# PROBABILITY FOR FINANCE

**Anno accademico 2020/2021**- 2° anno

**Docente:**

**Antonino Damiano Rossello**

**Crediti:**9

**Organizzazione didattica:**225 ore d'impegno totale, 165 di studio individuale, 60 di lezione frontale

**Semestre:**2°

ENGLISH VERSION

## Obiettivi formativi

**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’.**Applying knowledge and understanding**: Probability theory gradually learned should be applied to model (selected) financial problems and then to solve them, acting as a practitioner working in the financial industry. To this end, real world cases are discussed and critically analyzed during the classroom.**Making judgments**: The interaction between students and the instructor aims to stimulate their ability to judge the treated probabilistic models of risk management and financial engineering. Students should be able to revise them by the aid of information sources such as journal articles, working papers, empirical studies dataset, etc., available on the web.**Communication skills**: The learning process (with a modular structure) is intended to provide students with proper probabilistic language and notation. Students are expected to critically understanding and to circulate them as they acted in a real financial context.**Learning skills**: The course features typical aspects of applied mathematics. A certain degree of mathematical sophistication is also required. Students are provided with exercises, whose solutions are discussed during the classroom. Students are strongly required to ask questions concerning theoretical and practical aspects of the probabilistic financial models.

## 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. 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

**1st MODULE (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.

**2nd MODULE (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.

**3rd MODULE (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

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

**Additional Textbooks (not mandatory)**

- Statistics and Finance – D. Ruppert – Springer 2004
- Statistics of Financial Markets – J. Franke, W.K. Hardle, C.M. Hafner – Springer 2015
- Probability Essentials – J. Jacod, P. Protter – Springer 2004
- Elementary Sotchastic Calculus (With Finance in View) – T. Mikosch – World Scientific 1998
- Essential Mathematics for Market Risk Management – S. Hubbert – Wiley 2012
- Mathematical Techniques in Finance – A. Cerny – Princeton University Press 2009
- Statistical Methods for Financial Engineering – B. Rémillard – CRC Press 2013

## Programmazione del corso

Argomenti | Riferimenti testi | |
---|---|---|

1 | 1 *Random experiments. Events and their manipulation. Probability measure and its properties. | Bertsekas-Tsitsiklis Ch 1. Jacod-Protter Ch 1. |

2 | 2 * Conditional probability. Independence of events. Law of total probability. Bayes Theorem. | Bertsekas-Tsitsiklis Ch 1. Jacod-Protter Chs 1, 2. |

3 | 3 *Discrete random variables and probability measures. Probability mass and distribution functions. Bernoulli scheme. Some useful distributions (binomial, Poisson). | Bertsekas-Tsitsiklis Chs 1, 2. Jacod-Protter Ch 4. Instructor’s notes. |

4 | 4 *Continuous random variables and probability measures on R via distribution functions. Density function. Some useful distributions (uniform, exponential, normal, gamma, log-normal). | Bertsekas-Tsitsiklis Chs 2, 3. Jacod-Protter Chs 5, 6, 7, 8, 11. Instructor’s notes. |

5 | 5 *Random vectors and probability measures on Rn. Discrete and continuous joint distributions. | Bertsekas-Tsitsiklis Chs 2, 3. Jacod-Protter Ch 12. Instructor’s notes. |

6 | 6 *Sigma-algebra generated by random variables and random vectors. Independent random quantities. Risk and expected return of a portfolio. | Bertsekas-Tsitsiklis Chs 2, 3. Jacod-Protter Ch 12. Mikosch Ch 1. Instructor’s notes |

7 | 7 *Expectation: discrete and continuous random variables. General expectation and Stieltjes integral. Quantile function. | Bertsekas-Tsitsiklis Chs 2, 3. Jacod-Protter Ch 9. Instructor’s notes. |

8 | 8 *Monotone convergence Theorem. Change of variable Theorem. Radon-Nykodim Theorem. | Bertsekas-Tsitsiklis Ch 3. Jacod-Protter Ch 9. Instructor’s notes. |

9 | 9 *Some useful inequalities: Jensen, Holder, Minkowski, Cauchy-Schwartz, Chesychev. | Bertsekas-Tsitsiklis Ch 7. Jacod-Protter Ch 23. Mikosch Appendix A. Instructor’s notes. |

10 | 10 *Conditional distributions and Conditional expectation. Product measures (time permitting). | Bertsekas-Tsitsiklis Chs 3, 4. Jacod-Protter Chs 12, 23. Mikosch Ch 1. Instructor’s notes. |

11 | 11 *Moments and Lp classes of random variables. Variance and covariance. Correlation. | Bertsekas-Tsitsiklis Ch 4. Jacod-Protter Chs 5, 9. Mikosch Ch 1. Instructor’s notes. |

12 | 12 *Symmetric distributions. Summary statistics. Moments of random vectors. Equality in distribution. | Bertsekas-Tsitsiklis Chs 2, 3. Jacod-Protter Chs 11, 12. Hubbert Chs 9, 11. Ruppert Ch 2. Franke et al. Ch 3. Instructor’s notes. |

13 | 13 *Transformation of random variables: Moment generating function and characteristic function. | Bertsekas-Tsitsiklis Ch 4. Jacod-Protter Chs 11, 12, 13. Hubbert Ch 11. Instructor’s notes. |

14 | 14 *Introduction to stochastic processes: Paths, filtrations, mean and auto-covariance function. FIDIS. Kolmogorov extension Theorem (time permitting). | Bertsekas-Tsitsiklis Chs 5, 6. Ruppert Chs 3, 4. Franke et al. Chs 4, 5. Mikosch Ch 1. Instructor’s notes. |

15 | 15 *Some useful models of stochastic processes: white noise; Bernoulli; random walk; martingale; Brownian motion (Wiener); Markov. | Bertsekas-Tsitsiklis Chs 5, 6. Ruppert Chs 3, 4. Franke et al. Chs 3, 5, 11, 12. Mikosch Ch 1. Hubbert Ch 15. Instructor’s notes. |

16 | 16 *Convergence of random variables: almost surely; in distribution. Convergence theorems: WLLN; CLT. | Bertsekas-Tsitsiklis Ch 7. Jacod-Protter Chs 17, 20, 21. Franke et al. Ch 17. Mikosch Appendix A. Instructor’s notes. |

17 | 17 Copula and dependence in finance (selected). | Franke et al. Chs 11, 17. Instructor’s notes. |

18 | 18 * Regression and Expected value. Some relevant financial examples | Ruppert Chs 6, 7. Franke et al. Ch 11. Instructor’s notes |

19 | 19 *Log-normal model of financial returns. Value-at-Risk and coherent risk measures. Performance indexes. | Mikosch Ch 4. Hubbert Ch 5, 9, 10, 11, 13. Franke et al.Chs 11, 16, 18. Instructor’s notes. |

20 | 20 Binomial model of stock price, and its convergence to the Geometric Brownian motion. | Bertsekas-Tsitsiklis Ch 5. Ruppert Chs 3, 8. Franke et al. Chs 2, 4, 5. Instructor’s notes. |

21 | 21 Diffusive processes. Application: time evolution of a replicating portfolio. Self-financing portfolio. Ito’s integral and trading gains (time permitting). | Ruppert Ch 8. Franke et al. Chs 2, 5. Mikosch Chs 3, 4. Hubbert Ch 13. Instructor’s notes. |

22 | 22 Introduction to Stochastic Differential Equations (SDE). Ito’s Lemma. | Franke et al. Ch 5. Mikosch Ch 3. Instructor’s notes. |

23 | 23 Pricing of simple derivative instruments: Risk-neutral approach. Black-Scholes formula. | Ruppert Ch 8. Franke et al. Chs 2, 6, 7. Mikosch Ch 4. Instructor’s notes. |

24 | 24 Empirical distribution function and the plug-in estimator. Glivenko-Cantelli Theorem (time permitting). An idea of robustness. | Instructor’s notes |

25 | 25 Expectiles and Expected Shortfall, as coherent risk measures. | Instructor’s notes. |

26 | 26 Some stochastic orders. The very basics of VBA for excel (time permitting). | Instructor’s notes. |

27 | 27 Monte-Carlo simulation: pricing of a vanilla option and of a path-dependent option. Simulating some relevant distributions. | Instructor’s notes. |

28 | 28 Introduction to stochastic optimization over time. | Cerny Chs 3, 4, 9. Instructor’s notes. |

29 | 29 Discrete-time dynamic programming: pricing of American options. | Instructor’s notes. |

30 | A star, *, indicates those topics needed to pass the exam |

## Verifica dell'apprendimento

### Modalità di verifica dell'apprendimento

**Written class test:** is mandatory and gives rise up to 30% of the final mark (9/30). It consists of 8 questions with multiple choices. Students pass the test whenever they answer at least 4 questions rightly.

**Oral examination: **students who passes the class test must be further examined through 4 to 6 oral questions. This is also mandatory, and gives rise up to 70% of the final mark (21/30).

No partial exams.

A dedicated final exam will be reserved to regularly attending students, at the end of the course. It consists of a written and oral examination, the latter is carried out some days after the former during the scheduled exam periods.

### 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?