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Research in Mathematical & Computational Finance

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The Oxford Mathematical and Computational Finance Group is one of the leading academic research groups  in the world focused on mathematical modeling in finance and offers a thriving research environment, with experts covering multiple  areas of quantitative finance. Our group maintains close links with the Data Science , Stochastic Analysis and  Numerical Analysis groups as well as the Institute for New Economic Thinking (INET), the Alan Turing Institute (Machine Learning in Finance ) ,  DataSig ,  the Oxford-Man Institute of Quantitative Finance  and the Oxford Probability Group ,  enabling cross-fertilisation of ideas and techniques.

Research activities of the group cover a wide spectrum of topics in Quantitative Finance , ranging from  market microstructure and high-frequency modeling  to macro-financial modeling and systemic risk, as well as more traditional topics such as  portfolio optimisation, derivative pricing, credit risk modeling, using a variety of methods: stochastic analysis, probability, partial differential equations, optimisation, numerical simulation, statistics and machine learning.

Mathematical Foundations and Continuous-time finance

Positioned within Oxford's Mathematical Institute, the group has developed a unique expertise in the mathematical foundations underlying quantitative finance and pioneered new approaches in mathematical modeling.

Sam Cohen , Rama Cont , Ben Hambly , Blanka Horvath ,  Jan Obloj   and Zhongmin Qian explore topics in stochastic analysis  -stochastic calculus, backward stochastic differential equations, interacting particle systems, Malliavin calculus, Functional Ito calculus, rough path theory, pathwise methods in stochastic analysis, optimal transport- and their applications to the design of robust models for the pricing and hedging of derivatives in presence of model uncertainty.   Michael Monoyios works on duality methods for optimal investment and consumption problems, and on valuation and hedging problems in incomplete markets. He has worked on models with transaction costs, and with partial and inside information on asset price evolution. He has interests in Fernholz's stochastic portfolio theory, and on the geometric interpretation of functionally generated portfolios that arise in this theory.  Jan Obloj works on robust formulations of classical problems -- pricing, hedging, risk management, optimal investment – and seeks to understand and quantify the effects of model uncertainty. Blanka Horvath focusses on implied volatility modelling, rough volatility models, stochastic volterra equations and stochastic volatility models their short -time asymptotic properties as well as their numerical properties for pricing, hedging and simulation.

Statistical modeling and Machine Learning in Finance

Our group is one of the few academic research teams in the world with an active research agenda at the interface of machine learning and quantitative finance. Several group members are Fellows of the Alan Turing Institute. Hanqing Jin is Director of the Oxford-Nie Big Data Lab , where Ning Wang has developed algorithms for sentiment analysis based on social media data. Sam Cohen  is exploring applications of Deep Learning  to continuous-time finance as well as issues related to model robustness and its interaction with statistical modelling and optimal control. Rama Cont ,  Blanka Horvath   and Justin Sirignano investigate the use of Deep Learning and  data-driven modelling in finance. Terry Lyons and his team investigate the use of rough path signatures for machine learning. Jan Obloj   employs tools from the optimal transport theory to develop data-driven estimators for risk measures, and to quantify robustness of deep neural networks to adversarial attacks. Blanka Horvath   develops deep learning tools for option pricing, (deep) calibration and hedging and for data-driven simulation of asset price dynamics and data-driven portfolio choice problems.

Market microstructure and algorithmic finance

Álvaro Cartea   focuses on mathematical models of algorithmic trading and  the design of optimal trade execition strategies in electronic markets.

Rama Cont pioneered the analytical study of stochastic models for limit order books and intraday market modeling, and investigates the impact of algorithmic trading on market stability and liquidity.

Leandro Sanchez-Betancourt studies the equilibrium between makers and takers of liquidity with continuous-time models and tools from stochastic control and machine learning.

Macro-financial modeling: financial stability and systemic risk

Our group is actively engaged in the development of mathematical models of large-scale financial systems with the goal of providing quantitative insights on financial stability and systemic risk to regulators and policy makers. Rama Cont and Ben Hambly   investigate the link between micro- and macro-behavior in stochastic models of direct and indirect contagion in financial markets, using network models and analogies with interacting particle systems.

Rama Cont ,Research Fellow at the  Institute for New Economic Thinking (INET), have developed network models and simulation-based approaches for macro stress-testing and monitoring systemic risk in banking systems, in liaison with central banks and international organisations such as the Bank of England, the European Central Bank, IMF and Norges Bank.

Rama Cont   is  Director of the Oxford Martin Programme on Systemic Resilience , an interdisciplinary programme aimed at exploring solutions for managing stress scenarios with the potential for major and prolonged economic disruption, severe human or economic impacts, and contagion. 

Computational Finance

Our group is a leader in the development of advanced numerical methods and  high performance computiing for high-dimensional problems in finance: Mike Giles  is a pioneer  on multilevel Monte-Carlo methods and their applications in finance, and a leading expert on the use of GPU and high performance computing methods in finance. Raphael Hauser has developed robust numerical methods for portfolio optimisation and high-dimensional optimisation problems in finance. Jan Obloj develops numerical methods for martingale optimal transport problems which yield bounds for option prices and optimal transport techniques for model calibration. Justin Sirignano has pioneered the use of Deep Learning methods for various applications in finance ranging from credit risk modeling to limit order book modeling. Christoph Reisinger develops novel and efficient numerical methods for stochastic control problems and high-dimensional (S)PDEs and their applications in finance; Terry Lyons devised cubature methods in Wiener space for solving stochastic differential equations. Sam Howison and Jeff Dewynne were among the pioneers in the development of advanced  partial differential equation methods in finance, the use of asymptotic methods for their solution and their application to various markets such as energy and commodities. Blanka Horvath   develops numerical solutions for pricing, hedging and optimal investment problems and analytic- and asymptotic methods for a wide variety of stochastic models for equity, FX and interest rate modelling. The numerical methodologies explore path-dependent   data-driven machine learning solutions as well as quantum machine learning algorithms.

Behavioural finance

Hanqing Jin   develops quantitative models of investor behaviour, building on the fundamental work of Kahneman and Tversky's prospect theory and Lopes' SP/A theory. Ning Wang  is working on sentiment analysis based on social media data, as well as on using data to establish metrics for learning and identification purposes. Jan Obloj works on optimal decision problems for cumulative prospect theory agents and understanding their actions in dynamic environments, such as casino gambling.

For more information on research activities of our group please visit the individual websites of group members .

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Mathematical Finance

Mathematical Finance is the field of mathematics that studies financial markets. Topics in financial markets studied include market trading mechanisms, called market microstructure, corporate management decision making, called corporate finance, investment management, and derivative securities. In each of these areas, sophisticated mathematics is utilized for modeling purposes. The theory of stochastic processes, stochastic optimization, partial differential equations, and simulation methods are just some of the mathematical tools employed. For example, in the area of derivatives, stochastic calculus is used to price a call option on a common stock. A call option is a financial security that gives its owner the right to buy a common stock at a fixed price on or before a fixed future date. Using stochastic calculus, the price of a call option can be characterized as the expected value of a nonlinear and random payoff at a future date. Numerical methods, such as Monte Carlo simulation, are often used to compute these expected values.

Department of Mathematics

Mathematical finance and stochastic analysis

Our research interests span a broad range of topics in continuous and discrete time.

In mathematical finance our areas of research activity include:

• arbitrage and option pricing in markets with friction and incomplete markets
• entropy and financial value of information
• optimal investment strategies in markets, with prices depending on the volume of trading
• robust arbitrage and model-independent pricing
• discrete time models and their continuous time limits in the presence of market imperfections
• numerical methods for pricing financial derivatives
• applications of optimal stopping, singular control, and game theory to investment problems in the real economy ("real options").

In stochastic analysis our research focuses on:

• infinite dimensional stochastic analysis, including stochastic differential equations on infinite dimensional manifolds
• stochastic partial differential equations (especially stochastic Navier-Stokes and Euler equations arising in the context of turbulence phenomena)
• stochastic analysis on Riemannian and Finslerian manifolds
• rough paths and their applications to modelling probabilistic phenomena and numerical analysis (for example non-linear filtering)
• Feynman path integrals and more broad applications to mathematical physics, biology and finance.

We welcome PhD applications across a range of mathematical finance and stochastic analysis topics.

[email protected]

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We run regular seminars and host talks by external speakers from the UK and overseas, covering a wide range of topics of current interest in stochastic analysis and mathematical finance.

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Research topics.

The Center faculty are highly research-active, publishing many articles each year. They also regularly recruit new graduate students to their groups. Among  themes that are presently investigated are: Mean Field Games for Systemic Risk; Stochastic Portfolio Theory; Gaussian Process Regression for Portfolio Risk Management; Limit Order Book modeling; Contagion in Random Financial Networks; Stochastic Volatility models;  Monte Carlo methods for Stochastic Control; Stochastic Games.

Jean-Pierre Fouque (Distinguished Professor and Co-Director of the CFMAR) Stochastic processes. Financial Mathematics. Volatility modeling. Systemic risk, Mean-field Games Publications

Mike Ludkovski (Professor and Co-Director of the CFMAR) Monte Carlo simulation; Machine Learning for Stochastic Control; Energy Markets & Stochastic Games; Modeling of Renewable Power Generation; Longevity Risk. Publications

Tomoyuki Ichiba (Associate Professor PSTAT) Probability Theory, Stochastic Processes and their applications. Stochastic Differential Equations, Collisions of Brownian Particles, Local Time of Semimartingales, Mathematical Economics & Finance (Stochastic Portfolio Theory), and Statistics in Finance

Nils Detering (Assistant Professor PSTAT) Financial Mathematics: Systemic risk, energy markets and model risk; Probability theory: Stochastic Analysis and Random graphs, especially percolation on random graphs

Alex Shkolnik  (Assistant Professor PSTAT)

Quantification and management of credit risk; factor models for portfolio selection; simulation of jump-diffusion processes

Ruimeng Hu  (Assistant Professor PSTAT and MATH)

Machine learning, financial mathematics, and game theory: Deep learning algorithms and theory for stochastic differential games; mean-field portfolio games; portfolio optimization;  and optimal switching problems; systemic risk and central counterparty.

Recent advances in mathematical methods for finance

• Open access
• Published: 04 April 2024
• Volume 336 , pages 1–2, ( 2024 )

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• Giorgia Callegaro 1 ,
• Claudio Fontana 1 ,
• Martino Grasselli 1 ,
• Wolfgang J. Runggaldier 1 &
• Tiziano Vargiolu 1  

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In recent years, Mathematical Finance has witnessed the emergence of new research directions spurred by developments of financial markets, technological advances, and societal challenges. On the one hand, financial markets have seen the introduction of new financial products, regulatory frameworks, and trading infrastructures. On the other hand, artificial intelligence and machine learning techniques are introducing revolutionary changes in numerical methods in finance, overcoming computational challenges considered insurmountable until recently. In addition, new types of risks, such as climate-related and cyber-risks, have gained prominence, significantly impacting financial institutions and society at large.

This special issue on Recent Advances in Mathematical Methods for Finance provides a comprehensive overview of some of the latest developments in Mathematical Finance. We decided to launch this special issue on the occasion of the 10th General AMaMeF Conference, organised by the Guest Editors at the University of Padova and held in a virtual format on June 22–25, 2021. AMaMeF is the acronym for Advanced Mathematical Methods for Finance, and was born as a programme network of the European Science Foundation from 2005 to 2010, under the Sixth Framework Program for research and technological development of the European Union. AMaMeF now represents a European network of research promoting the exchange and diffusion of knowledge in the field of Mathematical Finance, spanning more than 20 countries. The biannual general conference stands as the flagship event of the AMaMeF network. The 10th General AMaMeF Conference spanned a broad range of topics in mathematical finance, including algorithmic trading and financial technologies, asset pricing under market frictions, collateralization and XVA, credit risk and interest rate modeling, energy and commodity markets, equilibrium and principal-agents models, climate risk, green and sustainable finance, machine learning and computational methods in finance, market microstructure, mean-field games and McKean–Vlasov equations, model uncertainty, model risk and robust finance, risk measures, stochastic control and portfolio optimization, stochastic volatility modeling, systemic risk and financial networks. These topics were specifically targeted by the call for papers for the special issue, which was open to the entire scientific community and not restricted to papers presented at the conference.

The special issue contains 44 papers, which underwent a rigorous peer review process under the supervision of the Guest Editors. Coherently with the title of the special issue, in the selection of the submitted papers emphasis was placed on the originality and interest of the mathematical methods employed, alongside the relevance of their financial applications. The selected papers encompass theoretical contributions as well as more applied research, offering a comprehensive view of promising research directions in mathematical finance.

We are thankful to Prof. Endre Boros, Editor-in-Chief of Annals of Operations Research , for giving us the opportunity to edit this special issue and to the Springer staff for their assistance throughout the production process. We are grateful to the referees for their valuable feedback and constructive criticisms, which aided in the selection of the submissions and enhanced the quality of accepted papers. Finally, our most sincere gratitude goes to the authors of the submitted papers, for contributing their work to this special issue. We hope that this collection of papers will stimulate further research on several emerging topics in Mathematical Finance.

Open access funding provided by Università degli Studi di Padova within the CRUI-CARE Agreement.

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Giorgia Callegaro, Claudio Fontana, Martino Grasselli, Wolfgang J. Runggaldier & Tiziano Vargiolu

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Callegaro, G., Fontana, C., Grasselli, M. et al. Recent advances in mathematical methods for finance. Ann Oper Res 336 , 1–2 (2024). https://doi.org/10.1007/s10479-024-05959-w

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Mathematical finance research topics ideas [ms phd].

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Long-Memory Models in Mathematical Finance

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About this Research Topic

In this Research Topic, we study long-memory models in mathematical finance. The classical models for financial time-series, especially those connected to pricing and hedging of financial derivatives, are Markovian or semimartingales. However, in recent 20 years it has been demonstrated that some financial ...

Keywords : mathematical finance, financial engineering, time-series analysis, long-range dependence, stochastic modeling

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251+ Math Research Topics [2024 Updated]

Mathematics, often dubbed as the language of the universe, holds immense significance in shaping our understanding of the world around us. It’s not just about crunching numbers or solving equations; it’s about unraveling mysteries, making predictions, and creating innovative solutions to complex problems. In this blog, we embark on a journey into the realm of math research topics, exploring various branches of mathematics and their real-world applications.

How Do You Write A Math Research Topic?

Writing a math research topic involves several steps to ensure clarity, relevance, and feasibility. Here’s a guide to help you craft a compelling math research topic:

• Identify Your Interests: Start by exploring areas of mathematics that interest you. Whether it’s pure mathematics, applied mathematics, or interdisciplinary topics, choose a field that aligns with your passion and expertise.
• Narrow Down Your Focus: Mathematics is a broad field, so it’s essential to narrow down your focus to a specific area or problem. Consider the scope of your research and choose a topic that is manageable within your resources and time frame.
• Review Existing Literature: Conduct a thorough literature review to understand the current state of research in your chosen area. Identify gaps, controversies, or unanswered questions that could form the basis of your research topic.
• Formulate a Research Question: Based on your exploration and literature review, formulate a clear and concise research question. Your research question should be specific, measurable, achievable, relevant, and time-bound (SMART).
• Consider Feasibility: Assess the feasibility of your research topic in terms of available resources, data availability, and research methodologies. Ensure that your topic is realistic and achievable within the constraints of your project.
• Consult with Experts: Seek feedback from mentors, advisors, or experts in the field to validate your research topic and refine your ideas. Their insights can help you identify potential challenges and opportunities for improvement.
• Refine and Iterate: Refine your research topic based on feedback and further reflection. Iterate on your ideas to ensure clarity, coherence, and relevance to the broader context of mathematics research.
• Craft a Title: Once you have finalized your research topic, craft a compelling title that succinctly summarizes the essence of your research. Your title should be descriptive, engaging, and reflective of the key themes of your study.
• Write a Research Proposal: Develop a comprehensive research proposal outlining the background, objectives, methodology, and expected outcomes of your research. Your research proposal should provide a clear roadmap for your study and justify the significance of your research topic.

By following these steps, you can effectively write a math research topic that is well-defined, relevant, and poised to make a meaningful contribution to the field of mathematics.

251+ Math Research Topics: Beginners To Advanced

• Prime Number Distribution in Arithmetic Progressions
• Diophantine Equations and their Solutions
• Applications of Modular Arithmetic in Cryptography
• The Riemann Hypothesis and its Implications
• Graph Theory: Exploring Connectivity and Coloring Problems
• Knot Theory: Unraveling the Mathematics of Knots and Links
• Fractal Geometry: Understanding Self-Similarity and Dimensionality
• Differential Equations: Modeling Physical Phenomena and Dynamical Systems
• Chaos Theory: Investigating Deterministic Chaos and Strange Attractors
• Combinatorial Optimization: Algorithms for Solving Optimization Problems
• Computational Complexity: Analyzing the Complexity of Algorithms
• Game Theory: Mathematical Models of Strategic Interactions
• Number Theory: Exploring Properties of Integers and Primes
• Algebraic Topology: Studying Topological Invariants and Homotopy Theory
• Analytic Number Theory: Investigating Properties of Prime Numbers
• Algebraic Geometry: Geometry Arising from Algebraic Equations
• Galois Theory: Understanding Field Extensions and Solvability of Equations
• Representation Theory: Studying Symmetry in Linear Spaces
• Harmonic Analysis: Analyzing Functions on Groups and Manifolds
• Mathematical Logic: Foundations of Mathematics and Formal Systems
• Set Theory: Exploring Infinite Sets and Cardinal Numbers
• Real Analysis: Rigorous Study of Real Numbers and Functions
• Complex Analysis: Analytic Functions and Complex Integration
• Measure Theory: Foundations of Lebesgue Integration and Probability
• Topological Groups: Investigating Topological Structures on Groups
• Lie Groups and Lie Algebras: Geometry of Continuous Symmetry
• Differential Geometry: Curvature and Topology of Smooth Manifolds
• Algebraic Combinatorics: Enumerative and Algebraic Aspects of Combinatorics
• Ramsey Theory: Investigating Structure in Large Discrete Structures
• Analytic Geometry: Studying Geometry Using Analytic Methods
• Hyperbolic Geometry: Non-Euclidean Geometry of Curved Spaces
• Nonlinear Dynamics: Chaos, Bifurcations, and Strange Attractors
• Homological Algebra: Studying Homology and Cohomology of Algebraic Structures
• Topological Vector Spaces: Vector Spaces with Topological Structure
• Representation Theory of Finite Groups: Decomposition of Group Representations
• Category Theory: Abstract Structures and Universal Properties
• Operator Theory: Spectral Theory and Functional Analysis of Operators
• Algebraic Number Theory: Study of Algebraic Structures in Number Fields
• Cryptanalysis: Breaking Cryptographic Systems Using Mathematical Methods
• Discrete Mathematics: Combinatorics, Graph Theory, and Number Theory
• Mathematical Biology: Modeling Biological Systems Using Mathematical Tools
• Population Dynamics: Mathematical Models of Population Growth and Interaction
• Epidemiology: Mathematical Modeling of Disease Spread and Control
• Mathematical Ecology: Dynamics of Ecological Systems and Food Webs
• Evolutionary Game Theory: Evolutionary Dynamics and Strategic Behavior
• Mathematical Neuroscience: Modeling Brain Dynamics and Neural Networks
• Mathematical Physics: Mathematical Models in Physical Sciences
• Quantum Mechanics: Foundations and Applications of Quantum Theory
• Statistical Mechanics: Statistical Methods in Physics and Thermodynamics
• Fluid Dynamics: Modeling Flow of Fluids Using Partial Differential Equations
• Mathematical Finance: Stochastic Models in Finance and Risk Management
• Option Pricing Models: Black-Scholes Model and Beyond
• Portfolio Optimization: Maximizing Returns and Minimizing Risk
• Stochastic Calculus: Calculus of Stochastic Processes and Itô Calculus
• Financial Time Series Analysis: Modeling and Forecasting Financial Data
• Operations Research: Optimization of Decision-Making Processes
• Linear Programming: Optimization Problems with Linear Constraints
• Integer Programming: Optimization Problems with Integer Solutions
• Network Flow Optimization: Modeling and Solving Flow Network Problems
• Combinatorial Game Theory: Analysis of Games with Perfect Information
• Algorithmic Game Theory: Computational Aspects of Game-Theoretic Problems
• Fair Division: Methods for Fairly Allocating Resources Among Parties
• Auction Theory: Modeling Auction Mechanisms and Bidding Strategies
• Voting Theory: Mathematical Models of Voting Systems and Social Choice
• Social Network Analysis: Mathematical Analysis of Social Networks
• Algorithm Analysis: Complexity Analysis of Algorithms and Data Structures
• Machine Learning: Statistical Learning Algorithms and Data Mining
• Deep Learning: Neural Network Models with Multiple Layers
• Reinforcement Learning: Learning by Interaction and Feedback
• Natural Language Processing: Statistical and Computational Analysis of Language
• Computer Vision: Mathematical Models for Image Analysis and Recognition
• Computational Geometry: Algorithms for Geometric Problems
• Symbolic Computation: Manipulation of Mathematical Expressions
• Numerical Analysis: Algorithms for Solving Numerical Problems
• Finite Element Method: Numerical Solution of Partial Differential Equations
• Monte Carlo Methods: Statistical Simulation Techniques
• High-Performance Computing: Parallel and Distributed Computing Techniques
• Quantum Computing: Quantum Algorithms and Quantum Information Theory
• Quantum Information Theory: Study of Quantum Communication and Computation
• Quantum Error Correction: Methods for Protecting Quantum Information from Errors
• Topological Quantum Computing: Using Topological Properties for Quantum Computation
• Quantum Algorithms: Efficient Algorithms for Quantum Computers
• Quantum Cryptography: Secure Communication Using Quantum Key Distribution
• Topological Data Analysis: Analyzing Shape and Structure of Data Sets
• Persistent Homology: Topological Invariants for Data Analysis
• Mapper Algorithm: Method for Visualization and Analysis of High-Dimensional Data
• Algebraic Statistics: Statistical Methods Based on Algebraic Geometry
• Tropical Geometry: Geometric Methods for Studying Polynomial Equations
• Model Theory: Study of Mathematical Structures and Their Interpretations
• Descriptive Set Theory: Study of Borel and Analytic Sets
• Ergodic Theory: Study of Measure-Preserving Transformations
• Combinatorial Number Theory: Intersection of Combinatorics and Number Theory
• Additive Combinatorics: Study of Additive Properties of Sets
• Arithmetic Geometry: Interplay Between Number Theory and Algebraic Geometry
• Proof Theory: Study of Formal Proofs and Logical Inference
• Reverse Mathematics: Study of Logical Strength of Mathematical Theorems
• Nonstandard Analysis: Alternative Approach to Analysis Using Infinitesimals
• Computable Analysis: Study of Computable Functions and Real Numbers
• Graph Theory: Study of Graphs and Networks
• Random Graphs: Probabilistic Models of Graphs and Connectivity
• Spectral Graph Theory: Analysis of Graphs Using Eigenvalues and Eigenvectors
• Algebraic Graph Theory: Study of Algebraic Structures in Graphs
• Metric Geometry: Study of Geometric Structures Using Metrics
• Geometric Measure Theory: Study of Measures on Geometric Spaces
• Discrete Differential Geometry: Study of Differential Geometry on Discrete Spaces
• Algebraic Coding Theory: Study of Error-Correcting Codes
• Information Theory: Study of Information and Communication
• Coding Theory: Study of Error-Correcting Codes
• Cryptography: Study of Secure Communication and Encryption
• Finite Fields: Study of Fields with Finite Number of Elements
• Elliptic Curves: Study of Curves Defined by Cubic Equations
• Hyperelliptic Curves: Study of Curves Defined by Higher-Degree Equations
• Modular Forms: Analytic Functions with Certain Transformation Properties
• L-functions: Analytic Functions Associated with Number Theory
• Zeta Functions: Analytic Functions with Special Properties
• Analytic Number Theory: Study of Number Theoretic Functions Using Analysis
• Dirichlet Series: Analytic Functions Represented by Infinite Series
• Euler Products: Product Representations of Analytic Functions
• Arithmetic Dynamics: Study of Iterative Processes on Algebraic Structures
• Dynamics of Rational Maps: Study of Dynamical Systems Defined by Rational Functions
• Julia Sets: Fractal Sets Associated with Dynamical Systems
• Mandelbrot Set: Fractal Set Associated with Iterations of Complex Quadratic Polynomials
• Arithmetic Geometry: Study of Algebraic Geometry Over Number Fields
• Diophantine Geometry: Study of Solutions of Diophantine Equations Using Geometry
• Arithmetic of Elliptic Curves: Study of Elliptic Curves Over Number Fields
• Rational Points on Curves: Study of Rational Solutions of Algebraic Equations
• Galois Representations: Study of Representations of Galois Groups
• Automorphic Forms: Analytic Functions with Certain Transformation Properties
• L-functions: Analytic Functions Associated with Automorphic Forms
• Selberg Trace Formula: Tool for Studying Spectral Theory and Automorphic Forms
• Langlands Program: Program to Unify Number Theory and Representation Theory
• Hodge Theory: Study of Harmonic Forms on Complex Manifolds
• Riemann Surfaces: One-dimensional Complex Manifolds
• Shimura Varieties: Algebraic Varieties Associated with Automorphic Forms
• Modular Curves: Algebraic Curves Associated with Modular Forms
• Hyperbolic Manifolds: Manifolds with Constant Negative Curvature
• Teichmüller Theory: Study of Moduli Spaces of Riemann Surfaces
• Mirror Symmetry: Duality Between Calabi-Yau Manifolds
• Kähler Geometry: Study of Hermitian Manifolds with Special Symmetries
• Algebraic Groups: Linear Algebraic Groups and Their Representations
• Lie Algebras: Study of Algebraic Structures Arising from Lie Groups
• Representation Theory of Lie Algebras: Study of Representations of Lie Algebras
• Quantum Groups: Deformation of Lie Groups and Lie Algebras
• Algebraic Topology: Study of Topological Spaces Using Algebraic Methods
• Homotopy Theory: Study of Continuous Deformations of Spaces
• Homology Theory: Study of Algebraic Invariants of Topological Spaces
• Cohomology Theory: Study of Dual Concepts to Homology Theory
• Singular Homology: Homology Theory Defined Using Simplicial Complexes
• Sheaf Theory: Study of Sheaves and Their Cohomology
• Differential Forms: Study of Multilinear Differential Forms
• De Rham Cohomology: Cohomology Theory Defined Using Differential Forms
• Morse Theory: Study of Critical Points of Smooth Functions
• Symplectic Geometry: Study of Symplectic Manifolds and Their Geometry
• Floer Homology: Study of Symplectic Manifolds Using Pseudoholomorphic Curves
• Gromov-Witten Invariants: Invariants of Symplectic Manifolds Associated with Pseudoholomorphic Curves
• Mirror Symmetry: Duality Between Symplectic and Complex Geometry
• Calabi-Yau Manifolds: Ricci-Flat Complex Manifolds
• Moduli Spaces: Spaces Parameterizing Geometric Objects
• Donaldson-Thomas Invariants: Invariants Counting Sheaves on Calabi-Yau Manifolds
• Algebraic K-Theory: Study of Algebraic Invariants of Rings and Modules
• Homological Algebra: Study of Homology and Cohomology of Algebraic Structures
• Derived Categories: Categories Arising from Homological Algebra
• Stable Homotopy Theory: Homotopy Theory with Stable Homotopy Groups
• Model Categories: Categories with Certain Homotopical Properties
• Higher Category Theory: Study of Higher Categories and Homotopy Theory
• Higher Topos Theory: Study of Higher Categorical Structures
• Higher Algebra: Study of Higher Categorical Structures in Algebra
• Higher Algebraic Geometry: Study of Higher Categorical Structures in Algebraic Geometry
• Higher Representation Theory: Study of Higher Categorical Structures in Representation Theory
• Higher Category Theory: Study of Higher Categorical Structures
• Homotopical Algebra: Study of Algebraic Structures in Homotopy Theory
• Homotopical Groups: Study of Groups with Homotopical Structure
• Homotopical Categories: Study of Categories with Homotopical Structure
• Homotopy Groups: Algebraic Invariants of Topological Spaces
• Homotopy Type Theory: Study of Foundations of Mathematics Using Homotopy Theory

In conclusion, the world of mathematics is vast and multifaceted, offering endless opportunities for exploration and discovery. Whether delving into the abstract realms of pure mathematics or applying mathematical principles to solve real-world problems, mathematicians play a vital role in advancing human knowledge and shaping the future of our world.

By embracing diverse math research topics and interdisciplinary collaborations, we can unlock new possibilities and harness the power of mathematics to address the challenges of today and tomorrow. So, let’s embark on this journey together as we unravel the mysteries of numbers and explore the boundless horizons of mathematical inquiry.

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Here are recent ebooks to get you started with  mathematical finance and economics:

Here are some of the academic journals that USC Libraries subscribes to that are related to mathematical finance and economics:

• Mathematical Finance Mathematical Finance brings together work on the mathematical aspects of finance theory from such diverse fields as finance, economics, mathematics, and statistics. An essential resource for academic finance researchers and practitioners alike, the journal publishes clear and concise articles which present the latest theoretical developments. Modern finance is becoming increasingly technical, requiring the use of sophisticated mathematical tools in both research and practice. Mathematical Finance offers a forum for the publication of articles which employ these techniques, as well as providing a much-needed bridge between mathematical scientists and financial economists. Mathematical Finance has been ranked 3rd in the category of Social Sciences/Mathematical Methods, and 6th in the category of Business and Finance journals according to the latest ISI rankings.
• Applied Mathematical Finance The journal encourages the confident use of applied mathematics and mathematical modelling in finance. The journal publishes papers on the following: modelling of financial and economic primitives (interest rates, asset prices etc); modelling market behaviour; modelling market imperfections; pricing of financial derivative securities; hedging strategies; numerical methods; financial engineering.
• Quantitative Finance The frontiers of finance are shifting rapidly, driven in part by the increasing use of quantitative methods in the field. Quantitative Finance welcomes original research articles that reflect the dynamism of this area. The journal provides an interdisciplinary forum for presenting both theoretical and empirical approaches and offers rapid publication of original new work with high standards of quality. The readership is broad, embracing researchers and practitioners across a range of specialisms and within a variety of organizations. All articles should aim to be of interest to this broad readership.
• Journal of Mathematical Economics The primary objective of the Journal is to provide a forum for work in economic theory which expresses economic ideas using formal mathematical reasoning. For work to add to this primary objective, it is not sufficient that the mathematical reasoning be new and correct. The work should have real economic content. The economic ideas should be interesting and important. These ideas may pertain to any field of economics or any school of economic thought. The economic ideas may be well-known, provided they are expressed and developed in a novel way.
• Economic Systems Research conomic Systems Research is a double blind peer-reviewed scientific journal dedicated to the furtherance of theoretical and factual knowledge about economic systems, structures and processes, their interaction with the natural environment, and their change through time and space, at the subnational, national and international level. The journal contains sensible, matter-of-fact tools and data for modelling, policy analysis, planning and decision making aimed at solving contemporary economic and environmental questions and problems.

The USC Libraries has other research guides that may be helpful with mathematical finance and economics:

• Economics This guide provides basic links to some of the key resources in the area of Economics available through the USC Libraries.
• Financial Databases Help Guide "How to Guides' for selected Financial Databases including Bloomberg, Capital IQ, Thomson One, WRDS.
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• Last Updated: May 7, 2024 9:06 AM
• URL: https://libguides.usc.edu/mathematics

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Topics in mathematical finance, econ 690-82, cross-listed as.

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Research Areas

At the Financial Mathematics Research Group, we advance mathematics in the context of financial markets. Our work has direct industrial application and we also work extensively on theory-oriented problems in order to accurately capture nuanced characteristics of financial markets. Unlike other scientific fields, our mathematical research projects are diverse and change constantly, including working on the latest developments in machine learning and fintech. We explore data, develop new mathematical models and improve on existing models. 

Funding : Our research is supported by a wide variety of grants, including MITACS, NSERC Discovery, NSERC USRA, NSERC Engage, and NSERC CRD.

To discuss research opportunities with our group, find your interest area and  contact a supervising professor .

Portfolio Optimization

Our work in this area involves developing investment strategies that maximize returns and minimize risks. Members of our group have experience in building portfolios for financial institutions that are large (e.g. banks) and small (e.g. hedge funds). By studying how investments move, we can predict the risk involved and potentially provide the basis for employing alternate asset allocation strategies to optimize returns.

Supervising professors:  Dr. Rubtsov and Dr. Xu

Risk Management

A prime part of our work involves understanding risk and creating models to measure and mitigate its potential impact. In the area of risk management we have been working with market risk, interest rate risk, climate change risk, systemic risk (risk of collapse of the entire financial system), among others. The results of our research were published in top-tier journals in Mathematical Finance.

Supervising professors:  Drs. Ferrando, Gao, Olivares, Rubtsov, Xanthos and Xu

Derivative Pricing

In this research area, we develop approaches for pricing derivative contracts based on future valuations of its underlying assets. A core component of our work is to improve on shortfalls in standard pricing models such as Black-Scholes-Merton model. We develop a variety of models to reflect real-world phenomena such as heavy-tail distributions or running jumps. The more accurate our models, the better investors can hedge in buying or selling.

Supervising professors:  Drs. Ferrando, Olivares, Rubtsov, Xanthos and Xu

Environmental Finance

We are currently one of only few mathematics groups with expertise in Environmental Finance – a growing and increasingly important research area. Our work in this field encompasses two main subjects:

Mitigation of climate change risk

In this research area we are providing answers to the following questions. How can financial institutions mitigate adverse impacts of climate change? How can financial institutions help in transitioning to lower carbon economies? How to optimize climate taxation and quantify the cost of delay in addressing potentially devastating consequences of global climate change?

Supervising professor : Dr. Olivares

Weather Derivative Contracts

Financial markets are renewing their interest in contracts involving weather-related impacts on pricing. Environmental factors such as rainfall can affect production in weather-dependent industries such as agriculture. We develop mathematical models to more accurately price weather derivative contracts within the context of climate change.

Supervising professor : Dr. Rubtsov

Emerging Topics in Finance

Our group stays current with some of the latest developments within Finance. We’re currently studying and modeling the characteristics of such advancing areas as:

• Blockchain, cryptocurrencies
• Data mining
• Artificial intelligence and Machine learning
• Behavioural finance

Supervising professors : Dr. Rubtsov and Dr. Xu

Browse Course Material

Course info, instructors.

• Dr. Peter Kempthorne
• Dr. Choongbum Lee
• Dr. Vasily Strela
• Dr. Jake Xia

Departments

• Mathematics

As Taught In

• Applied Mathematics
• Probability and Statistics

Learning Resource Types

Topics in mathematics with applications in finance, course description.

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