## Financial Math

Currently research in financial mathematics at Stanford is in two broad areas. One is on mathematical problems arising from the analysis of financial data; it involves statistical estimation methods for large data sets, often using random matrix theory and in particular dynamic or time-evolving large random matrices. The other is multi-agent stochastic control problems that model interacting markets. Mean field games are an example that give rise to mathematical problems at the interface between differential equations and stochastic analysis.

© Stanford University . Stanford , California 94305 .

## Center for Financial Mathematics and Actuarial Research - UC Santa Barbara

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.

## Search form

- Travel & Maps
- Our Building
- Supporting Mathematics
- Art and Oxford Mathematics
- Equality, Diversity & Inclusion
- Undergraduate Study
- Postgraduate Study
- Current Students
- Research Groups
- Case Studies
- Faculty Books
- Oxford Mathematics Alphabet
- Oxford Online Maths Club
- Oxford Maths Festival 2023
- It All Adds Up
- Problem Solving Matters
- PROMYS Europe
- Oxfordshire Maths Masterclasses
- Maths Week England
- Outreach Information
- Mailing List
- Key Contacts
- People List
- A Global Department
- Research Fellowship Programmes
- Professional Services Teams
- Conference Facilities
- Public Lectures & Events
- Departmental Seminars & Events
- Special Lectures
- Conferences
- Summer Schools
- Past Events
- Alumni Newsletters
- Info for Event Organisers & Attendees

## 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, sample topics for the final paper.

Twenty-five percent of the course grade is based upon a final paper on a math finance topic of the student’s choice. Below are some sample topics. Students may propose other topics as well.

## Portfolio Management

Based on what you learned in class, research further and come up with your own views in portfolio risk management.

## Regime-Shift Modeling

Detail one or more approaches to regime-shift modeling, addressing the statistical modeling methodology and its use in a specific, real-world application.

## Low-Volatility Investing

Critically review the rationales of low-volatility investing strategies in the U.S. equity market and their connection to the portfolio theory covered in class; evaluate the performance of such strategies as implemented in exchange-traded funds and / or mutual funds.

## Modeling Financial Bubbles

Detail one or more approaches to modeling asset bubbles; e.g., the work of Didier Sornette.

## Relationship between Black-Scholes and Heat Equations

- Go through the change of variables to get from Black-Scholes PDE to Heat Equation.
- Go through calculations verifying that a European call option price for a lognormaly distributed stock is in fact a discounted expected value of the pay-off under risk neutral measure.
- Explore possible numerical methods for the solution with various boundary conditions.
- Go through computations showing that Black-Scholes price of a digital option is a partial derivative of the call option price with respect to strike.

## Hybrid products

- Price zero coupon bonds in USD and EUR in this jump–diffusion model.
- Determine the dynamic hedging strategy. There are two sources of risk, so need at least 2 hedging instruments. FX forwards are a great candidate.

## HJM vs Short-Rate Interest Rate Models.

- Start from the equation for forward rates df tT = μ tT dt + σ tT dB t and derive the no-arbitrage condition for drift μ tT .
- Derive drift at for the short rate Ho-Lee Model dr t = a t dt + σdB t . Next, show that the Ho-Lee model can be written in the HJM form. Remember that r t = f tt .
- Add a mean reversion to the Ho-Lee model dr t = (a t - κr t *)dt + σdB* t and write it in the HJM form.

## Ross Recovery

- Try to offer financial intuition for the Perron Forbenius theorem for positive matrices.
- Try to extend Ross recovery to a countable state space for a Markov chain.

## A Few Topics Chosen by Students Last Year

- Transformation of Black-Scholes PDE to Heat Equation
- From Black-Scholes-Merton model to heat equation: Derivations and numerical solutions
- Solving Black-Scholes equation with Initial conditions by change of variables
- Derive HJM no arbitrage condition
- HJM model and Ho-Lee model
- Pricing zero-coupon USD and EUR bonds in the FX jump diffusion model
- Pricing Asian options
- On the Minimal Entropy Martingale Measure in Finite Probability Financial Market Model
- Principal Component Analysis on Oil, Gas, Power and Currency Swap Curves before and after the 2008 Financial Crisis
- A review of finite grid summation method and Monte-Carlo method for a three-legged spread option integration
- Monte-Carlo option pricing using the heston model for stochastic volatility

- Frontiers in Applied Mathematics and Statistics
- Mathematical Finance
- Research Topics

## Stochastic Calculus for Financial Mathematics

Total Downloads

Total Views and Downloads

## About this Research Topic

Stochastic calculus is the area of mathematics that deals with processes containing a stochastic component and thus allows the modeling of random systems. Many stochastic processes are based on functions which are continuous, but nowhere differentiable. This rules out differential equations that require the use of derivative terms, since they are unable to be defined on non-smooth functions. Instead, a theory of integration is required where integral equations do not need the direct definition of derivative terms. In quantitative finance, the theory is known as Ito calculus. Over the past four decades, stochastic calculus has represented a rapidly growing area of research, both in terms of the theory and its application to practical problems arising in such varied fields as econophysics and mathematical finance, in which self-similar processes are used – including Brownian motion, stable Lévy processes, and fractional Brownian motion. Brownian motion was first applied in finance by Bachelier in 1900. In 1973, the log-price of a stock was modeled using Brownian motion in an approach named the Black–Scholes–Merton model. Stable Lévy processes are widely used in financial econometrics to model the dynamics of stock, commodities, and currency exchange prices, etc. Fractional Brownian motion is a centered Gaussian process that extends Brownian motion, and has attracted interest from researchers in a number of fields due to, among other things, its long-range dependence. The primary use of stochastic calculus in finance is for modeling the random motion of an asset price in the Black–Scholes model. The physical process of Brownian motion (specifically geometric Brownian motion) is used to model asset prices via the Weiner process. This process is represented by a stochastic differential equation, which despite its name is in fact an integral equation. This Research Topic encourages the submission of original research and review articles exploring stochastic processes, stochastic partial differential equations and integration, and their application to finance. Potential topics of interest include, but are not limited to: - The rough volatility model; - Stochastic processes applied in finance and other fields; - Stochastic differential equations; - Stochastic partial differential equations; - Fractional diffusion; - Transform methods applied in stochastic differential equations; - Numerical methods for stochastic partial differential equations; - Fractional operators; - Asset pricing; - Energy market pricing by stochastic models.

Keywords : mixed Gaussian processes, fractional Brownian motion, sub-fractional Brownian motion, stochastic partial differential equations, financial applications

Important Note : All contributions to this Research Topic must be within the scope of the section and journal to which they are submitted, as defined in their mission statements. Frontiers reserves the right to guide an out-of-scope manuscript to a more suitable section or journal at any stage of peer review.

## Topic Editors

Topic coordinators, submission deadlines.

Submission closed.

## Participating Journals

Total views.

- Demographics

No records found

total views article views downloads topic views

## Top countries

Top referring sites, about frontiers research topics.

With their unique mixes of varied contributions from Original Research to Review Articles, Research Topics unify the most influential researchers, the latest key findings and historical advances in a hot research area! Find out more on how to host your own Frontiers Research Topic or contribute to one as an author.

## IMAGES

## COMMENTS

Currently research in financial mathematics at Stanford is in two broad areas. One is on mathematical problems arising from the analysis of financial data; it involves statistical estimation methods for large data sets, often using random matrix theory and in particular dynamic or time-evolving large random matrices.

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

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...

"Financial Mathematics" is a Section of the open access, peer-reviewed journal Mathematics, which publishes original papers of a high scientific level in all areas of quantitative, computational, and theoretical finance.

Financial Mathematics is the field of applied mathematics that involves defining problems in finance and providing solutions using methods that draw from probability, statistics, differential equations, optimization, numerical methods, and data science. The primary emphasis in financial mathematics is the derivation of the mathematical models ...

Financial Mathematics - Science topic. Mathematicial Modelling. Aug 2023. Harry Poveda. Aug 2023. Peter Pflaumer. Aug 2023. Peter Pflaumer. Ángeles María Longarela-Ares.

Operations Research & Financial Engineering 609-258-0100. Digital Accessibility. Financial mathematics concerns mathematical models and problems arising in financial markets and applies tools from probability, optimization, stochastic analysis and statistics. Specific areas of research include risk management, pricing and hedging in incomplete ...

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 ...

An introduction to the principles and methods of financial mathematics, with a focus on discrete-time stochastic models. Topics include no-arbitrage pricing of financial derivatives, risk-neutral probability measures, the Cox-Ross-Rubenstein and Black-Scholes-Merton options pricing models, and implied volatility. Prerequisites: MATH 233, Math ...

Learn more about Research Topics Explores mathematical and quantitative finance in the financial market environment, providing useful mathematical tools for scientists who need deep theories of statistics and applied mathematical ...

This is an attempt to review some of the breakthroughs in economic research as they impacted the nascent field of financial mathematics over the last 25 years. Because of the prominent role of Finance and Stochastics in the definition of this emerging field, I try to view things through the lens of its published papers, and I try to stay away from financial engineering applications.

The aim of this Special Issue is to bridge the gap between the classical models of mathematics and statistics and their applications in real cases of finance. Authors can submit their contributions concerning applications or developments of mathematical and statistical models in finance topics such as risky assets, discrete time market models ...

The aim of this Special Issue is to contribute to the enrichment of Mathematical Finance by broadening the knowledge of this area with research papers on the following potential topics: Stochastic analysis and control theory in finance; fPDE and fractional models in finance; PIDE and Lévy processes in finance;

Econometric methods used in finance and accounting related research include single equation multiple regression, simultaneous regression, panel data analysis, time series analysis, spectral analysis, non-parametric analysis, semi-parametric analysis, GMM analysis, and other methods.

Traditionally, financial mathematics has been used to solve financial problems. With globalization, financial transactions require new analysis based on tools of probability, statistics, and economic theory. Global research trends in this topic during the period 1935-2019 have been analyzed.

Le livre intitulé « Financial Numeracy in Mathematics Education: Research and Practice (La numératie financière dans l'enseignement des mathématiques: la recherche et la pratique)» propose des idées originales et novatrices liées à l'usage de la numératie financière dans l'apprentissage des mathématiques en classe.

List of Research Topics and Ideas of Mathematical Finance for MS and Ph.D. Thesis. A class of mesh-free algorithms for mathematical finance, machine learning and fluid dynamics A Mathematical Finance Database By Marek Rutkowski and Marek Musiela

Financial mathematics describes the application of mathematics and mathematical modeling to solve financial problems. it is sometimes referred to as quantitative finance, financial engineering, and computational finance. The discipline combines tools from statistics, probability, and stochastic processes and combines it with economic theory.

The purpose of the class is to expose undergraduate and graduate students to the mathematical concepts and techniques used in the financial industry. Mathematics lectures are mixed with lectures illustrating the corresponding application in the financial industry. MIT mathematicians teach the mathematics part while industry professionals give the lectures on applications in finance.

Abstract. The following issues are discussed: introduction to basic concepts of financial mathematics; protection of fixed income with localization theorems for determination of all positive roots ...

Twenty-five percent of the course grade is based upon a final paper on a math finance topic of the student's choice. Below are some sample topics. Students may propose other topics as well. Portfolio Management. Based on what you learned in class, research further and come up with your own views in portfolio risk management. Regime-Shift Modeling

Stochastic calculus is the area of mathematics that deals with processes containing a stochastic component and thus allows the modeling of random systems. Many stochastic processes are based on functions which are continuous, but nowhere differentiable. This rules out differential equations that require the use of derivative terms, since they are unable to be defined on non-smooth functions ...

We would like to show you a description here but the site won't allow us.