Thursday, November 23, 2006

Books on Financial Mathematics

Books on Financial Mathematics

  For General Reading
  
  Peter L. Bernstein
  
  Capital Ideas: The Improbable Origins of Modern Wall Street
  
  Free Press, 1992
  
  Every student who is interested in financial mathematics should read this
book. It explains the historical development of some important ideas in
economics related to financial mathematics.
  
  
  
  Peter L. Bernstein
  
  Against the Gods: The Remarkable Story of Risk
  
  John Wiley & Sons, 1996
  
  This is a sequel to Capital Ideas, and it is worth reading. It traces the
history of concept of probability in the point of view of risk.
  
  
  
  Emanuel Derman
  
  My Life as a Quant: Reflections on Physics and Finance
  
  John Wiley & Sons, 2004
  
  This is a personal history of financial mathematics.
  
  
  
  
  
  For Financial Mathematics
  
  - In the alphabetical order of author names -
  
  
  
  Martin Baxter and Andrew Rennie
  
  Financial Calculus: An Introduction to Derivative Pricing
  
  Cambridge University Press, 1996
  
  A concise introduction to option pricing based on martingale approach. This is
one of the early readable books that introduced martingale approach. No rigorous
mathematics is presented.
  
  
  
  Jamil Baz and George Chacko
  
  Financial Derivatives: Pricing, Applications, and Mathematics
  
  Cambridge University Press, 2004
  
  An introduction to financial mathematics for business majors, but it is also
useful for mathematics majors who want gain to some insight into how economists
use mathematics. The use of mathematics in the book relies more on intuition in
the beginning but in the last part of the book more rigorous mathematics is
introduced.
  
  
  
  Thomas Björk
  
  Arbitrage Theory in Continuous Time, 2nd ed.
  
  Oxford University Press, 2004
  
  An introduction to option theory based on martingale approach. Optimal
stochastic control is also presented.
  
  
  
  Z. Brzeźniak and T. Zastawniak
  
  Basic Stochastic Processes
  
  Springer, 1998
  
  A student trying to understand the Itô calculus should read this book. All the
exercise problems have solutions.
  
  
  
  Sasha Cyganowski, Peter Kloeden and Jerzy Ombach
  
  From Elementary Probability to Stochastic Differential Equations with MAPLE
  
  Springer, 2001
  
  If you are not good at programming or if you want to visualize data in
numerical simulations, this book may be helpful.
  
  
  
  Alison Etheridge
  
  A Course in Financial Calculus
  
  Cambridge University Press, 2002
  
  A concise introduction to option theory based on martingale approach.
  
  
  
  Hélyette Geman
  
  Commodities and Commodity Derivatives
  
  John Wiley & Sons, 2005
  
  A comprehensive and readable introduction to commodity market including
electricity.
  
  
  
  Desmond Highham
  
  An Introduction to Financial Option Valuation: Mathematics, Stochastics and
Computation
  
  Cambridge University Press, 2004
  
  An introduction to option pricing theory at undergraduate level with Monte
Carlo simulations in Matlab. It contains many anecdotes.
  
  
  
  John C. Hull
  
  Options, Futures, and Other Derivatives, 5th ed.
  
  Prentice Hall, 2002
  
  A general introduction to financial mathematics with minimum amount of
mathematics.
  
  
  
  Peter James
  
  Option Theory
  
  John Wiley & Sons, 2003
  
  A concise but thorough introduction to option theory including the numerical
method and martingale method. Practically oriented compared with other books,
but in the last part of the book rigorous and advanced mathematics is
introduced.
  
  
  
  Mark Joshi
  
  The Concepts and Practice of Mathematical Finance
  
  Cambridge University Press, 2003
  
  An introduction to financial mathematics including the numerical method and
martingale method. The book is more practically oriented compared with other
books.
  
  
  
  Marek Musiela and Marek Rutkowski
  
  Martingale Methods in Financial Modelling, 2nd ed.
  
  Springer, 2004
  
  A comprehensive and mathematically rigorous introduction to financial
mathematics based on martingale approach. For people with strong mathematical
background, this book is recommended.
  
  
  
  Salih N. Neftci
  
  Principles of Financial Engineering
  
  Academic Press, 2004
  
  An introduction to financial mathematics.
  
  
  
  Bernt Oksendal
  
  Stochastic Differential Equations: An Introduction with Applications, 6th ed.
  
  Springer, 2003
  
  This is a standard textbook on SDE at graduate level.
  
  
  
  Stanley R. Pliska
  
  Introduction to Mathematical Finance: Discrete Time Models
  
  Blackwell Publishers, 1997
  
  An introduction to financial mathematics using the discrete time models. There
are many concrete examples in the book.
  

 
  Paul Wilmott, Sam Howison and Jeff Dewynne
  
  The Mathematics of Financial Derivatives: A Student Introduction
  
  Cambridge University Press, 1995
  
  An introduction to financial mathematics in terms of numerical solution of the
Black-Scholes equation. It is advised to learn numerical analysis first before
starting to read this book.

其中,John Hull's "Options, Futures and other derivatives" 是textbook.
  
  Emanuel Derman's "Life as a Quant" mentions both the career and the human
sides of a quant。平时读来for fun。可能要有physics background的人读来会比较亲切,里面提到很多物理学的big
names,我自己是一直看到merton-black-scholes才有共鸣。
  
  另外用的一本是:
  Jaksa Cvitanic and Fernando Zapatero (2004). Introduction to the Economics and
Mathematics of Financial Markets.
  MIT Press, Cambridge Massachusetts.
  里面 Options, futures and forward contracts, other derivative securities,
valuation, stochastic differential equations 什么都讲了一点,还包括
  Ito's rule 和 martingale approach,是本不错的入门书。用的math比John Hull那本稍微深一点。
  
  另外有个问题:
  Financial Mathematics里面常用的软件是什么?
  学校里只教过S-plus 和 R,不过好像在统计方面用得比较多。

软件主要好象有C++,Excel,VBA等等。去国外大学网站查查就知道了。
一般用matlab比较多

Wednesday, November 22, 2006

A Math Prof's Students Are in Demand at Banks

A Math Prof's Students Are in Demand at Banks



  By CARRICK MOLLENKAMP and CHARLES FLEMING
  Staff Reporters of The Wall Street Journal
  
  From The Wall Street Journal Online
  
  (See Corrections & Amplifications item below.)
  
  
  
  When Xavier Charvet applies for a job at an investment bank next year, he thinks he'll have an advantage. The 24-year-old French student's resume begins with the phrase: "DEA d'El Karoui."
  
  That stands for the postgraduate degree he is studying for under Nicole El Karoui, a math professor in Paris. She teaches skills required to create and price derivatives, the complex financial instruments based on stocks, bonds or loans. "When I talk about El Karoui's master's, everyone knows" about the degree, says Mr. Charvet.
  
  As derivatives have become one of the hottest areas for the world's biggest banks, Ms. El Karoui, 61 years old, has become an unlikely player in the business. Her courses at the prestigious Ecole Polytechnique and a state university, in such rarefied subjects as stochastic calculus, have become an incubator for experts in the field. A resume with her name on it "is a shortcut because you don't need to train the person on the basics of derivatives," says Rachid Bouzouba, a former student who is now head of European equity trading at the London office of Lehman Brothers Holdings Inc.
  
  The derivatives departments at banking giants J.P. Morgan Chase & Co., Deutsche Bank AG, Dresdner Kleinwort Wasserstein, and France's BNP Paribas SA and Societe Generale SA include many of her protégés.
  
  The high demand for her students reflects big changes in the global banking industry. Investment banks used to make much of their money from underwriting and trading stocks and bonds, or providing mergers-and-acquisitions advice. They hired people with a wide range of academic experience, including liberal-arts and science graduates.
  
  In recent years, profits from trading and selling derivatives have come to rival those from stocks and bonds at many banks. On average, revenue from derivatives based on stocks now accounts for about 30% of an investment bank's total revenue from stock-related businesses, according to a Citigroup Inc. report issued in January.
  
  As a result, banks are hiring an increasing number of recruits who understand derivatives. Inside banks, they are known as "quantitative analysts," or "quants" for short. They are able to marry stochastic calculus -- the study of the impact of random variation over time -- with the realities of financial trading.
  
  Derivatives are financial contracts, often exotic, whose values are derived from the performance of an underlying asset to which they are linked. Companies use them to help mitigate risk. For example, a company that stands to lose money on fixed-rate loans if rates rise can mitigate that risk by buying derivatives that increase in value as rates rise. Increasingly, investors are also using derivatives to make big bets on, say, the direction that interest rates will move. That carries the possibility of large returns, but also the possibility of large losses.
  
  The 75 or so students who take Ms. El Karoui's "Probability and Finance" course each year are avidly sought by recruiters. Three years ago, Joanna Cohen, a specialist in quant recruitment at Huxley Associates in London traveled to Paris to meet Ms. El Karoui to ensure her search firm was in the loop when students hit the job market. Today, Ms. Cohen says she carefully checks resumes with Ms. El Karoui's name to make sure applicants aren't overstating their interaction with the professor.
  
  "French quant candidates know that Nicole El Karoui's name has real clout, so many of them put her name on their [curriculum vitae] even if they've just taken one course with her. They want to give the impression that she has supervised their Ph.D.," Ms. Cohen says. "It'd be impossible for any one person to supervise the number of students who put her name on their CV."
  
  Rama Cont, a former student and now a research fellow at the Ecole Polytechnique, describes a degree with Ms. El Karoui's name on it as "the magic word that opened doors for young people."
  
  Headhunters say Ms. El Karoui's graduates can expect to earn up to about $140,000 a year in their first job, including a bonus, once they complete an internship that constitutes part of her course. After five years, they could be earning at least three times as much.
  
  In BNP Paribas's offices in London, the fixed-income interest rates derivatives research team, which totals six, includes three of her former students. On a recent day, Fahd Belfatmi, who took Ms. El Karoui's course in 2003, was working at the bank on a model to predict long-term interest rates. For help, he keeps handy a beat-up, paperback copy of Ms. El Karoui's French-language textbook, "Stochastic Models in Finance."
  
  Ms. El Karoui's only hands-on banking experience in her 38-year career was a six-month stint about two decades ago at a French retail bank. "I'm still a theoretician. My knowledge of markets is patchy and I've never spent a year in a trading room," she says. "On many counts, I probably have a fairly naive vision of things."
  
  Carving Out a Niche
  
  But she was one of the first in the world to carve out an academic niche studying the underpinnings of derivatives transactions, starting courses in the late 1980s. About two dozen universities have moved into that field, setting up their own mathematical-finance departments, including Stanford University, Carnegie Mellon University and the Massachusetts Institute of Technology.
  
  One of eight children in a middle-class family, Ms. El Karoui grew up a Protestant in a predominantly Catholic town in eastern France. Today she attributes her nonconformity to that background. "Protestants are rebels by nature," she says. Though her mother thought France's elite colleges were better suited for boys, her father, an engineer, encouraged her to take the tough entrance exams for Ecole Normale Supérieure, where she was accepted to study math. In 1968, around the time she was protesting the Vietnam War, she married a Muslim Tunisian economics professor, Faycal El Karoui.
  
  "If you'd told the left-winger that I was then that I was going to end up working in finance, I'd never have believed it," Ms. El Karoui says.
  
  France, the land of Descartes and Fermat, has a storied tradition in the study of math. Over the years, its engineering schools, including Ecole Polytechnique, a 212-year-old institution transformed by Napoleon into a military academy, have produced a steady stream of math students. Louis Bachelier's work in 1900 at the Sorbonne is considered the earliest effort to grasp how the markets work.
  
  Ms. El Karoui first branched into finance in 1987. The government had just closed down the elite Ecole Normale Superieure in Saint Cloud, a Paris suburb, where she had been teaching. She took a six-month sabbatical to work in the research department of consumer credit bank Compagnie Bancaire.
  
  At the time, many French mathematicians tended to deem the world of finance beneath them. "Finance meant selling your soul to the devil," she says. Her break with the French math establishment "took a lot of courage," says Marek Musiela, a leading figure in financial mathematics and the global head of fixed-income quant research at BNP Paribas.
  
  At first, Ms. El Karoui felt out of her depth. "I didn't even know what a bond is. I took a dictionary to look up the financial words," she recalls.
  
  But she soon realized that employees on the bank's newly formed derivatives desk were facing problems similar to those of stochastics scholars in trying to build models to predict the impact of interest-rate changes.
  
  After her time at the bank, she took a post teaching at the Paris VI, officially known as the University of Pierre and Marie Curie. She and another academic, Hlyette Geman, launched a postgraduate mathematical-finance course. Demand for know-how in derivatives was growing rapidly among banks at that time, sparked by the development of specialized exchanges that could trade derivative products, such as futures.
  
  "I said 'That's beautiful mathematics and it's teachable as a theoretical course,'" Ms. El Karoui says.
  
  Amine Belhadj, head of BNP Paribas's U.S. equity and derivatives department in New York, says Ms. El Karoui played a crucial role in finding interns when the bank began handling derivatives for clients in 1989. "There was nobody on the options desk with a mathematical-financial background," he says. "Having someone like Nicole who was making a specialty of it was pretty timely."
  
  Today, four of her five children have pursued careers in math and sciences, two as academics and two still as students. In her spare time, Ms. El Karoui plays classical piano, with a preference for Brahms sonatas.
  
  She earns about 80,000, or about $95,000, a year as a professor, plus a smaller amount for consulting fees -- a fraction of what her students can make. She drives around Paris in a small Renault.
  
  A Warning
  
  Lately, Ms. El Karoui has been vocal in warning students to use derivatives carefully. She says she is perturbed that an instrument that began primarily as a hedge for banks and financial firms against market risk is increasingly being used as a way to make a profit. Investors can profit, for example, by betting that the prices of stocks or bonds will increase. Ms. El Karoui worries that those looking for quick speculative gains could ramp up their bets on derivatives, but lose sight of the underlying financial instruments on which they're based, actually increasing their risk exposure.
  
  "Some clients aren't mature enough to understand the risks of products that are too complex," she says. "It's better to do business with those people responsibly, either taking the time to teach them or selling them a less complex product."
  
  Some big banks are being criticized for selling derivatives to institutions that may not understand the risks. Last year, for instance, Bank of America Corp. and Barclays PLC of the United Kingdom each agreed to settle claims that they had missold or mismanaged derivatives that were purchased by smaller banks in Italy and Germany. The banks said the matters were settled amicably.
  
  One recent afternoon in her classroom, Ms. El Karoui ran through a series of dense formulas designed to price derivatives. In class were about 50 students studying for the DEA, or "Diplome d'Etudes Approfondies," as a French master's degree leading to a doctorate is known.
  
  Ms. El Karoui talked softly toward the blackboard as much as she faced her students. There were few questions. Only near the end of the two-hour class did she raise a faint titter as she gestured to a full page of equations headed "General Pricing Formula." "There might be some of you brave enough to go through this," she said, then continued on, breezing through arcane jargon such as "smile risk," "volatility of volatility" and "Vega hedging."
  
  To some, Ms. El Karoui has been almost too successful in placing her students in top international banks. Ryan Taylor, a headhunter specializing in quantitative-finance candidates at Napier Scott Executive Search Ltd. in London, says some investment bankers are now starting to question how many French-trained quants are in the field. "France has got what borders on a monopoly of quant candidate production and we'd love to hear from quants in other countries," he says.
  
  
  
  Email your comments to cjeditor@dowjones.com.
  
  --March 10, 2006
  
  Corrections & Amplifications:
  French math professor Nicole El Karoui attended Ecole Normale Supérieure. An earlier version of this article incorrectly said she attended Ecole Nationale Supéieure. Also, the French government in 1987 closed the Ecole Normale Supérieure in Saint Cloud, a Paris suburb, moving it to Lyon. The earlier version of the article incorrectly stated the government closed the Ecole Normale Supérieure in Paris.

Tuesday, November 21, 2006

Introduction of Financial Mathematics

金融数学简介

金融数学(Financial Mathematics),又称数理金融学、数学金融学、分析金融学,是利用数学工具
研究金融,进行数学建模、理论分析、数值计算等定量分析,以求找到金融学内在规律并用以指导实践。金融数学也可以理解为现代数学与计算技术在金融领域的应用,因此,金融数学是一门新兴的交*学科,发展
很快,是目前十分活跃的前言学科之一。

金融数学是一门新兴学科,是“金融高技术 ”的重要 组成部分。研究金融数学有着重要的意义。


金融数学总的研究目标是利用我国数学界某些方面的优势,围绕金融市场的均衡与有价证券定价的数学理论进行深入剖析,建立适合我国国情的数学模型,编写一定的计算机软件,对理论研究结果进行仿真计算,对实际数据进行计量经济分析研究,为实际金融部门提供较深入的技术分析咨询。

主要的研究内容和拟重点解决的问题包括:

(1)有价证券和证券组合的定价理论

发展有价证券(尤其是期货、期权等衍生工具)的定价理论。所用的数学方法主要是提出合适的随机微分方程或随机差分方程模型,形成相应的倒向方程。建立相应的非线性Feynman一Kac公式,由此导出非常一般的推广的Black一Scho1es定价公式。所得到的倒向方程将是高维非线性带约束的奇异方程。

研究具有不同期限和收益率的证券组合的定价问题。需要建立定价与优化相结合的数学模型,在数学工具的研究方面,可能需要随机规划、模糊规划和优化算法研究。

在市场是不完全的条件下,引进与偏好有关的定价理论。

(2)不完全市场经济均衡理论(GEI)

拟在以下几个方面进行研究:

1.无穷维空间、无穷水平空间、及无限状态

2.随机经济、无套利均衡、经济结构参数变异、非线资产结构

3.资产证券的创新(Innovation)与设计(Design)

4.具有摩擦(Friction)的经济

5.企业行为与生产、破产与坏债

6.证券市场博奕。

(3)GEI 平板衡算法、蒙特卡罗法在经济平衡点计算中的应用,
GEI的理论在金融财政经济宏观经济调控中的应用,不完全市场条件下,持续发展理论框架下研究自然资源资产定价与自然资源的持续利用。
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数学一样成为任何一门科学发展过程中的必备工具。美国花旗银行副总裁柯林斯(Collins)1995年3月6日在英国剑桥大学牛顿数学科学研究所的讲演中叙述到:“在18世纪初,和牛顿同时代的著名数学家伯努利曾宣称:‘从事物理学研究而不懂数学的人实际上处理的是意义不大的东西。’那时候,这样的说法对物理学而言是正确的,但对于银行业而言不一定对。在18世纪,你可以没有任何数学训练而很好地运作银行。过去对物理学而言是正确的说法现在对于银行业也正确了。于是现在可以这样说:‘从事银行业工作而不懂数学的人实际上处理的是意义不大的东西’。”他还指出:花旗银行70%的业务依赖于数学,他还特别强调,‘如果没有数学发展起来的工和技术,许多事情我们是一点办法也没有的……没有数学我们不可能生存。”这里银行家用他的经验描述了数学的重要性。在冷战结束后,美国原先在军事系统工作的数以千的科学家进入了华尔街,大规模的基金管理公司纷纷开始雇佣数学博士或物理学博士。这是一个重要信号:金融市场不是战场,却远胜于战场。但是市场和战场都离不开复杂艰深,迅速的计算工作。然而在国内却不能回避这样一个事实:受过高等教育的专业人士都可以读懂国内经济类、金融类核心期刊,但国内金融学专业的本科生却很难读懂本专业的国际核心期刊《Journal
of Finance》,证券投资基金经理少有人去阅读《Joural of Portfolio
Management》,其原因不在于外语的熟练程度,而在于内容和研究方法上的差异,目前国内较多停留在以描述性分析为主着重描述金融的定义,市场的划分及金融组织等,或称为描述金融;而国外学术界以及实务界则以数量性分析为主,比如资本资产定价原理,衍生资产的复制方法等,或称为分析金融,即使在国内金融学的教材中,虽然涉及到了标的资产(Underlying
asset)和衍生资产(Derivative
asset)定价,但对公式提出的原文证明也予以回避,这种现象是不合理的,产生这种现象的原因有如下几个方面:首先,根据研究方法的不同,我国金融学科既可以归到我国哲学社会科学规划办公室,也可以归到国家自然科学基金委员会管理科学部,前者占主要地位,且这支队伍大多来自经济转轨前的哲学和政治学队伍,因此研究方法多为定性的方法。而西方正好相反,金融研究方向的队伍具有很好的数理功底。其次是我国的金融市场的实际环境所决定。我国证券市场刚起步,也没有一个统一的货币市场,投资者队伍主要由中小投资者构成,市场投机成分高,因此不会产生对现代投资理论的需求,相应地,学术界也难以对此产生研究的热情。然而数学技术以其精确的描述,严密的推导已经不容争辩地走进了金融领域。自从1952年马柯维茨(Markowitz)提出了用随机变量的特征变量来描述金融资产的收益性,不确定性和流动性以来,已经很难分清世界一流的金融杂志是在分析金融市场还是在撰写一篇数学论文。再回到Collins的讲话,在金融证券化的趋势中,无论是我们采用统计学的方法分析历史数据,寻找价格波动规律,还是用数学分析的方法去复制金融产品,谁最先发现了内在规律,谁就能在瞬息万变的金融市场中获取高额利润。尽管由于森严的进入堡垒,数学进入金融领域受到了一定的排斥和漠视,然而为了追求利润,未知的恐惧显得不堪一击。于是,在未来我们可以想象有这样一个充满美好前景的产业链:金融市场--金融数学--计算机技术。金融市场存在巨大的利润和高风险,需要计算机技术帮助分析,然而计算机不可能大概,左右等描述性语言,它本质上只能识别由0和1构成的空间,金融数学在这个过程中正好扮演了一个中介角色,它可以用精确语言描述随机波动的市场。比如,通过收益率状态矩阵在无套利的情形下找到了无风险贴现因子。因此,金融数学能帮助IT产业向金融产业延伸,并获取自己的利润空间。
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