Introduction to the Mathematics of Finance: Arbitrage and Option Pricing (Undergraduate Texts in Mathematics)

The first eight chapters + the two appendices introduce a finite model of a market and discuss pricing derivatives in this model. This part of the book is rigorous, elegant and lucid and a real joy to read. It is self-contained and almost all the results are proven in full with a handful of easy proofs left as exercises to the reader. If the book ended here, I would give it 5 stars. The last two chapters discuss the famous Black-Scholes formula in continuous time. The topic is not presented rigorously and I did not find the discussion as compelling as the discussion of the finite model. If you have absolutely no clue about finance and wish to get a mathematically rigorous taste of it and an introduction to the buzz-words as "option", "arbitrage", "pricing", then the first eight chapters of this book fit the bill excellently. Just be aware that the finite model is very simplistic, unrealistic and doesn't reflect the models actually used by financial analysts. But it's a great starting point.