Can anyone recommend books that explain the math used in quantitative finance academic papers?
If you need a primer covering various domains of math then Dan Stefanica's text will do the job. The text covers multivariable calculus, lagrange multipliers, black scholes PDF, greeks & hedging, newton's method, bootstrapping, taylor series, numerical integration, and risk neutral valuation. It also includes a mathematical appendix.
If you want an introduction to risk analysis complete with geometric interpretations check out ATillio Meucci's Risk and Asset Allocation.
Hull's Options, Futures, and Derivatives is a classic that includes stochastic calculus and the topics in the title.
Here are the best applied statistics books:
Rene Carmona's "Statistical Analysis of Financial Data in S-Plus" covers a lot of ground with examples compatible with R. He starts with foundations and builds towards more complex models.
If you want ready-to-apply solutions Eric Zivot's "Modeling Financial Time Series with S-Plus" is encyclopedic in the range of topics covered. Whereas Carmona will focus on various modeling techniques, Zivot will cover portfolio optimization, factor analysis, and many other topics. It makes for a great reference rather than a cover-to-cover read.
If you want to focus on time-series specifically with an applied bent - Shumway and Stoffer's Time Series Analysis and Applications is also great. The solutions are compatible with R.
There are various theoretical statistics books (Hamilton, Ruey Tsay) but those will assume you understand the math.
I doubt you will find one book that covers everything you need, but here are a few that I continually come back to whenever I have some questions on the mathematics.
- Analysis of Financial Time Series by Ruey Tsay
- An Introduction to High-Frequency Finance by Dacorogna et al
- Probability and Statistics by DeGroot and Schervish
- Statistical Inference by Casella and Berger
- Econometric Analysis by Greene
- Options, Futures, and Other Derivatives by Hull
- Financial Calculus: An Introduction to Derivative Pricing by Baxter and Rennie
- An Introduction to the Mathematics of Financial Derivatives by Salih Neftci
If you asked me for a single book as a starting point I'd probably go for:
- Frequently Asked Questions in Quantitative Finance by Paul Wilmott
Some more references. Here are three starting books:
- for generic knowledge: Theory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Management, by Bouchaud and Potters;
- for risk + statistical approach: Risk and Asset Allocation, by Meucci;
- for microstructure: Market Microstructure in Practice, by Lehalle and Laruelle.