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Paulo Rocha
PT

Paulo Rocha

Mathematician and university lecturer

  • BScPhysics2012
  • MScMathematics2014
  • PhDMathematics2026

Faculty of Sciences, University of Lisbon

Research on optimal control of stochastic differential equations and stochastic volatility models, with applications to finance. Teaching Mathematics, Statistics and Programming at university level since 2021.

Simulated one-year asset price paths in the α-hypergeometric stochastic volatility model, where W and Z are correlated Brownian motions. This is the model studied in the two most recent papers.

News

Papers, projects and other news

Journal of Applied Mathematics and Computing Pricing European-options with α-hypergeometric stochastic volatility model under parameter uncertainty João Boto · Fernanda Cipriano · Paulo Rocha Abstract 72, 232 · 2026
New paperSeptember 2026

Pricing European-options with α-hypergeometric stochastic volatility model under parameter uncertainty

Journal of Applied Mathematics and Computing 72, 232. With João Pedro Boto and Fernanda Cipriano.

Abstract

In this paper we study the pricing of European options in a financial market modeled by the α-hypergeometric stochastic volatility model. We consider a setting that accommodates for parameter uncertainty, which is introduced by assuming that the model parameters are not known exactly but rather estimated from statistical data. The uncertainty is represented by a compact set of admissible parameters and we derive conservative bounds for the option price by treating the problem as an optimal control problem.

Read for free Journal page
Cite this article

APA

Boto, J., Cipriano, F., & Rocha, P. (2026). Pricing European-options with α-hypergeometric stochastic volatility model under parameter uncertainty. Journal of Applied Mathematics and Computing, 72(9), Article 232. https://doi.org/10.1007/s12190-026-02888-8

BibTeX

@article{Boto2026,
  author  = {Boto, Jo{\~a}o and Cipriano, Fernanda and Rocha, Paulo},
  title   = {Pricing {European}-options with $\alpha$-hypergeometric stochastic volatility model under parameter uncertainty},
  journal = {Journal of Applied Mathematics and Computing},
  year    = {2026},
  volume  = {72},
  number  = {9},
  pages   = {232},
  doi     = {10.1007/s12190-026-02888-8},
  url     = {https://doi.org/10.1007/s12190-026-02888-8}
}
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Research

Stochastic analysis and its applications to finance

Current work addresses stochastic optimal control problems in finance, such as portfolio choice with consumption and European option pricing under parameter uncertainty, in the α-hypergeometric stochastic volatility model.

Earlier work dealt with the stochastic modelling of non-stationary financial time series, in particular the evolution of volume-price distributions in stock markets.

Recent papers

Research interests

  • Stochastic analysis
  • Optimal control
  • Mathematical finance
  • Numerical analysis
  • Dynamical systems
  • Mathematical physics
  • Linear algebra
  • Computer science

Publications

Peer-reviewed journal articles and conference proceedings

  1. 2026

    Pricing European-options with α-hypergeometric stochastic volatility model under parameter uncertainty

    Boto, J., Cipriano, F., Rocha, P.

    Journal of Applied Mathematics and Computing 72, 232

    DOIFree to readQuartilesSJRScopusJCR

  2. 2025

    Portfolio problem for the α-hypergeometric stochastic volatility model with consumption

    Boto, J., Cipriano, F., Rocha, P.

    Journal of Mathematical Analysis and Applications 543 (2), 128891

    DOIQuartilesSJRScopusJCR

  3. 2017

    Stochastic modelling of non-stationary financial assets

    Estevens, J., Rocha, P., Boto, J. P., Lind, P. G.

    Chaos 27, 113106

    DOIarXivQuartilesSJRScopusJCR

  4. 2016

    Uncovering the evolution of nonstationary stochastic variables: the example of asset volume-price fluctuations

    Rocha, P., Raischel, F., Boto, J. P., Lind, P. G.

    Physical Review E 93, 052122

    DOIarXivQuartilesSJRScopusJCR

  5. 2015

    Stochastic evolution of stock market volume-price distributions

    Rocha, P., Raischel, F., Cruz, J., Lind, P. G.

    Proceedings of the 3rd SMTDA Conference, pp. 619-627

    arXiv

  6. 2014

    Optimal models of extreme volume-prices are time-dependent

    Rocha, P., Raischel, F., Boto, J. P., Lind, P. G.

    Journal of Physics: Conference Series 574, 012148

    DOIarXivQuartilesSJRScopus

Conference talks

Ten talks and one poster at international conferences in Portugal, Spain, Germany and Tunisia.

  • July 2025

    BIP 2025

    BIP Tech4Society

    Sintra, Portugal

  • July 2025

    JIM 2025

    Summer JIM Days

    Coimbra, Portugal

  • July 2022

    EUROPT 2022

    19th Workshop on Advances in Continuous Optimization

    Caparica, Portugal

  • November 2020Online

    Hammamet 2020

    Recent Developments in Stochastics

    Online

  • October 2019

    Hammamet 2019

    International Conference on Control, Games and Stochastic Analysis

    Hammamet, Tunisia

  • September 2018

    Graph Drawing 2018

    26th International Symposium on Graph Drawing and Network Visualization

    Barcelona, Spain

  • June 2017

    CHAOS 2017

    10th Chaotic Modeling and Simulation International Conference

    Barcelona, Spain

  • July 2015

    Summer School 2015

    Novos Talentos em Matemática

    Lisbon, Portugal

  • August 2014

    IC-MSQUARE 2014

    International Conference on Mathematical Modeling in Physical Sciences

    Madrid, Spain

  • June 2014

    SMTDA 2014

    3rd Stochastic Modeling Techniques and Data Analysis International Conference

    Lisbon, Portugal

  • April 2014Poster

    DPG 2014

    DPG-Frühjahrstagung

    Dresden, Germany

Theses

Master's and doctoral theses, in open access

PhD in Mathematics · 2026

Open access

Optimal Control of Stochastic Differential Equations in Finance

Asset allocation and option pricing under the α-hypergeometric stochastic volatility model. It solves the portfolio problem with consumption through the Hamilton-Jacobi-Bellman equation and derives price bounds for European options when the model parameters are estimated.

Supervisors
João Pedro Boto and Fernanda Cipriano
Specialty
Mathematical Analysis
Institution
Faculty of Sciences, University of Lisbon

MSc in Mathematics · 2014

Open access

Stochastic evolution of parameters defining probability density functions: application to the New York stock market

Evolution of non-stationary time series and extraction, from empirical data, of the stochastic differential equations that describe it, applied to volume-price distributions in the New York stock market.

Supervisors
Pedro G. Lind and João Pedro Boto
Institution
Faculty of Sciences, University of Lisbon

Academic profiles

Publications and citations in bibliographic databases

Name in citations
Rocha, P.
Scientific domain
Mathematics and Physics

Teaching

Mathematics, Statistics and Programming in bachelor's and master's programmes

Academic year 2026/27

Teaching now · 1st semester
  • Álgebra LinearEscola Superior Náutica Infante D. Henrique
  • Matemática AplicadaIscte, University Institute of Lisbon
  • Tópicos de Matemática Elementar IIscte, University Institute of Lisbon
  • Tópicos de Matemática Elementar IIIscte, University Institute of Lisbon
  • Introduction to Python for Data AnalysisNova School of Business and Economics, master's
2nd semester
  • Geometria e EstatísticaIscte, University Institute of Lisbon
  • Álgebra Linear e GeometriaIscte, University Institute of Lisbon

Teaching materials

  • Escola Superior Náutica Infante D. Henrique · 2026/27

    Álgebra Linear

    Lecture notes, exercise sheets, slides, animations and quizzes, written from scratch for the course (in Portuguese).

  • Iscte, University Institute of Lisbon · 2026/27

    Tópicos de Matemática Elementar I

    Lecture notes and exercise sheets for the preparatory mathematics modules, with step-by-step worked examples and solutions (in Portuguese).

  • Iscte, University Institute of Lisbon · 2026/27

    Tópicos de Matemática Elementar II

    Syllabus of the preparatory mathematics modules, with lecture notes and exercise sheets in preparation (in Portuguese).

  • Iscte, University Institute of Lisbon · 2025/26

    Geometria e Estatística

    Interactive online notes with demos, step-by-step examples and quizzes, and a textbook with solved exercises (in Portuguese).

  • Iscte, University Institute of Lisbon · 2025/26

    Estatística e Probabilidades

    Textbook and collection of solved exercises.

  • Iscte, University Institute of Lisbon · 2024/25

    Moodle exercise bank

    About 900 exercises for Matemática Aplicada, Introdução à Estatística, Estatística e Probabilidades, Geometria e Estatística and Complementos de Matemática.

Courses

Bachelor's degrees

  • Álgebra Linear, course coordinator2026/27

Bachelor's degrees

  • Matemática para Farmácia2023/24 to 2025/26
  • Elementos de Álgebra Linear2024/25
  • Álgebra Linear e Geometria Analítica2023/24
  • Análise Matemática IV2023/24

Bachelor's degrees

  • Matemática Aplicada2024/25 to 2026/27
  • Tópicos de Matemática Elementar I2022/23, 2024/25 to 2026/27
  • Tópicos de Matemática Elementar II2022/23, 2024/25 to 2026/27
  • Álgebra Linear e Geometria2026/27
  • Geometria e Estatística2023/24 to 2026/27
  • Fundamentos de Matemática2025/26
  • Introdução à Estatística2024/25
  • Estatística e Probabilidades2024/25
  • Geometria e Trigonometria2022/23, 2023/24
  • Lógica, Probabilidades e Combinatória2022/23, 2023/24
  • Funções e Números Complexos2022/23, 2023/24
  • Derivadas de Funções e Aplicações2021/22 to 2023/24
  • Optimization2021/22

Bachelor's degrees

  • MS Office Specialist/Expert Excel2022/23 to 2024/25
  • Linear Algebra with Programming2021/22, 2022/23
  • Calculus II2021/22

Master's degrees

  • Introduction to Python for Data Analysis2026/27

Master's degrees, grading assistant

  • Data Curation for Business Analytics2021/22, 2025/26
  • Introduction to Python for Data Analysis2025/26
  • Numerical Methods for Economics and Finance2025/26
  • Computational Thinking2022/23

Background

Education and employment

Education

  1. 2026

    PhD in Mathematics, Mathematical Analysis

    Faculty of Sciences, University of Lisbon

  2. 2014
  3. 2012

    BSc in Physics

    Faculty of Sciences, University of Lisbon

    Final grade 16/20

Employment

  1. since 2026

    Invited Lecturer

    Escola Superior Náutica Infante D. Henrique

    Course coordinator, Linear Algebra

  2. since 2022
  3. since 2021
  4. 2023 to 2026
  5. 2016 to 2018

    Research Assistant

    CMAF-CIO, University of Lisbon

    Linear Algebra and Computational Mathematics, with Christian Barbarosie

  6. 2015

    Research Assistant

    ForWind, University of Oldenburg, Germany

    Computational and Theoretical Physics, with Pedro G. Lind

  7. 2014 to 2015

    Research Assistant

    CMAF-CIO, University of Lisbon

    Linear Algebra and Computational Mathematics, with Christian Barbarosie

  8. 2012

    Research Assistant

    CFTC, University of Lisbon

    Computational and Theoretical Physics, with Pedro G. Lind

Other projects

Granymede

Author and developer · Vue 3, TypeScript and Pyodide

  • Web app for grading Jupyter (.ipynb) and Python (.py) assignments securely, right in the browser.
  • Suggests grades, runs student code, detects plagiarism and invisible Unicode characters, and exports grades to CSV.

Earth Quiz

Author and developer

  • Trivia game about world rankings, where you guess in which category each country ranks best (population, GDP, tourism, football and more).
  • Three game modes (classic, reverse and time attack), for iPhone and iPad.

Bullfight Finder

Founder and lead software developer

  • App for iPhone and Android, released on the App Store in 2022, with the calendar and alerts for the street bullfights of Terceira island.
Finalist · Hack to ReEmerge 2 2022

Localy

Founder and lead software developer

  • Tourism entrepreneurship project.
Regional winner · Tourism Explorers 2019 National finalist · Tourism Explorers 2019 Finalist · Hack to Emerge 2020 Finalist · Hack To ReEmerge 2021 Present at Web Summit 2019

Training and tutoring

Private tutoring

Private tutor

Mathematics

  • Mathematical Analysis
  • Calculus
  • Linear Algebra
  • Statistics

Physics

  • Mechanics
  • Electromagnetism

Programming

  • Python

Training

Instructor outside higher education

  1. 2023

    Microsoft 365

    Chamber of Commerce of Angra do Heroísmo

  2. 2022

    Programação para Gamers and Programação de Aplicações

    Academia J, Regional Youth Directorate of the Azores

Skills

Languages, programming and software

Languages

  • PortugueseNative
  • EnglishAdvanced
  • SpanishIntermediate
  • ItalianBasic