Sort of a bad title, but here are the stages, courses, and best textbooks that I find are necessary to advance oneself through the world of finance. Mastery of a stage can be assumed by passing certain standardized tests or through the more obvious path of passing the class in school. Standardized tests have the advantage of being uniform where academic standards can be different between schools or professors, while school or work provides a better structure to prove your competence or find a niche. Most university curricula are centered around pushing students through each level in a year or so. Students are expected to develop their learning capacity and discipline so as to accommodate the more difficult and larger workloads that higher levels require.
This is really just for me to create a structure for myself and hopefully lay the groundwork for a neat tree diagram which will make my goals clearer. It borrows heavily from a guide to being a pure mathematician, Amazon, and a variety of other sources on Google. I’ll use the level system to provide a basis for difficulty. Level E is fresh out of high school level, going up to A, which would be doctorate level. There will also be level S for those extraordinarily difficult subjects that are unnecessary unless you’re a masochist, an inventor, or a fraud.
Level E/D – Elementary stuff: calculus, probability, economics
This is for the intro level courses where professors are correcting bad habits learned from high school, where reasoning was usually just a lot of hand-waving and magic. For most students, this means taking a modestly formal class that teaches complex numbers, systems of equations, limits, delta-epsilon proofs, exponential functions, independence, conditional probability, expectation, and the basics of supply and demand. For people who want to learn programming, discrete math will lay the foundation for logical thinking. Learning these topics is crucial later on – gaps in knowledge about calculus or probability will really hurt when the math gets much heavier.
Standardized tests: SAT, AP exams
Spivak – Calculus (probably the best and most rigorous book out there)
Stewart – Calculus: Early Transcendentals (the standard with pretty pictures and lots of problems, but not the most rigorous mathematically)
Hardy – A Course of Pure Mathematics (if you really want to sharpen your logical skills)
Anton and Rorres – Linear Algebra with Applications (first few chapters)
Chung – Elementary Probability Theory (the best, although Ross is also good)
Mankiw – Principles of Economics (good standard text, but lots are pretty good)
Halmos – Naive Set Theory (very difficult but good for proofs and set theory)
Graham, Knuth, and Patashnik – Concrete Mathematics: A Foundation for Computer Science
Level C – Beginning the Education: linear algebra, multivariable calculus, statistics, micro/macro-economics
Most universities structure their math requirements to encourage students to choose their specialty, but this allows them to neglect portions that may be important. A balance should be struck between finding one’s strengths to choose a major and building a proper foundation for higher education. This is the area that is most dangerous, because a lot of authors and professors like to go back to the bad habit of hand-waving rather than making students take the difficult transition to rigorous and formal thinking. If you got the books from Level E that I suggested, most of them are good enough to carry you through here too. Topics to focus on are vectors, linear transformations, matrix algebra, orthogonality, double/triple integrals, Green’s theorem, Fourier series, basic hypothesis testing, distributions, maximum likelihood estimators, marginal cost/revenue/benefit, market structure, business cycles, and exchange rates.
If you’re programming (and sooner or later you’ll have to learn this), you should be learning how to use Excel, Access, and maybe a programming language; you should be able to know what programs and components are good and bad for your computer and how to fix minor problems.
Standardized tests: real estate exams, GRE/GMAT (math portion)
Apostol – Calculus Vol 2 (very dry and technical, but very good – better than Stewart, which will get you here too)
Boyce and DiPrima – Elementary Differential Equations and Boundary Value Problems (the standard text along with Anton and Rorres)
Freedman, Pisani, and Purves – Statistics (excellent overview of all topics in probability and statistics)
Ott and Longnecker – Statistical Methods and Data Analysis (good overview of statistics with a hands-on approach)
Ross – A First Course in Probability (pushes you farther than Chung)
Varian – Microeconomics (not the most intense but the best conceptually)
Mas-Colell – Microeconomic Theory (the most intense, for serious economists only)
Mankiw – Macroeconomics (the easiest to read and understand)
Level B – Mature education: analysis, statistical inference/computing, finance, accounting
This is it. The honest start of your education. Don’t expect “plug and chug” to work any more, because you have to know what you’re doing and why. This is an intense part of the field, because finance and accounting involve cramming a LOT of stuff into your brain. At this point, authors will often skip steps and calculations with the dreaded note “trivial proof left to reader”. It sounds mean but the serious student will spend the time understanding why it’s trivial by filling in the gap. It is a tool that will serve well later on when presented with new ideas, which must be critically regarded. If you have been, you should stop being impressed with resumes and school names, because you do not want to be brow-beaten into beliefs just because someone from Harvard treats you like an idiot. If their ideas don’t flow logically, then you should learn to identify it and point it out. Mastering this level would provide a strong foundation for graduate school and most likely marks the end of an undergraduate career. Topics covered are metric spaces, compactness, measure theory, Markov chains, parametric and non-parametric tests, time value of money, financial ratios, capital budgeting, managerial decisions, and derivative products.
On the programming side, you should know how to build and maintain a network, how the internet works, and feel comfortable with a programming language and/or software packages such as R. You want to be very strong at C++, Java, Matlab, or R. Don’t spread yourself thin learning a little bit about all of them.
Standardized tests: CMA (first 2 levels), Actuarial exams P and FM, Series 7
Ross – Introduction to Probability Models (a more thorough version of his other book, esp. introducing stochastic processes)
Rice – Mathematical Statistics and Data Analysis (lots of good illustrations and lots of gaps in proofs – if you do not have a good grounding in statistics, you will get hammered)
DeGroot and Schervish – Probability and Statistics (detailed explanations and points out common misconceptions)
Rudin – Principles of Mathematical Analysis (Baby Rudin, but still not for the faint of heart)
Brealey – Principles of Corporate Finance (a Bible in the field for MBAs and finance students)
Hull – Options, Futures and Derivatives (a common text, weak on math but a good introduction to derivatives)
Joshi – Concepts and Practice of Mathematical Finance (best intro book out there)
Horngren and Harrison – Accounting (another great intro book, but this is a wide open field)
Garrison, Noreen, and Brewer – Managerial Accounting (best budgeting text out there)
Level A – Grad school/Pro: regression, time series, martingales, modeling
These are the bread and butter tools that analysts and bankers use to make financial decisions. It can be mathematically rigorous but the real skill is to turn these advanced concepts into readable facts by the average person or more importantly, convenient packets of numbers for accounting purposes. These are extremely powerful in the right hands, as can be seen with examples like Enron and the housing bust, and if you didn’t follow my advice before about being impressed with resumes, it’s easy to be convinced that what you’re doing is very profitable, very effective, and very safe. The necessity for rigorous thinking here is absolute, because the translation from tremendous mathematical data into simple fact means that there’s a lot of room for error, misconception, and poor understanding. Remember that if statistical results seem to fly in the face of common sense, they’re probably wrong. Always moderate your faith in science with a healthy dose of non-mathematical skepticism. But if you’ve reached this level, you’ll have a graduate degree and/or a well-paying job.
The topics I listed above are the ones that best relate to advanced statistics and mathematical finance, but it’s very possible to have gone the less mathematical route through business school or law school. If you’re still programming at this point, you should build your own start-up or sell your programs.
Standardized tests: CPA, CFA, EA, NASD Series exams, bar exam
Casella and Berger – Statistical Inference (text used widely in grad school)
Kutner, Nachtscheim, Neter, and Li – Applied Linear Statistical Models (THE book on regression)
Tsay – Analysis of Financial Time Series (not easy, but a great book)
Chung – A Course in Probability Theory (the adult version of his elementary work)
Williams – Probability with Martingales (not an intro prob book, but an intro martingales book)
Chung – Introduction to Stochastic Integration (read and digest Williams first, or you will regret it)
Level S – Mathematicians/Physicists with a career change: interest rate modeling, credit derivatives, numerical techniques
The list is pretty much over, and you’d only learn this stuff if you were a quant, Einstein, or trying to commit suicide with math. At this point, you would be beyond exams because you would almost certainly have a PhD. Proving yourself at this point is a matter of accomplishment, publications, or millions of dollars. If you wanted to swindle the investing public, it is probably not difficult for you to scare up billions of dollars. You don’t fear Congressional subpoenas or the Supreme Court of the United States. Even for the so-called Masters of the Universe on Wall Street, the prospect of seeing those things makes their testicles shrivel into their abdomen.
Just remember that nothing hits harder than the ground. If something goes wrong and you melt down the US financial system, your career is effectively over and you should plan to die before you ever see the inside of a federal penitentiary (e.g. Ken Lay).
Brace – Engineering BGM (Brace is the “B” in BGM, because he invented the LIBOR market model that forms the basis for global capital flows. Brace believes in “less is more” when it comes to explaining things, which is why he’s smarter than you are.)
Schonbucher – Credit Derivatives Pricing Models: Models, Pricing and Implementation (an explanatory text for pricing credit derivatives, a booming industry before the financial system went belly-up, so either it’s all junk or everyone was doing it wrong)
Glasserman – Monte Carlo Methods in Financial Engineering (definitive reference on Monte Carlo methods in finance)
Level 0 – Nonessential reading and movies
If you want to know about the trials and tribulations of working in finance, here are some great books. A lot of these are cautionary tales about taking mathematical models too far and the excess of both work and play in the financial world.
Derman – My life as a quant: reflections on physics and finance (takes you through his career in physics and finance, and he lived through some very interesting times and was an interesting character. You see the egos in Wall Street and the vicious schadenfreude.)
Lewis – Liar’s Poker (shows the excess of Wall Street in the 1980s. Read by everyone who’s ever wanted to get in the money game)
Lowenstein – When Genius Failed: The Rise and Fall of Long Term Capital Management (story of statistical models blowing up a hedge fund. Not an exact sequel but a lot of overlap with characters from Liar’s Poker)
Taleb – Fooled by Randomness/Black Swan (two books, first details that most skill is masking extraordinary luck, second shows you that most mathematical theory and science is a house of cards built on sand. Writing is turgid and repeats itself ad nauseum, but it’s worth reading if nothing else to provide an exercise in why you think Taleb is an incredible douchebag)
Enron: Smartest Guys in the Room (documentary about the end of Enron. You get to see that incredible assholes are common in finance and how tenuous the system is between statistics, economics, and law when placed in the hands of very smart and very greedy people. The money game attracts sharks and most regulators and lawmakers are just blubber)