Statistics for Finance

Analyze how statistics and probability models future asset values and forecast returns using time series data such as prices and returns, addressing investor risk and horizons.

Explore why returns are the fundamental form of financial data, and learn to compute gross and net returns, multi-period returns, and continuously compounded (logarithmic) returns.

Master numpy arrays, indexing, slicing, and elementwise operations that underpin pandas structures, covering one- and two-dimensional arrays, shapes, dtypes, range function, reshape, and common ufuncs.

Learn how to initialize and manipulate pandas series, including indexing, slicing, and arithmetic, compare with NumPy arrays, and perform boolean indexing with a U.S. population dataset.

Learn to use pandas to work with financial time series, reading the S&P 500 data, extracting a close series, and computing and plotting logarithmic returns with date-based indexing.

Explore the foundations of probability by defining experiments, sample spaces, and events, with examples like coin tosses, dice, cards, surveys, and stock price paths.

Explore the probability function that assigns probabilities to events in the sample space, using axioms, complement, disjoint events, and finite additivity, with a two-die example.

Compute exclusive or of events A and B via A union B minus A intersect B using inclusion exclusion, and apply this to bacterial infection probabilities to find neither infection.

Learn combinatorial probability using counting techniques under the uniform distribution, with tools like factorials and n choose k, to solve problems such as birthdays, poker hands, and sampling.

Learn conditional probability in statistics for finance: apply probability to examples, including a two-child example, compute P(A|B), handle A contained in B and disjoint events, with complement and inclusion-exclusion rules.

Explore urn models as a finite combinatorial probability framework that illustrates sampling with and without replacement, conditional probability, and updating urn contents after each draw.

Explore the concept of independence in probability, define independent events, and apply independence to multiple trials from drawing marbles and dice to coin flips, emphasizing partitioning and the product rule.

Apply independence to coin tosses and die rolls, deriving the distribution of heads in three tosses and the probability the first die exceeds the second (5/12).

Define random variables as functions on the sample space with real values, illustrating the shift from space to the variable, and introduce discrete and continuous cases plus the unifying cdf.

Explore the cumulative distribution function as the universal tool for expressing a random variable’s probability distribution, with discrete and continuous cases.

Learn how to model discrete random variables with the probability mass function, link it to the cdf, and compute pmf-based distributions and example probabilities including biased coins and loaded dice.

Explore the discrete uniform distribution, its constant probability over a finite set, how to compute the constant via normalization, and practical probability calculations using cardinality.

Explore the Bernoulli and binomial distributions as fundamental named models of success and failure, their link to coin tossing, and the binomial pmf with practical calculations.

Explore the binomial distribution through sample calculations, including sampling with replacement (p=0.3, n=10) to compute P(X>3) via the complement, and a quality-control example with 50 parts and 0.04 defect rate.

Explain how independence extends from events to random variables, and define independent random variables; apply the product rule to discrete cases and compute sums like x plus y.

Extend independence to finite sets of random variables and show sums of independent Bernoulli trials are binomial; preserve independence under 1-1 transformations, then compute P(X^2+Y^3>30)=0.117.

Explore the expected value, or mean, of discrete distributions as a weighted average of outcomes, with examples from Bernoulli, uniform, and binomial cases and key formulas.

Compute the expected value of a function of a random variable from its distribution, illustrated with g(x)=x^2-2x+2 on a binomial X and E[1/X] for a custom mass function.

Master the linearity of the expected value, proving that the expected value of a sum equals the sum of expected values, and extending to linear combinations.

Examine the variance as the dispersion of a distribution, defined by the expected value expression E[(X−E[X])^2], and connect it with standard deviation for practical interpretation.

Compute the expected value and variance of a binomial distribution by summing independent Bernoulli trials, then apply the formulas to 50 tosses with p=0.3, yielding 15 and 10.5.

Explore continuous random variables, their probability density function, and how to compute probabilities via integration, including uniform distributions, CDF relations, and normalization.

Compute the expected value and variance of continuous random variables using pdfs and integrals, applying linearity and independence with practical examples.

Moments and moment generating functions summarize distributions with mean and variance. Derivatives of moment generating function at zero yield moments and enable transformations, sums of independent variables, and binomial derivations.

Explore discrete bivariate distributions by building and interpreting the joint distribution and joint pmf, examine independence, and practice reading probabilities from two way tables.

Sum the joint pmf over the other variable to obtain the marginal distributions for x and y, and use marginals to assess independence and expectations.

Learn to analyze continuous bivariate distributions by using joint pdfs, verify nonnegativity and normalization via double integrals, and compute probabilities, marginal pdfs, and independence.

Master discrete conditional distributions by deriving the conditional mass function from a joint distribution, using f(x|y)=f(x,y)/f_Y(y). Learn through step-by-step examples and data conversions.

defines the continuous conditional pdf, using joint and marginal pdfs; demonstrates with continuous examples, including P(X<1/2|Y=1/2)=3/8 and a uniform x, y|x joint distribution, with notes on zero-probability conditioning.

Explore conditional expectation and conditional probability, deriving definitions for discrete and continuous cases using conditional pmf and pdf. Apply the law of iterated expectation and conditional variance in finance contexts.

Explore covariance and correlation as measures of dependence between two random variables, with Cov(X,Y)=E[XY]-E[X]E[Y] and rho = Cov(X,Y)/sqrt(Var(X)Var(Y)); zero correlation does not imply independence.

Extend the concept of a random variable to multivariate distributions, analyze joint pdfs and CDFs, and compute normalization, probabilities, marginals, and the expected value in a triple example.

Define the normal distribution and its standard form, the standard normal, and explain its pdf, cdf, symmetry, and how mu and sigma generate the general normal.

Explore the normal distribution using the Python distribution library to compute pdfs, cdfs, and quantiles, including standard normal, and apply to a reliability problem with independent lifetimes.

Show that the sum of independent normal random variables is normal, with mean equal to the sum of means and variance equal to the sum of variances, with applications.

Examine the exponential distribution as a gamma special case, define its pdf and cdf, derive mean 1/lambda and variance 1/lambda^2, and discuss waiting times and memorylessness.

Explore the gamma distribution, a two-parameter family with shape alpha and rate beta. Derive its pdf, compute the normalizing constant via the gamma function, and review mean and variance.

Explore the gamma distribution's scaling and convolution properties, show how sums of independent gamma variables remain gamma, relate gamma to exponential, and apply to sample means.

Explore the chi-square distribution as a gamma special case with its link to the normal. Learn its pdf, mean, variance, mgf, degrees of freedom, and sums of chi-squares for sampling.

Explore location parameter shifts of random variables, showing how X plus mu yields Y with same shape but shifted mean; distributions shift in CDF and PDF, variance stays the same.

Explore scale transformations by multiplying a variable by a positive alpha to stretch or compress its distribution; derive CDF and PDF relations and moments, with uniform and normal examples.

Combine shift and scale into location scale transformations, y = alpha x + mu, and see how they move the distribution and transform the pdf, cdf, mean, and variance.

Explore location-scale families, showing how normal distributions share the same shape under location-scale transforms, and how uniform distributions also form a location-scale family, including the standard uniform U(0,1).

Explore transformations of continuous random variables beyond location-scale, derive the change of variable formula, and apply it to pdfs via increasing or decreasing mappings, with practical examples.

This lecture demonstrates change of variable techniques using a Weibel distribution with y equals 1/x, comparing first principles and formula-based approaches, and then derives the Chi-square(1) distribution from Z^2.

Explore transformations of discrete random variables, derive the distribution of Y = H(X) using 1-to-1 and non-1-to-1 cases, and illustrate with examples like binomial and uniform X.

Explore how quintiles and the quantile function, as the generalized inverse of the CDF, characterize distributions, and compute medians and quartiles for continuous distributions.

Explore quantiles of discrete distributions by applying the generalized inverse quantile function to a stepwise cdf, and compute medians, quartiles, and selected percentiles.

Explore how location and scale parameters shift and scale distributions, and how shape parameters capture what remains after adjusting for mean and variance, using standardized moments to compare shapes.

Explore skewness, the third standardized moment, as a measure of asymmetry in return distributions, illustrating positive and negative skewness and the need for broader finance distributions beyond normal.

Learn to manufacture skewed distributions from symmetric ones to model financial data, using the skew-normal construction with an alpha parameter and the Fernandez and Steel method to induce real-line skewness.

Explore kurtosis, the fourth standardized moment, and fat tails, and compare standard normal with a mixed normal to see extreme value behavior beyond variance.

Explore the student’s t distribution, a fat-tailed alternative to the normal distribution in finance, defined by T = Z / sqrt(W/m) with Z ~ N(0,1) and W ~ chi-square(m).

Explore normal mixture distributions as a versatile tool for modeling fat-tailed finance data, deriving two-component normal mixtures, and comparing their heavier tails and excess kurtosis to the normal distribution.

Random samples are independent, identically distributed observations, and the sample mean estimates mu with variance sigma^2/n, linking sampling with replacement, coin tossing, and Bernoulli models to inference.

Explore Markov's and Chebyshev's inequalities and their role in bounding tail probabilities, linking random sampling to inferential methods, with a sample application showing E[X] ≥ 1 when P(X≥2)=1/2.

Explore the law of large numbers and convergence in probability, showing how the sample mean converges to mu with growing n, guiding estimation and hypothesis testing.

Explore the central limit theorem and its link to the law of large numbers, illustrating convergence in distribution via the sample mean, binomial to normal, and a coin-toss example.

Learn to construct and interpret order statistics from samples, derive their pdfs, and apply calculations with uniform and exponential data, including the fourth order statistic.

Explore the sample mean and the sample variance as unbiased estimators of the population mean and variance, their independence under normal sampling, and the chi-square link for inference.

Explain how the t distribution arises from sampling normal data via central limit theorem, with independence of x bar and sigma hat, and use it to obtain 0.95 quantile (n=14).

Transform price time series into returns, analyze financial time series with Python, extract the S&P 500 close and Euro USD data, and compute and plot log returns.

Explore time series as stochastic processes and learn how to model them with mean and covariance functions. Examine white noise, random walk, and practical simulations for finance applications.

Explore the core idea of stationarity in time series, from strict stationarity to the role of constant mean and lag-dependent covariance, with iid white noise, AR, MA, and GARCH examples.

Explore the autocovariance and autocorrelation functions of stationary time series, defining gamma_x(h) and rho_x(h), and illustrate with white noise, ma1, and ar1 models.

Learn to compute the sample mean, autocovariance, and autocorrelation for financial time series and interpret their role as estimators of the underlying process, with Python tools and Arma simulations.

explore moving from time series to the marginal distribution by treating financial returns as a random sample, and compute moments like mean, variance, skewness, and kurtosis with the dataset module.

Explore autoregressive (AR) models within ARMA frameworks, derive stationary AR(1) moments and autocorrelation function, discuss random walk and explosive cases, and illustrate AR simulations vs white noise.

Explore moving average models (ma q) as arma zero q process with no autoregressive terms, offering stationarity for any theta. Autocorrelations vanish after q lags, with gamma(1)=theta1 sigma^2.

Study the arma(1,1) model, a blend of autoregressive and moving average parts, as a building block for finance applications and GARCH models. Analyze its stationarity, mean, and autocorrelation.

Compute the sample mean, covariance, and autocorrelation for financial time series; analyze S&P 500 returns, including absolute and squared values, to illustrate volatility clustering and stylized facts.

Explore stationarity and structural breaks in financial time series by analyzing S&P 500 returns and absolute returns, using autocorrelation to identify stationary sub-periods like the mid-1990s.

Explore market efficiency, the efficient markets hypothesis, and how prices absorb information; learn martingales, discounted prices, and the random walk path for asset prices.

Explore the random walk model for asset prices, with a deterministic drift and a mean-zero random walk, and compare its empirical fit to real prices like the S&P 500.

Explore the geometric random walk model for asset prices, deriving price dynamics from i.i.d. log returns, calibrating drift and volatility to the S&P 500, and assessing nonstationarity.

Explore the lognormal model, a geometric random walk with normally distributed returns, and derive its lognormal price distribution, moments, and practical calibration to market data.

Explore random walk tests for asset prices by using autocorrelation, Ljung-Box, McLeod-Lee, and turning point tests on log returns and squared returns from the mid 2000s S&P 500.

Apply the runs test to log s&p 500 prices to assess randomness and the random walk hypothesis, noting strong rejection for 1960–72, borderline for mid-2000s, and consistency for mid-1990s.

Explore statistical inference as the core of classical statistics, learning to select models that explain data and make probabilistic statements about unknown parameters using estimation, interval estimation, and hypothesis testing.

Explore parametric estimation by selecting finite-parameter model families and finding the parameter values that best explain data, using estimators like the sample mean and the method of moments.

Explore maximum likelihood estimation as a preeminent technique for constructing estimators, using a coin toss example to maximize the likelihood and obtain p-hat 0.7.

Explore maximum likelihood estimation with one-parameter examples, deriving lambda hat as 1 over sample mean for an exponential model and p hat as s over n for a binomial proportion.

Explore maximum likelihood estimation for normal models, deriving mu hat equals x bar and sigma squared hat equals (1/n) sum (xi minus x bar)^2, with one- and two-parameter cases.

This lecture introduces the method of moments as a simpler alternative to maximum likelihood, illustrating moment matching with a one-parameter model and a two-parameter normal model.

Explore one-sided confidence bounds for normal means and derive delta minus and delta plus from x-bar, known sigma, and z-quantiles. Learn to bound mu with a chosen confidence level.

Learn to compute lower and upper confidence bounds for a normal mean using the sample mean, known sigma, and z quantiles, with practical 90% and 95% examples.

Learn how to build two-sided confidence intervals for a normal mean by combining lower and upper one-sided bounds, using symmetric tails and standard normal quantiles.

Explains hypothesis testing as a binary inference within parametric statistics, using a concrete mutual fund example to compare a 1.5% historical mean to a 2.7% sample mean.

Explore the formal framework of hypothesis testing in statistics, distinguishing null and alternative hypotheses, simple and composite forms, types I and II errors, and how data inform parameter inference.

Learn how test statistics and p-values decide whether to reject the null by comparing data to its null distribution, using x-bar and z-standardized statistics.

Design hypothesis tests by choosing a significance level, formulating null and alternative hypotheses, and using P values and a critical region for the test statistic Z.

Define the size of the test for a composite null, set the 0.1 rejection boundary at x-bar > 9.64, and conclude with a 7.28 sample mean that we do not reject.

Design and evaluate hypothesis tests by balancing type I and II errors using the power function, set significance levels, and compare one-sided test strategies.

Assess the white noise hypothesis for financial returns, considering weak and strong forms. Analyze mid eighties S&P 500 returns using autocorrelation, confidence bands, and lung box and McLeod Lee tests.

Explore why white noise fails for financial returns and how arch/garch models link past squared returns and past volatility to volatility, with alpha plus beta less than one ensuring stationarity.

Learn to fit a GARCH(1,1) ARCH(1) model using the method of moments. Derive alpha, beta, and omega from sample variance, kurtosis, and autocorrelations.

Learn exploratory data analysis through visualization, focusing on histograms as a robust univariate tool to visualize data, estimate the empirical density, and understand marginal distributions.

Learn how probability plots, including normal and t-quantile plots, test distributional fit using order statistics, and reveal tail behavior in financial data.

Explore stationary time series models for finance, focusing on white noise and iid returns, and assess autocorrelation, variance, and structural breaks in financial data.

Assess the normality of financial returns using exploratory plots, the Jarque-Bera test, and maximum likelihood fitting. Investigate the S&P 500 mid-2000s returns and compare to a normal distribution.

Analyzing the normality of S&P 500 returns across the mid-1990s and mid-2000s, the lecture uses graphical tests and the Jack Barrett test, showing greater deviations in the mid-1990s.

Use the method of moments to fit a generalized t distribution with mu and rho to financial returns, addressing fat tails and kurtosis, and compare with the normal.

Use maximum likelihood estimators to fit the generalized student t distribution (mu, scale, nu) to financial returns. Compare these to method of moments and analyze tail behavior with probability plots.

Describe normal mixture distributions for modeling financial returns and fat tails, and demonstrate fitting two- and three-component mixtures to S&P 500 data using maximum likelihood.