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.
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Center for Financial Mathematics and Actuarial Research - UC Santa Barbara
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.
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Course info, instructors.
- Dr. Peter Kempthorne
- Dr. Choongbum Lee
- Dr. Vasily Strela
- Dr. Jake Xia
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- 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.
Based on what you learned in class, research further and come up with your own views in portfolio risk management.
Detail one or more approaches to regime-shift modeling, addressing the statistical modeling methodology and its use in a specific, real-world application.
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.
- 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.
- 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
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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
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