Proof-Oriented Textbooks for Mathematical Finance
Summary
The discussion recommends mathematical finance texts for a reader who wants rigorous proofs alongside financial applications. Björk’s *Arbitrage Theory in Continuous Time* and Shreve’s *Stochastic Calculus for Finance* are presented as core options: the former emphasizes formal definitions and proofs, while Shreve’s volumes move from binomial models to continuous-time finance. Both still leave some technical results unexplored.
For deeper probability foundations, the answers point to Karatzas and Shreve’s *Brownian Motion and Stochastic Calculus* and Williams’s *Probability with Martingales*. These are more mathematically rigorous but offer less direct financial content. Joshi, Neftci, and Baxter and Rennie are also listed as alternatives. The evidence is a set of recommendations rather than a comparative review, so readers should assess each book’s level and coverage against their own background and goals.
Key ideas
- Björk’s text presents arbitrage theory in a formal definition, proposition, and proof style.
- Shreve’s books cover financial models ranging from binomial settings to continuous time.
- Karatzas and Shreve and Williams provide deeper treatment of probability and stochastic calculus.
- The more rigorous probability texts may contain little finance-specific material.
- Joshi, Neftci, and Baxter and Rennie are additional mathematical finance references.
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Full text
# Proof oriented introductory text? # Proof oriented introductory text? I'm currently taking this Coursera course on financial engineering, which is fine but it's very focused on applications instead of proofs. They recommend Investment Science as a companion text, but some of the comments on Amazon seemed to indicate that this book also isn't very proof oriented. I've also heard of Options, Futures and Other Derivatives as a recommended text, but again the Amazon comments indicate that this might not be very rigorous. Are there any texts which require a similar amount of knowledge about finance but are more math focused? If it's relevant, my background is in pure math so I'm decently skilled mathematically but I may have forgotten some particulars about calculus and probability. ## Answer by Olaf (score 8, accepted) https://quant.stackexchange.com/a/15029 The first book that comes to mind that is written in the style of Definition - Proposition - Proof is: - Bjork - Arbitrage Theory in Continuous Time It's pretty well written and can get quite technical. Probably a more common reference is the two-volume set: - Shreve - Stochastic Calculus for Finance I & II The first part deals with the binomial model, which is the technique used in the Coursera course. Usually when people mention Shreve they refer to the second part, which is a lot bigger and goes into the world of continuous time. You'll find that many Master's in quantitative finance or financial mathematics will incorporate one of these two books in their curriculum. Still, in both these books you will encounter plenty of examples where they do not go into more technical details. For that you could look at: - Karatzas, Shreve - Brownian motion and Stochastic Calculus - Williams - Probability with Martingales which is pretty much written for a math audience already. In particular, pretty much all mathematical results proven and not proven in the first two books I mention, are rigorously examined in Williams. On the other hand you will not learn anything financial from this book. Some other books worth mentioning: - Joshi - The concept and practice of mathematical finance - Neftci - An introduction to the mathematics of financial derivatives - Baxter, Rennie - Financial Calculus
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