapter 6Francis X. Diebold, Elements of Forecasting, 4th Edition (Mason, Ohio: CengageLearning, 2006).

æ Describe the sources of seasonality and how to deal with it in timeseries analysis.

æ Explain how to use regression analysis to model seasonality.

æ Explain how to construct an h-step-ahead point forecast.



QA-12 Characterizing Cycles


à Define covariance stationary, autocovariance function,autocorrelation function, partial autocorrelation function andautoregression.

à Describe the requirements for a series to be covariance stationary.

à Explain the implications of working with models that are notcovariance stationary.

à Define white noise, independent white noise, and normal(Gaussian) white noise.



QA-12 Characterizing Cycles (cont’d)

à Explain the characteristics of the dynamic structure of white noise.

à Explain how a lag operator works.

à Describe Wold’s theorem.

à Define a general linear process.

à Relate rational distributed lags to Wold’s theorem.

à Calculate the sample mean and sample autocorrelation, anddescribe the Box-Pierce Q-statistic and the Ljung-Box Q-statistic.

à Describe sample partial autocorrelation.



QA-13 Modeling Cycles: MA, AR, and ARMAModels


á Describe the properties of the first-order moving average (MA(1))process, and distinguish between autoregressive representationand moving average representation.

á Describe the properties of a general finite-order process of order q(MA(q)) process.

á Describe the properties of the first-order autoregressive (AR(1))process, and define and explain the Yule-Walker equation.

á Describe the properties of a general p-th order autoregressive(AR(p)) process.



QA-13 Modeling Cycles: MA, AR, and ARMAModels (cont’d)

á Define and describe the properties of the autoregressive movingaverage (ARMA) process.

á Describe the application of AR and ARMA processes.



Learning Outcomes of QA14

Chapter 10.John C. Hull, Risk Management and Financial Institutions, 4th Edition (Hoboken, NJ:John Wiley & Sons, 2015).

â Defne and distinguish between volatility, variance rate, and impliedvolatility.

â Describe the power law.â Explain how various weighting schemes can be used in estimating

volatility.â Apply the exponentially weighted moving average (EWMA) model

to estimate volatility.â Describe the generalized autoregressive conditional

heteroskedasticity (GARCH(p, q)) model for estimating volatilityand its properties.



Learning Outcomes of QA14 (cont’d)

â Calculate volatility using the GARCH(1,1) model.â Explain mean reversion and how it is captured in the GARCH(1,1)

model.â Explain the weights in the EWMA and GARCH(1,1) models.â Explain how GARCH models perform in volatility forecasting.â Describe the volatility term structure and the impact of volatility

changes.



QA-15 Correlations and CopulasChapter 11.John Hull, Risk Management and Financial Institutions, 4th Edition (Boston: PearsonPrentice Hall, 2015).

Ä Define correlation and covariance, differentiate betweencorrelation and dependence.

Ä Calculate covariance using the EWMA and GARCH (1,1) models.Ä Apply the consistency condition to covariance.Ä Describe the procedure of generating samples from a bivariate

normal distribution.Ä Describe properties of correlations between normally distributed

variables when using a one-factor model.Ä Define copula, describe the key properties of copula and copula

correlation.Ä Explain one tail dependence.Ä Describe Gaussian copula, Student t-copula, multivariate copula

and one factor copula.Christopher Ting QF 603 October 14, 2017 23/25


QA-16 Simulation Modeling

Chapter 13.Chris Brooks, Introductory Econometrics for Finance, 3rd Edition (Cambridge, UK:Cambridge University Press, 2014).

ã Describe the basic steps to conduct a Monte Carlo simulation.ã Describe ways to reduce Monte Carlo sampling error.ã Explain how to use antithetic variate technique to reduce Monte

Carlo sampling error.ã Explain how to use control variates to reduce Monte Carlo

sampling error and when it is effective.ã Describe the benefits of reusing sets of random number draws

across Monte Carlo experiments and how to reuse them.ã Describe the bootstrapping method and its advantage over Monte

Carlo simulation.



QA-16 Simulation Modeling (cont’d)

ã Describe the pseudo-random number generation method and howa good simulation design alleviates the effects the choice of theseed has on the properties of the generated series.

ã Describe situations where the bootstrapping method is ineffective.ã Describe disadvantages of the simulation approach to financial

problem solving.


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