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Introductory books for prospective quant students

Three books come to mind (there may be others I can't think of at the moment). In ascending order of sophistication they are:

1) An Introduction to the Mathematics of Money, by Lovelock, Mendel and Wright, published by Springer (294 pages);

2) Investment Mathematics, by Adams, Booth, Bowie, and Freeth, published by Wiley (419 pages); and

3) Mathematics for Finance, by Capinski and Zastawniak, published by Springer (310 pages) .

The first book explicitly assumes a one-year calc background. The same background should be more than enough for the second book as well -- indeed, while the financial coverage seems to be greater than the first, the mathematical sophistication is less. Only some sections towards the the end use calc ideas and provided the reader doesn't faint when he sees an integral sign or a partial derivative, he should be okay for those parts. The third assumes calculus, some linear algebra and some probability theory. In fact, it might complement Dan Stefanica's book (A Primer for the Mathematics of Financial Engineering) rather well.

An Introduction to the Mathematics of Money was written by three academics at the University of Arizona and was published in 2007. According to the authors, the aim of the book is both to introduce some math (recurrence relations, proof by induction, inequalities, probability) and to cover the rudiments of investing. The chapters are as follows:

Chapter 1: simple interest
Chapter 2: compound interest
Chapter 3: inflation and taxes
Chapter 4: annuities
Chapter 5: loans and risks
Chapter 6: amortisation
Chapter 7: credit cards
Chapter 8: bonds
Chapter 9: stocks and stock markets;
Chapter 10: stock market indexes, pricing, and risk
Chapter 11: options

The chapter on bonds covers, inter alia, duration and convexity. The chapter on options covers, inter alia, put-call parity, hedging, and a derivation of the Black-Scholes formula.

A basic book? Yes. All the more reason to recommend it as one doesn't lose sight of the wood for the trees. It provides an overview of finance for people coming from other backgrounds. The arcane mathematical material can come later -- for people coming from math or physics backgrounds, the math isn't the problem; understanding the basic financial concepts and financial worldview is.

Investment Mathematics was written by a British team, three of whom are academics at different universities. According to the authors, the book is meant for investment practitioners as well as those studying for professional exams. Broadly speaking, there's a more detailed coverage of topics covered in the previous book as well as four chapters devoted to basic probability and statistical theory. In the chapter on options, there's also coverage of the Greeks (the previous book only covers delta). A final chapter is devoted to stochastic investment models (including brief coverage of ARIMA, ARCH, and the Vasicek model).

Mathematics for Finance was written by a couple of academics, one in England and the other in Poland. The authors put forward coverage of two main ideas as their objective: Markowitz portfolio optimisation and CAPM on the one hand, and the Black-Scholes methodology on the other. The chapters are as follows:

Chapter 1: a simple market model
Chapter 2: risk-free assets
Chapter 3: risky assets
Chapter 4: discrete time market models
Chapter 5: portfolio management
Chapter 6: forward and futures contacts
Chapter 7: options: general properties
Chapter 8: option pricing
Chapter 9: financial engineering
Chapter 10: variable interest rates
Chapter 11: stochastic interest rates

The first chapter quickly introduces the no-arbitrage principle, the one-step binomial model, forwards and options. The second, the time value of money, compound interest, and bonds (zero-coupon and coupon). The third, multi-step binomial trees, risk-neutral probability, and martingales. The fourth develops the ideas of chapters 1 and 4 a bit further. The fifth, the CAPM and beta. Chapters 7, 8, and 9 cover the usual material on derivatives with the difference that here American options are also considered (in passing, the Black-Scholes PDE is presented but not derived). Chapters 10 and 11 look at some of the theory behind interest rates.

Why do I like this book? Its relative mathematical sophistication. Anyone who works through this book (which means taking notes, making summaries, and working through the exercises) will hit the ground running in any real quant program. I'm assuming, of course, that general math and programming skills are up to scratch. The book is designed for a year-long course but an earnest autodidact can probably cover it thoroughly in two months.

I'll come back to this post again, clean it up a bit, and perhaps add some more comments.
Are these your personal suggestions?

I'm afraid so. I've seen at least a couple of threads on the subject of introductory books over the last few weeks and I've found it difficult to agree with the recommendations. Provided my caveats are heeded -- that the first two are not exactly advanced and abstruse treatises -- they should serve readers coming from other backgrounds well.