金融数学学习建议及书单

2024年4月16日发(作者：裕安区小升初数学试卷分析)

亦转自国大论坛

金融数学是一门应用性极强的学科,其特殊之处在于,与许多其他应用学科如生物相比,它的难度更类

似于数学物理,而另一方面,它的应用性可以和engineering相提并论,因为好的结果必须是\"有利可图\"

的,you may cheat a Journal, but you cannot cheat 而更加独特的是,它要求一个人

有极其博杂的知识,所以一份好的书单很重要

大体而言,所需要的知识分为三类

1.数量

2.经济金融

3.编程,这方面我比较弱,至今还算不上professional programmer

LEVEL的书籍:

ng in C++ Vol 1 & 2

C++ Programming Language

另外,还需要data structure & alogrithms的知识

好在编程高手尽多,这方面也不太需要我业余的意见,呵呵

现在我列一下数量方面的书单

1.概率论

很不幸的事实是,概率论基本上没有好的中文教材(1998之前,之后我就不清楚了)

Ross的书适合本科和硕士生,胜在例子详尽

Billingsley的概率论和弱收敛的两本教材是非常好的入门书

chung的概率论教材很严格,读起干巴巴的来会有点累,不过是真长工夫的密籍

Durrett的书很流行,不过里面的小错误很多

如果你真的想理解概率论,feller的两本书是不可不读的,可以说,从高中水平到博士以上学位的读者,都

会从中获益---如果要推选概率论里面最有影响的教材,feller的书无可比拟,不过读时要一路自己算,fell

er书里面错误非常多,虽然都显然是笔误

Breiman的书也是经典,概率味比chung的浓

loeve的书可以作为工具书使用

2.随机分析

黄志远的随机分析入门是一本很好的书

严加安的鞅论可以做工具书用

Ross的Inrto to probability model可以做本科生随机过程入门,例子很多

Karlin & Taylor的两本书非常适合硕士生用

resnick的几本书概率味很不错,应用性也很强

oksendal的书是SDE里面最简单的

Karatzas Shreve有好几本书,金融数学的博士不可不读

大致上来说,一个人需要吃透如下

Revuz Yor的连续鞅是很好的书

Protter的书是严格随机分析里面最容易读的,文笔很好

williams的书深入浅出,入门很合适

Chung Williams的书比oksendal稍微难一点,作为应用随机分析的标准教材很不错

3-控制论

控制论在portfolio selection problem和risk management里面有很多的应用,optimal stopping在美

式derivative非常重要

金融数学里面用的主要是随机控制,和粘性解(因为operator is often degenerate)

经典的随机控制书是

G and RISHEL, (1975) Deterministic and Stochastic Optimal Control.

, (1980) Controlled diffusion processes

, (1989) Optimal control of diffusion processes.

SSAN and LIONS, (1982) Controle Impulsionnel et Inequations Variationnelles

粘性解的标准文献是

1. Crandall, Ishii and Lions, User\'\'s guide to viscosity solutions of second order partial differe

ntial equations, Bull. Amer. Math. Soc. 27 (1992),

g and Soner, Controlled Markov Processes and Viscosity Solutions, 1992.

4.数值算法

首先,finite difference是极其常用的算法,这方面书籍很多,比如Ames的经典教材

计算矩阵: Golub and Van Loan, Matrix Computations, 1996

Kushner and Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time,

1992. Kushner\'\'s Markov chain approximation method是控制论里最有用的算法

ROGERS and TALAY, Numerical Methods in Financial Mathematics. 1997.论文集

Kloeden and Platen, Numerical Solution of Stochastic Differential Equations, 1997. 偏理论,实用

性差一点

Glasserman, Monte Carlo Methods in Financial Engineering, 2003这本书非常非常实用,可以说是

金融数学数值算法的最新经典

5-时间序列

当然,学习时间序列之前,统计特别是多变量统计要先学好

A Guide to Econometrics: by Peter Kennedy可能是最通俗易懂的入门书

Econometric Analysis,by William H. Greene和Time Series Analysis by James Douglas Hamilton

是非常标准的教材,许多学校都在用

Box Jerkins的Time Series Analysis: Forecasting & Control,当之无愧的经典

Time Series and Dynamic Models by Christian Gourieroux,Gourieroux写了许多书,但似乎他的书

不如他的研究文章水准高

The Econometrics of Financial Markets,by John Y. Campbell, Andrew W. Lo, A. Craig MacKinl

ay,新经典

现在我们来看一下经济金融方面的书单

首先要强调,金融不是经济,经济考虑的是国计民生,环球宇宙之类的大问题,而金融考虑的是money m

aking, risk control之类的充满铜臭味的小问题

当然,经济背景也是需要的,比如说

Varian: Microeconomic Analysis(1992)

Samuelson: Economics

如果有时间,最有价值的书大概是Keynes的general principle,

看的时候的感觉会跟第一次学微积分差不多

现在我们进入金融书单

1.理论金融

Merton: Continuous time finance

Huang Litzenberger: Foundation for financial economics

Ingersoll: Theorey of financial decision making

Ross: Neoclassical Finance

Ross, Westerfield, Jaffe: Corporate Finance

Duffie: security market

Duffie: Dynamic Asset Pricing Theory

当然,金融文献浩如烟海,上面的书单是针对ASSET PRICING一块的,因为这一块最为定量化.至于做u

nderwriting, M&A,一般不是很需要数量出身的人,至少到目前为止:)

2.入门和综合类

然后就要开始看一些实际的入门书了

Hull, Options, Futures and Other Derivatives

Baxter and Rennie, Financial Calculus

Shreve:Stochastic Calculus Models for Finance vol 1 & 2

Wilmott: quantitative finance

然后

Bjork: Arbitrage theory in continuous time

Cvitanic, Zapatero: Introduction to the economics and mathematics of financial markets

Elliott, Kopp: Mathematics of Financial markets

Karatzas Shreve: Method of math finance

Musiela and Rutkowski: martingale method for finance

Bielecki, Rutkowski: Credit Risk : Modeling , Valuation and Hedging

Duffie Singleton: Credit Risk

Amman: Credit risk valuation

Taleb

3. Fixed income

Tuckman: Fixed Income Securities: Tools for Today\'\'s Markets是入门的最佳选择

然后,就不得不面对Fabozzi的无数厚书乐:)

Fixed Income Mathematics

Fixed Income Securities

Bond Markets : Analysis and Strategies

The Handbook of Fixed Income Securities,

Handbook of Mortgage Backed Securities

Collateralized Debt Obligations: Structures and Analysis

Interest Rate, Term Structure, and Valuation Modeling

Jessica James, Nick Webber Interest Rate Modelling: Financial Engineering,这本书乱而全

Brigo, Mercurio:Interest Rate Models 数学上难一些

Tavakoli: Collateralized Debt Obligations and Structured Finance

Tavakoli: Credit Derivatives & Synthetic Structures: A Guide to Instruments and Applications

Hayre: Salomon Smith Barney Guide to Mortgage-Backed and Asset-Backed Securities

4:其他类

Rebonato有几本很好的书:

Volatility and Correlation : The Perfect Hedger and the Fox

Modern Pricing of Interest-Rate Derivatives : The LIBOR Market Model and Beyond

Interest-Rate Option Models : Understanding, Analysing and Using Models for Exotic Interest-

Rate Options

Sch?nbucher:Credit Derivatives Pricing Models: Model, Pricing and Implementation写得很乱但

是无可替代

GENCAY: An Introduction to High-Frequency Finance第一本关于high frequency的书

O\'\'Hara:Market Microstructure Theory

Harris:Trading and Exchanges: Market Microstructure for Practitioners

补注:

我个人觉得要学好financial maths / fin eng，首先要学好probability / stochastic process，之后是

学stochastic calculus，学了之后读任何introductory financial maths book都会很轻松，容易理解。

ynamic Hedging

所以我推荐下列一些书来学好prob / stochastic processes

1. Intoroduction to probability models by Sheldon Ross ( i used that for my 2nd / 3rd year

probability / stochastic process courses, very good book )

2. Weighing the odds by David Williams ( 1的替代品，同样非常好 )

3. Probability with martingales by David Williams ( measure-theoretic intro to probability and

discrete-time martingales )

读了这3本里面的2本（1/2 或者 2/3）后，可以很流畅的读完Financial Calculus by Rennie and

Baxter，这本书introduces stochastic calculus and brownian motion from an intuitive way。之后

再读Stochastic calculus for finance vol II by Steven Shreve从而了解financial maths from a m

ore technical way。

当然，读哪些概率书还需要数学基础，需要懂得简单的analysis: e.g. convergence, limit, continuit

y etc.

总之，金融数学是由随机学+数学+编程组成的。

运用数学+随即过程随机积分学来建立模型，在于用编程来实施模型到现实中。学的东西真的很多。