数学金融博士
发展数学、金融和技术交叉领域的专业知识
PhD in Mathematical Finance Course Requirements
The PhD in Mathematical Finance curriculum is tailored to each incoming student, based on their academic background. 学生可以从奎斯特罗姆和波士顿大学其他学院提供的一系列课程中进行选择，以建立数学和金融的坚实基础，重点是它们如何在金融世界中相互作用。
The Math Finance Curriculum
The PhD in Math Finance attracts students with a deep interest in the creation of complex models and financial instruments as well as a passion for in-depth analysis. 如果这听起来像你，点击下面的链接下载数学金融课程表。
Given the growing importance of technology in financial modeling, computer science is also integrated into the coursework. All first-year students participate in a shared academic experience by enrolling in DS906 Philosophy and Science of Research in the fall semester.
以下是典型的课程时间表：
第一年：秋季
新古典一般均衡理论。Topics covered include consumption, production, existence of competitive equilibrium, fundamental welfare theorems, externalities, and uncertainty.
Measure theory and integration on measure spaces, specialization to integration on locally compact spaces, and the Haar integral. Lp空间，对偶性和表示定理。Introduction to Banach and Hilbert spaces, open mapping theorem, spectral theorem for Hermitian operators, and compact and Fredholm operators.
Introduction to probability with measure theoretic foundations. Fundamentals of measure theory. 概率空间。可测量函数和随机变量。Expectation and conditional expectation. 零点定律和Borel-Cantelli引理。Characteristic functions. Modes of convergence. Uniform integrability. Skorokhod representation theorem. Basic limit theorems.
本课程向学生介绍澳门威尼斯人注册网站研究。The class provides a brief introduction to the philosophy of science and debates about the nature of theory before diving thoroughly into different research methods. Students are exposed to research methods from their own and adjacent fields ranging from causal inference and experiments to qualitative research methods. 课程的最后一部分向学生介绍澳门威尼斯人注册网站研究中的多样性、伦理和公平问题。As part of the class students will complete the introductory ethics modules that are required by the university. Students will be graded on their class participation, a research proposal which is due at the end of the class, and their feedback to other students on their research proposals.
第一年：春天
非合作的博弈理论。Economics of information: adverse selection, signaling, principal agent problem, moral hazard and introduction to mechanism design.
在应用和澳门威尼斯人注册网站研究中重要的概率主题。大数定律。Three series theorem. Central limit theorems for independent and non-identically distributed random variables. Speed of convergence. 大的偏差。Laws of the iterated logarithm. 稳定且无限可分的分布。Discrete time martingales and applications.
本课程全面深入地介绍了现代资产定价理论。连续时间随机过程、随机微积分和最优控制得到了广泛的应用。特别是，鞅方法用于解决以下主题：(i)最优消费组合政策和（ii）一般均衡模型中的资产定价。涉及不可分偏好、不完全信息和主体多样性的澳门威尼斯人注册网站研究进展将被讨论。
增加一门选修课。
第二年：秋季
介绍宏观经济学的主题和工具。Dynamic programming and rational expectations; 新古典增长与真实商业周期模型；investment and financial markets; analysis of frictional labor markets.
基于证据的随机过程澳门威尼斯人注册网站研究方法。布朗运动。Continuous martingales. 随机整合。Ito formula. Girsanov’s Theorem. Stochastic differential equations. Feynman-Kac formula. Markov Processes. 当地时间。Levy processes. Semimartingales and the general stochastic integral. 稳定的过程。分数布朗运动。
本博士课程旨在为学生提供金融经济学的入门知识。课程内容包括无套利条件、偏好和风险规避、投资组合选择、资本资产定价模型、资产定价和动态资产定价。除了讲课外，本课程还包括阅读和作业。Open to MBA students with faculty member’s permission. 必须有较强的量化背景和几门金融或经济学课程。
本课程全面深入地介绍了现代资产定价理论。连续时间随机过程、随机微积分和最优控制得到了广泛的应用。Particular emphasis will be placed on (i) stochastic calculus with jumps; （二）具有跳跃的资产定价模型；（iii） Hamilton-Jacobi-Bellman方程和随机控制；（四）金融随机控制问题的数值方法。（数学金融课程为数学金融专业的学生保留。）
第二年：春天
税收和政府支出的影响；monetary non-neutrality and nominal rigidities; 最优财政和货币政策。
选择三门选修课：
A concise introduction to recent results on optimal dynamic consumption- investment problems is provided. Lectures will cover standard mean-variance theory, dynamic asset allocation, asset-liability management, and lifecycle finance. The main focus of this course is to present a financial engineering approach to dynamic asset allocation problems of institutional investors such as pension funds, mutual funds, hedge funds, and sovereign wealth funds. 还将介绍实现资产配置模型的数值方法。课程还涵盖了动态投资组合问题的经验特征和实际应用。（数学金融课程为数学金融专业的学生保留。）
本课程全面深入地介绍了衍生证券的估值方法。连续时间随机过程、随机微积分和鞅方法得到了广泛的应用。The main topics to be addressed include (i) European option valuation, (ii) Exotic options, (iii) Multiasset options, (iv) Stochastic interest rate, (v) Stochastic volatility, (vi) American options and (vii) Numerical methods. 根据时间限制，可能会涉及其他主题。（数学金融课程为数学金融专业的学生保留。）
随着信用风险成为促进快速金融创新的主要因素，衍生品市场在过去十年中经历了巨大的增长。This course will provide an in-depth approach to credit risk modelling for the specific purpose of pricing fixed income securities and credit-risk derivatives. 本课程将探讨信用风险背后因素的本质，并建立包含违约风险的模型。将介绍和讨论信用衍生品的类型和结构。将推导出流行的信用衍生品的估值公式。Numerical methods, for applications involving credit derivative structures and default risks, will be presented. （数学金融课程为数学金融专业的学生保留。）
This course explores algorithmic and numerical schemes used in practice for the pricing and hedging of financial derivative products. 本课程的重点在于数据分析。它涵盖了这样的主题：随机模型跳跃，先进的模拟方法，优化例程，和基于树的方法。它还介绍了机器学习的概念和方法，包括交叉验证、降维、随机森林、神经网络、聚类和支持向量机。（数学金融课程为数学金融专业的学生保留。）
主题和方法结合宏观经济学和金融，重点是发展和测试涉及金融市场和宏观经济之间联系的理论。
本课程着重于固定收益证券的估值、套期保值和管理。介绍了理论和实证的期限结构概念。提出了短期利率模型和Heath-Jarrow-Morton方法。详细讨论了市场模型及其在远期、掉期、上限、下限和掉期以及其他利率衍生品估值中的应用。（数学金融课程为数学金融专业的学生保留。）
毕业先决条件：（GRSMA711）或同等学历。-Banach和Hilbert空间理论，以及Hahn-Banach和分离定理。Dual spaces. 巴拿赫收缩映射定理。Reflexivity and Krein-Milman theorem. Operator theory. Brouwer-Schauder fixed-point theorems. Applications to probability, dynamical systems, and applied mathematics.
课程内容包括：Feynman-Kac公式和Fokker-Plank方程，有跳跃的随机微积分，金融中的Levy过程和跳跃扩散模型，Bellman的动态规划原理和Hamilton-Jacobi- Bellman方程，金融中最优控制的经典问题（Merton问题等），具有交易成本的投资-消费决策，pde的资产定价与自由边界问题之间的联系，optimal stopping problems and the exercise of American-style derivatives, capital structure and valuation of real options and corporate debt, exchange options, stochastic volatility models, and Dupire’s formula. （数学金融课程为数学金融专业的学生保留。）
This course introduces common algorithmic and numerical schemes that are used in practice for pricing and hedging financial derivative products. Among others, the course covers Monte-Carlo simulation methods (generation of random variables, exact simulation, discretization schemes), finite difference schemes to solve partial differential equations, numerical integration, and Fourier transforms. Special attention is given to the computational requirements of these different methods, and the trade-off between computational effort and accuracy. （数学金融课程为数学金融专业的学生保留。）
本课程将介绍电子市场的概念，以及统计和最优控制技术来模拟和交易这些市场。我们将首先描述电子市场的基本要素，数据的一些特征，其经验含义和简单的微观经济模型。接下来，我们将澳门威尼斯人注册网站研究统计工具来估计和预测高频价格的价格和波动率。Then we will investigate algorithmic trading problems from the stochastic optimal control perspective, including the optimal execution problem and show how to modify the classical approaches to include order-flow information and the effect that dark pools have on trading. Trading pairs of assets that mean-revert is another important algorithmic strategy, and we will see how stochastic control methods can be utilized to inform agents how to optimally trade. （数学金融课程为数学金融专业的学生保留。）
这是数学金融课程中计量经济学序列的第二门课程。本课程快速回顾OLS、GLS、最大似然原理（MLE）。然后，课程的核心集中在贝叶斯推理，现在是金融计量经济学不可避免的支柱。在学习了贝叶斯推理的原理之后，我们澳门威尼斯人注册网站研究了它们在金融关键模型中的实现，特别是与投资组合设计和波动率预测相关的模型。我们还简要讨论了Lasso和Ridge方法，并将它们与贝叶斯方法进行了比较。在过去的二十年中，模拟方法的激进发展，如马尔可夫链蒙特卡罗（MCMC）扩展了贝叶斯方法的功能。因此，在学习了直接蒙特卡罗模拟方法之后，本课程将涵盖非平凡的模拟方法，如马尔可夫链蒙特卡罗（Markov Chain Monte Carlo， MCMC），并将其应用于随机波动等模型的实现。（数学金融课程为数学金融专业的学生保留。）
介绍金融市场分析的计量经济学理论和方法。主题包括资产收益的横截面和时间序列属性、参数和非参数波动率测量、隐含波动率、资产定价模型的估计、连续时间模型、系统风险和模型不确定性。
总学分：48-64
年3 - 5
在完成所有课程和综合考试后，学生进入候选资格。此时，重点转移到论文澳门威尼斯人注册网站研究。学生将组成一个委员会，制定澳门威尼斯人注册网站研究计划，并最终为他们的工作辩护。During this time, students will also serve as Teaching Assistants.