Teaching code

4S009132

Academic staff

Luca Di Persio,

Coordinator

Luca Di Persio

Credits

6

Language

Italian

Scientific Disciplinary Sector (SSD)

MAT/06 - PROBABILITY AND STATISTICS

Period

II semestre dal Mar 2, 2020 al Jun 12, 2020.

Lessons timetable Moodle Seminars 0

Learning outcomes

The aim of the course, fully delivered in flipped-classroom mode, is to introduce students to the foundations of the modern theory of financial mathematics, with specific reference to the study of concrete models, currently used by major banking institutions, of market risk analysis, and insurance.

Program

Introduction
Notation
Linear algebra primer
Vector spaces
Linear transformations
Spectral decomposition
Metric and normed spaces
Inner product spaces
Matrix transpose-square-root
Matrix operations
Calculus
Differentiation
Numerical derivatives
Taylor expansion
Integration
Monotone functions
Convexity
Week 1
Introduction
About the ARPM Lab
About quantitative finance: P and Q
The "Checklist": executive summary
The Checklist: Step 1 - Risk drivers identification
Risk drivers identification
Equities
Fixed-income
Derivatives
Credit
Applications
Case studies - Checklist: summary
Step 1. Risk drivers identification - Historical
Step 1. Risk drivers identification - Monte Carlo
Week 2
The Checklist: Step 2 - Quest for invariance (univariate)
Quest for invariance
Simple tests
Efficiency: random walk
Mean-reversion (continuous state): ARMA
Mean-reversion (discrete state)
Volatility clustering
Distributions
Representations of a distribution
Normal distribution
Notable multivariate distributions
Elliptical distributions
Scenario-probability distributions
Exponential family distributions
Mixture distributions
Applications
Step 2. Quest for invariance - Historical
Step 2. Quest for invariance - Monte Carlo
Week 3
Location and dispersion
Expectation and variance
Expectation and covariance
L2 geometry
Generalized location-dispersion: variational principles
Copulas
Univariate results
Definition and properties of copulas
The Checklist: Step 2 - Quest for invariance (multivariate)
Order-one autoregression
Cointegration
The Checklist: Step 3 - Estimation
Estimation
Setting the flexible probabilities
Historical
Maximum likelihood
Robustness
Applications
Step 3. Estimation - Historical
Step 3. Estimation - Monte Carlo
Week 4
Linear factor models
Factor models and learning
Overview
Regression LFM's
Principal component LFM's
Factor-analysis LFM's
Cross-sectional LFM's
Application: principal component analysis of the yield curve
Week 5
Machine learning: foundations and prediction
Overview
Point vs. probabilistic statements
Inference and learning
Least squares regression
Week 6
Bias reduction
Quantile and non-least-squares regression
Applications
Machine learning for hedging: introduction
Machine learning for hedging: least squares regression
Machine learning for hedging: least absolute distance regression
Week 7
Machine learning: foundations and prediction
Classification
Least squares autoencoders
Probabilistic graphical models
Week 8
Machine learning: out of sample enhancements
Estimation risk assessment
Regularization and features selection
Bayesian estimation
Ensemble learning
Credit default classification
Dynamic models
Overview
Linear state space models
Spectral representation
Probabilistic state space models
Week 9
The Checklist: Step 4 - Projection
Projection
One-step historical projection
Monte Carlo
Historical
The Checklist: Step 5 - Pricing at the horizon
Pricing at the horizon
Exact repricing
Taylor approximations
Applications
Step 4. Projection - Historical
Step 4. Projection - Monte Carlo
Step 5. Pricing at the horizon - Historical
Step 5. Pricing at the horizon - Monte Carlo
Week 10
The Checklist: Step 6 - Aggregation
Aggregation
Returns
Static market/credit risk
Enterprise risk management
The Checklist: Step 7 - Ex-ante evaluation
Ex-ante evaluation
Stochastic dominance
Satisfaction/risk measures
Mean-variance trade-off
Expected utility and certainty-equivalent
Quantile (value at risk)
Spectral satisfaction measures/Distortion expectations
Applications
Step 6. Aggregation - Historical
Step 6. Aggregation - Monte Carlo
Step 7. Ex-ante evaluation - Historical
Step 7. Ex-ante evaluation - Monte Carlo
Week 11
The Checklist: Step 8a - Ex-ante attribution: performance
Ex-ante attribution: performance
Bottom-up exposures
Top-down exposures: factors on demand
The Checklist: Step 8b - Ex-ante attribution: risk
Risk budgeting: general criteria
Homogenous measures and Euler decomposition
Applications
Step 8. Ex-ante attribution - Historical
Step 8. Ex-ante attribution - Monte Carlo
Week 12
The Checklist: Step 9a - Construction: portfolio optimization
Construction: portfolio optimization - Overview
Mean-variance principles
Analytical solutions of the mean-variance problem
Continuous programming
Integer N-choose-K heuristics
Mean-variance pitfalls
The Checklist: Step 10 - Execution
Execution
High frequency risk drivers
Market impact modeling
Order scheduling
Applications
Step 9. Construction - Historical
Step 9. Construction - Monte Carlo
Step 10. Execution - Historical
Step 10. Execution - Monte Carlo

Examination Methods

The exam will consist in the presentation of a project, an in-depth study of one of the founding themes of the entire course, with particular reference to stochastic models for calculating risk. Furthermore, using the flipped-classroom mode, students will be asked to carry out weekly exercises relating to the individual topics covered during the lessons, so that the final grade will be expressed by mediating between the results obtained in solving these weekly exercises, and the grade obtained post presentation of the aforementioned project.