Paul Romer is onto something. The mathematization of economics is a double edged sword. On the one hand, mathematics is a powerful tool and language we can use to describe real world phenomenon. We are able to build and analyze models which encapsulate the important phenomenon we observe in the real world without distracting us from the rest. However, if we build into our models inaccurate assumptions, we shouldn't be surprised when they result in inaccurate predictions. Furthermore, there may be a point when our models are so mathematically complex that they are entirely intractable; should we still rely upon them? 

This is a really great introduction to behavioral economics. The premise is that economic phenomenon cannot be explained fully when assuming people are fully rational, profit or utility maximizing agents. That may sound obvious; people are complex, and are often driven by emotions. Yet much of economics and finance assumes that people are fully rational. 

When I was a graduate student, I asked my economics professor about this assumption: did he really think people were fully rational? He said obviously not, however, you cannot predict or model irrational behavior. I'm not sure that's entirely true. 

The efficient markets hypothesis has been the central proposition in finance since the 1970's. It states that securities prices in financial markets must equal fundamental values, either because all investors are rational or because arbitrage eliminates pricing anomalies. This book describes an alternative approach to the study of financial markets: behavioral finance. This approach starts with an observation that the assumptions of investor rationality and perfect arbitrage are overwhelmingly contradicted by both psychological and institutional evidence. In actual financial markets, less than fully rational investors trade against arbitrageurs whose resources are limited by risk aversion, short horizons, and agency problems. 

"Probably more so than any other chapter in the book, this review of the internet bubble seems inconsistent with the view that the stock market is rational and efficient. The lesson from this chapter, it seems to me, is not that markets occasionally can be irrational and, therefore, that we should abandon the firm-foundation theory. Rather, . . . in every case, the market did correct itself. The market eventually corrects any irrationality . . . ." 

"The greatest investment advisor of the twentieth century, Benjamin Graham, taught and inspired people worldwide. Graham's philosophy of "value investing" -- which shields investors from substantial error and teaches them to develop long-term strategies -- has made The Intelligent Investor the stock market bible ever since its original publication in 1949."

This is a must read for the serious student of mathematics thinking about going into quantitative finance. I think he accurately describes the frustration one feels when transitioning from math or physics to finance. In particular, in mathematics we prove theorems. In finance and economics we build models which are used to claim a deeper understanding of the real world. This is sometimes dangerous. 

I like this book to introduce the topic to MS students or advanced undergraduate students. 

This is a more rigorous and comprehensive treatment of the material presented in Shreve. He also has offered online classes which can be found on his website here. 

This is an excellent introductory text to the area of stochastic calculus. This is written for the beginning graduate student who has taken a sequence in real analysis or a rigorous course in probability. I would recommend this to any student interested in understanding the mathematics a little better. 

Fractional Brownian motion may be a better tool to develop models for stocks and bonds then classical Brownian motion. This book develops the calculus and compares different methods of integration.