ECONOMIA E IMPRESAFinanza AziendaleAnno accademico 2022/2023

# 1008302 - PROBABILITY FOR FINANCE

Docente: ANTONINO DAMIANO ROSSELLO

## Risultati di apprendimento attesi

1. Knowledge and understanding: The course addresses fundamental concepts of probability applied to finance, especially those that are most relevant to some aspects of risk management and financial engineering. Probabilistic ideas and language are tailored for a smooth transition from basic (calculus based) probability to a more advanced treatment with a modicum of measure theory, emphasizing financial applications as tools to enforce the critical understanding of probability (models and estimation) ‘jargon’.

## Modalità di svolgimento dell'insegnamento

The course is teached in English. The probabilistic toolkit will be developed through Lectures to be held during the classroom, and additionally by means of real world cases. This is beacause the relevant probability theory would be applied to problems arising from risk management and financial engineering (selected topics). Sometimes, real world cases consist of numerical evaluation of financial dataset, in term of probabilistic models, using spreadsheets developed during the classroom. Excel and a glimpse of Visual Basic for Application (VBA) are the main device to manipulate spreadsheets. Furthermore, students are asked to solve exercises to test their understainding of the main probabilistic definition, theorems and formulas.

## Prerequisiti richiesti

Knowledge of undergraduate calculus (differentiation and integration) and elementary financial mathematics (time value of money) is strongly recommended, albeit not mandatory. Some previous exposure to undergraduate statistics courses is useful though not necessary. Ordinary first order differential equations and convergence of functions will be discussed during the classroom, but students take advantage to learn them before attending the course.

## Frequenza lezioni

Not mandatory, but students are firmly suggested to attend the course.

## Contenuti del corso

PART #1 (3 CFU)

Topic: Review of basic probability theory.

Learning goals: Probabilistic tools from the calculus viewpoint, with some elements of measure theory.

Topic description: Discrete and general probability spaces. Conditional probability and independence. Discrete and continuous random variables and their distributions. Most frequently used probability distribution in finance (univariate). Moments and characteristic function. Location and dispersion indexes. Covariance. Quantiles. Conditional expectation. Inequalities. Hazard function and the probability integral transform.

PART #2 (3 CFU)

Topic: Multivariate (static and dynamic) probability models.

Learning goals: Probability distributions of random vectors and stochastic processes.

Topic description: Random vectors and joint distributions. Marginals and finite dimensional distributions of infinitely many random variables. Distribution of a stochastic process. Convergence theorems. Monte-Carlo simulation. Some commonly used stochastic processes in finance (Bernoulli, random walk, Wiener, AR(1), martingale, Markov. Poisson). A glimpse to stochastic calculus.

PART #3 (3 CFU)

Topic: Stochastic models in finance (selected).

Learning goals: Applying random distributions and other probabilistic tools to some typical financial models.

Topic description: Modeling market invariants (log-returns, risk and expected return of a portfolio). Coherent risk measures and Value-at-Risk. Binomial asset pricing model. Geometric Brownian motion and Black-Scholes model. Market probabilities vs risk-neutral probabilities. Fat tails in practice (time permitting). Copula and dependance concepts. Empirical distribution function and plug-in estimators (time permitting). Expectiles. Robustness (time permitting). Some stochastic orders and conditional risk measures.

## Testi di riferimento

1. Introduction to Probability – D.P. Bertsekas, J.N. Tsitsiklis – Athena Scientific, 2nd edition, 2008
2. Instructor’s notes

1. Essential Mathematics for Market Risk Management – S. Hubbert – Wiley 2012
2. Statistical Methods for Financial Engineering – B. Rémillard – CRC Press 2013

## Programmazione del corso

ArgomentiRiferimenti testi
1Random experiment, sample spaces and events. Random variables. Examples in finance: asset prices and returns.Bertsekas-Tsitsiklis Ch 1 and instructor's notes
2Sigma-algebra of events, probability measure; special kinds of events induced by rv’s.Bertsekas-Tsitsiklis Ch 1 and instructor's notes
3Numerical example. Stylized facts for asset returns. Anticipating correlation and variability of probability laws. CDF.instructor's notes
4Discrete rv’s. Continuous rv’s. Bertsekas-Tsitsiklis Chs 1, 2 and instructor's notes
5Indicator rv’s.  Bernoulli scheme. Binomial rv. Normal Standard rv.Bertsekas-Tsitsiklis Chs 1, 2 and instructor's notes
6Gaussian rv. Binomial model of asset price and binomial treesBertsekas-Tsitsiklis Chs 2, 3 and instructor's notes
7Transformation of rv’s: scalar-to-scalar and vector-to-scalar. Positive part and negative part. Financial examples  Equality a.s. and in distribution. Conuterexamples. Symmetric distributions round 1. Bertsekas-Tsitsiklis Chs 2, 3, 4 and instructor's notes
8Symmetric distributions round 2. Hazard function. Qunatile function.Bertsekas-Tsitsiklis Chs 2, 3 and instructor's notes
9Random vector. Special case: bi-variate. Joint CDF.Bertsekas-Tsitsiklis Chs 2, 3 and instructor's notes
10Properties of bi-variate CDF. Some geometric aspects. Discrete random vectors and joint masses.Bertsekas-Tsitsiklis Chs 2, 3 and instructor's notes
11Absolutely continuous random vector. Joint masses. Fubini’s Theorem. Stieltjes integrals. Examples. From joint distribution to marginals.Bertsekas-Tsitsiklis Chs 2, 3, 4 and instructor's notes
12Stochastic independence. Conditional probability. Frechét bounds and stochastic dependence. Block dependence.Bertsekas-Tsitsiklis Chs 1, 2, 3 and instructor's notes
13Expectation of discrete rv’s. Examples. Frequency interpretation. Expectation of abs. cont. Rv’s. Unifying (change of variable) formula via Stieltjes integral w.r.t. CDF. Bertsekas-Tsitsiklis Chs 2, 3 and instructor's notes
14Expectation over subsets and probability, via indicators. Properties of expectation.Bertsekas-Tsitsiklis Chs 2, 3 and instructor's notes
15Variance and standard deviation. Central moments. Moment generating function. Characteristic function round 1. First order stochastic order. Bertsekas-Tsitsiklis Chs 2, 3, 4 and instructor's notes
16Cauchy-Schwarz inequality: covariance and correlation. Jensen’s inequality. Chebychev’s inequality.  Summary statistics round 1Bertsekas-Tsitsiklis Chs 2,  3, 4 and instructor's notes
17Summary statistics round 2. Expectation of random vectors round 1. Convolution formula.Bertsekas-Tsitsiklis Chs 2, 3, 4 and instructor's notes
18Conditional expectation: discrete and continuous rv’s. General conditional expectation. Conditional cdf, masses and densities. Financial application: risk-neutral pricing and optimal forecast.Bertsekas-Tsitsiklis Chs 2, 3, 4, 9 and instructor's notes
19Prediction, minimal mean-square error and projection. Best linear forecast: regression. More on multivariate Gaussian.Bertsekas-Tsitsiklis Chs 2, 3, 4, 9 and instructor's notes
20Introduction to stochastic processes. Examples. Mean and auto-covariance functions. Filtration.Bertsekas-Tsitsiklis Chs 2, 3, 4 and instructor's notes
21Strictly and weak stationary processes. Stationary and independent increments. White Noise, Random Walk, AR(1)  and Wiener processes.Bertsekas-Tsitsiklis Chs 2, 3, 4 and instructor's notes
22Introduction to copula functions.Instructor's notes
23Introduction to risk measure via quantiles (VaR). Exercises Connection with portfolio optimization. Instructor's notes
24Expected shortfall (ES) as alternative to VaR. Sharpe ratio maximization and performance evaluation.Instructor's notes
25Some convergence concepts: almost surely, in distribution, in mean-square. Bertsekas-Tsitsiklis Chs 5 and instructor's notes
26Weak Law of Large Numbers, Central Limit Theorem. Application to log-returns.Bertsekas-Tsitsiklis Chs 5 and instructor's notes
27Monte Carlo simulation and elements of point estimation. Examples.Bertsekas-Tsitsiklis Chs 9 and instructor's notes
28Numerical examples of nonparametric estimation of VaR and ES.Instructor's notes

## Verifica dell'apprendimento

### Modalità di verifica dell'apprendimento

Oral examination: 4/5  oral questions.

No partial exams.

### Esempi di domande e/o esercizi frequenti

What is an event?

What is a probability measure?

What is a sigma-algebra?

What is the distribution of a random variable?

What is a joint distribution?

What is convergence in distribution?

What is a stochastic process?

What does the Central Limit Theorem tell?

What does the Weak Law of Large Numbers tell?

What is the Bayes’ theorem?

How does the log-normal model of stock-price characterized?

What are the Ito’s Integral and the Ito’s Lemma?

What is a coherent risk measure?

What is a quantile?

What are the moment generating function and the characteristic function of a random variable?

What is a copula function?

What is an empirical distribution function?

When a class of random variables is said to be independent?

What is a random walk?

What is a Bernoulli process?

What is an AR(1) process?

What is a martingale?

What is a Markov process?

English version