Risk Analysis and Management

Kockázatelemzés és -kezelés

A tantárgyleírás hatályossága

Hatályosság kezdete:

2026. March 21.

Hatályosság vége:

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Subject name (Hungarian, English)

Kockázatelemzés és -kezelés

Risk Analysis and Management

Subject code

BMEVIHIM279

Subject type

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Training Level

—

Course types and hours (weekly/semester)

Course type	lecture	tutorial	laboratory
hours (weekly)	3	0	1
type (linked/independent)	—	—	derived course

Assessment type

vizsga

Credits

Subject coordinator

DR. Telek Miklós József

position: egyetemi tanár

contact: telek.miklos@vik.bme.hu

Responsible department

Hálózati Rendszerek és Szolgáltatások Tanszék

Faculty

Villamosmérnöki és Informatikai Kar

Subject website

edu.vik.bme.hu

Primary curriculum type

—

Direct prerequisites – Strong prerequisite

none

Direct prerequisites – Weak prerequisite

none

Direct prerequisites – Parallel prerequisite

none

Direct prerequisites – Milestone prerequisite

none

Direct prerequisites – Exclusion

none

Objectives

Programme

Systems of linear equations and their solvability, special matrices (identity matrix, diagonal matrix, unitary matrix), commutativity of matrices

Singular value decomposition and its graphical interpretation, relationship between SVD and systems of linear equations, linear matrix equations, Kronecker product, Sylvester equation and its solution using the Kronecker product

Spectral decomposition, diagonalizability, dyadic form of spectral decomposition, computation of spectral decomposition, iterative methods for computing dominant eigenvalues and eigenvectors, application of spectral decomposition to matrix functions

Definition of random variables (distribution function, density function, probability mass function), key parameters (moments, variance, cumulants), law of total probability and its applications. Transformed functions (characteristic function, moment-generating function, etc.). Dependent random variables (joint distribution, joint density, joint mass function, marginal distributions). Normal distribution, standard normal distribution, multivariate normal distribution and its construction using independent normal distributions

Cash-flow management problem and its variants, brute-force solution. Inequalities for distributions and their applications: Markov inequality, Chebyshev inequality, moment-based inequalities

Inequalities for distributions and their applications: central moment inequalities, Chernoff bound, relationship between Chernoff bound and moment bounds, Cantelli inequality

Central limit theorem and its applications, sampling methods, Li-Sylvester method, random sampling (Monte Carlo simulation) and its application to risk estimation, stratified sampling and its effect on estimation uncertainty

Optimization of portfolio models, possible objective functions. Definition of the stationary multinormal case and determination of its parameters (expected value, covariance matrix). Portfolio optimization for risk minimization and its approximation

Estimation of expected value and covariance matrix of a portfolio based on samples, real-time (“on-the-fly”) estimation

Mean-reverting portfolio optimization: definition of the Ornstein–Uhlenbeck model, integral and differential formulations and their relationship. Autoregressive model as a stationary normal special case, stability condition, determination of covariance matrix based on time-domain representation

Optimal portfolio based on estimability, model identification, trading algorithm for mean-reverting portfolios

Fundamentals of mathematical models for pricing financial options: single-step binomial pricing model, assumptions and fundamental equation of the Black–Scholes model

General binary tree-based model, mathematical description of the binomial pricing model, computation of option prices, pricing without transaction costs for European and American options

Derivation of the Black–Scholes model, distribution and expected value of option prices

Detailed syllabus of exercises/labs

Python basics
Python practice
Jordan decomposition, application of the central limit theorem
Solving systems of linear equations, singular value decomposition
Solving the cash-flow management problem using brute-force methods
Approximate solution of the cash-flow management problem I – Markov inequality, central limit theorem
Approximate solution II – Chernoff bound
Approximate solution III – Li-Sylvester estimation
Comparison of solution methods for the cash-flow management problem
Portfolio risk minimization – approximate solution
Application of optimization methods to the mean-reverting portfolio model
Portfolio risk minimization – solution of the original problem
Identification of autoregressive models
Binomial pricing model

To provide comprehensive knowledge for future decision-makers about currently used risk analysis and risk management methodologies, as well as their application strategies. The course primarily focuses on mathematical tools suitable for identifying, managing, and avoiding risk-related problems arising in business practice.

Learning outcomes

Ez a tantárgy a KKK rendeletben meghatározott, következő kompetenciák fejlesztését szolgálja:

Knowledge

No learning outcomes recorded.

Skills

No learning outcomes recorded.

Attitudes

No learning outcomes recorded.

Autonomy and responsibility

No learning outcomes recorded.

Oktatási módszertan

Lecture, laboratory

Tanulástámogató anyagok

Online források

Krein: Survival Analysis, 3rd edition. Springer, 2005.; Wose: Risk Analysis: A Quantitative Guide, 2nd edition. Wiley, 2000.; Wosmer, Lemeshow, May: Applied Survival Analysis: Regression Modeling of Time to Event Data, 2nd ed., 2008.

Recommended preliminary knowledge for completing the subject

Knowledge type competencies

(azon előzetes ismeretek összessége, amelyek megléte nem kötelező, de a tantárgy eredményes teljesítését nagyban elősegíti)

basic mathematics, linear algebra, probability theory

Skill type competencies

(azon előzetes képességek és készségek összessége, amelyek megléte nem kötelező, de a tantárgy eredményes teljesítését nagyban elősegíti)

nincs

Recommended (non-compulsory) preliminary competencies

(azon ajánlott (nem kötelező) előzetesen megszerzendő kompetenciák összessége, amelyek jelentősen hozzájárulnak a tantárgy eredményes teljesítéséhez)

basic mathematics, linear algebra, probability theory

General rules

Requirements: During the semester: achieving at least a passing grade (40%) on the midterm test and attending at least 70% of the lab sessions is required for course completion. During the exam period: Achieving at least a passing grade (40%) on the written exam. The final grade is calculated as the average of the midterm, the exam, and the semester lab assignments (each weighted 1/3). Grading scale: 85%– excellent (5) 70–84% good (4) 55–69% satisfactory (3) 40–54% pass (2) 0–39% fail (1) Additional possibilities: One retake opportunity for the midterm is provided during the semester. For those who fail both the midterm and the retake, one additional opportunity is provided during the retake period. A signature (course completion) requires achieving at least a passing grade on one of these attempts (original, retake, or second retake).

Assessment methods

In-term assessments

No detailed assessments provided.

Weight of in-term assessments

No weights provided.

Exam-period assessments

No detailed assessments provided.

Weight of exam elements

No weights provided.

Grade calculation

No grade thresholds provided.

Attendance requirements

No attendance requirements provided.

Rules for retake and resubmission

Not provided.

Short description

Not provided.

Detailed description

Not provided.

Recommended courses

Not provided.

Workload to complete the subject

No workload breakdown provided.

Validity of subject requirements

Requirements valid from:

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Requirements valid until:

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Curriculum placement

No curriculum placements recorded for this subject version.