金融中的数学方法

金融中的数学方法
课程编号：02832420 授课对象：本科生
学分：3 任课教师：李辰旭
课程类型：必修开课学期：2014秋
先修课程：高等数学（A类或B类）、概率论
办公电话：62767295
电子邮箱：cxli@, cxli@
辅导、答疑时间：
一、Program Learning Goals and Objectives
Learning Goal 1: Graduates will possess a solid understanding of business and management and will be able to translate this knowledge into practice.
1.1O bjective 1 Our students will have a good command of fundamental theories and
knowledge.
1.2O bjective 2 Our students will have a good command of analytical methods and
decision-making tools.
1.3O bjective 3 Our students will be able to apply theories and methodologies in key
business functions.
Learning Goal 2: Our students will be able to think critically.
2.1O bjective 1 Our students will be able to identify and summarize problems
2.2O bjective 2 Our students will be able to collect data and analyze problems in a critical
manner
2.3O bjective 3 Our students will be able to put forward effective solutions to business
problems
Learning Goal 3:Our students will have a sense of social responsibility.
3.1O bjective 1 Our students will be aware of the importance of ethics.
3.2 Objective 2 Our students will be able to provide solutions that take account of
contrasting ethical standpoints.
Learning Goal 4: Our students will be effective communicators.
4.1O bjective 1 Our students will be proficient in oral and written communication.
4.2O bjective 2 Our students will possess good interpersonal skills.
4.3O bjective 3 Our students will be able to adapt to diverse learning environments. Learning Goal 5: Our students will have global perspectives.
5.1O bjective 1 Our students will be aware of social and cultural differences.
5.2O bjective 2 Our students will be aware of the impact of globalization on business
operations, opportunities, and challenges.
5.3O bjective 3 Our students will be proficient in English.
二、课程概述
课程将介绍金融学（特别是在金融衍生品定价及其风险管理领域）中的重要量化工具：例如，随机过程，随机微积分和偏微分方程，以及Monte Carlo模拟等模型的数值实现方法。

同时，本课程避免枯燥单一的数学推导，在重视方法的同时将以生动的实例佐证量化方法在金融建模中的应用。

本课程将为同学们从量化的角度理解金融学中的一些问题或从事量化研究打下基础（也将为同学们选修我院所开设的金融工程/金融衍生品定价等相关课程提供重要的“量化”工具）。

三、课程目标（包括学生所提高的技能要求）
通过课程的讲授，使同学们初步了解随机过程，随机微积分和偏微分方程，以及Monte Carlo模拟等多种模型的数值实现方法在金融衍生品定价和风险管理中的实践应用。

本课程在介绍量化工具同时，将联系金融建模中的实例并进行生动的分析，各部分穿插进行，整体课程自成体系。

我们将根据课程的进展选取如下所列举的内容：
数学量化工具部分主要介绍条件数学期望、随机过程，鞅、Markov过程，随机游动、Brownian运动、Poisson过程、以及Ito随机积分, Ito公式，随机分析中的一些重要工具（例如Girsanov变换测度等），随机微分方程；偏微分方程相关内容并以金融衍生品定价为动机介绍其应用，数学方法方面我们将初步介绍偏微分方程随机微积分的联系(Feynman-Kac定理) 等，抛物型方程初值问题的求解方法。

数值实现方法部分将穿插在理论工具的介绍中，主要介绍Monte Carlo模拟（随机数产生，重要分布的模拟，随机过程的模拟，提高模拟性能的方差降低方法，随机微分方程的离散模拟等），二项（或多项）格点方法，偏微分方程的数值解等。

量化方法在金融建模中的应用实例大致涉及随机建模和数值方法在金融衍生品定价中的应用。

如时间允许我们将从量化原理的角度探讨近期金融衍生品（例如Stocks Index Futures，Credit Default Swap, Options Index Futures等）在我国的发展。

Tentative topics on mathematical tools include：
conditional expectation, stochastic processes, martingales, Markov process, random walk, Brownian motion, Poisson process, stochastic integration, stochastic calculus (Ito’s lemma and some fundamental theorems, e.g. Girsanov change of measure), stochastic differential equations, application of partial differential equations in derivatives pricing, parabolic equations and its stochastic interpretation via the Feynman-Kac theorem, etc.
Tentative topics on computational tools include:
Monte Carlo simulation (random number generation, simulation of important distributions and the sample path of stochastic processes, variance reduction techniques, discretization methods for simulating the solutions to stochastic differential equations), binomial/multinomial lattice, finite difference methods for differential equations, etc.
Tentative topics on the applications of quantitative tools may include:
modeling and computing methods for pricing derivative securities on a wide variety of asset classes, such as equity, fixed-income, credit, commodity, foreign-exchange, etc. If time allowed, we will discuss about the development of the Chinese derivatives market, for example the stocks index futures and credit default swaps, from the quantitative perspective.
四、内容提要及学时分配
To be determined
五、教学方式
每周授课3学时, 每周留适量作业。

六、教学过程中IT工具等技术手段的应用
使用投影课件(Latex Beamer)，对于重要理论的现场推导，有时利用信息资源介绍一些相关内容。

对于Monte Carlo模拟，我们将通过Matlab现场演示。

七、教材
1. 讲稿(slides)，随课程进展讲稿和作业等相关信息将上传到BlackBoard系统。

2. S. E. Shreve. Stochastic calculus for finance, V olume I, II. Springer Finance. Springer-Verlag, New York, 2004. 影印版：/dp/enbk607357
八、参考书目
[1] T. Mikosch. Elementary Stochastic Calculus With Finance in View，World Scientific Publishing Company, 1999.
随机分析基础(英文版) 世界图书出版公司; 第1版(2009年8月1日)
/gp/product/B002NX0IQS?ver=gp&qid=1277632646&ref_=sr _1_1&sr=8-1&s=books
[2] P. Glasserman. Monte Carlo Methods in Financial Engineering, Springer; 2003.
影印版：
/mn/detailApp/ref=sr_1_1?_encoding=UTF8&s=books&qid=1 270096284&asin=B001D6DWFA&sr=8-1
九、教学辅助材料，如CD、录影等
穿插选用相关内容
十、课程学习要求及课堂纪律规范
要求同学按时上课，按时完成作业。

作业每周上课前交给助教，助教将对每周作业打分。

不交、缓交和抄袭将影响平时作业成绩。

十一、学生成绩评定办法（需详细说明评估学生学习效果的方法）
平时作业30％，期中考试30％，期末考试40％。

期末考试时间将由学院指定。