Mathematical References for Martingales and Arbitrage Pricing
Summary
The document collects recommendations for rigorous study of martingale theory, stochastic calculus, and arbitrage pricing. Suggested books include works by Karatzas and Shreve, Shreve’s two-volume finance series, and Musiela and Rutkowski. The answers distinguish resources suited to continuous-time methods from material on the binomial model, and one contributor notes that exhaustive proofs of the Fundamental Theorem of Asset Pricing can be technically demanding.
For formal results, the responses point readers to papers by Harrison and Pliska on risk-neutral measures and no-arbitrage or market completeness, as well as a general theorem by Delbaen and Schachermayer. One answer also recommends their book on the mathematics of arbitrage. These are peer recommendations rather than a systematic comparison or review; the contributors differ in what they have read and how accessible they found particular proofs. The document is therefore a bibliography and orientation guide, not a tutorial deriving the results.
Key ideas
- Karatzas and Shreve is recommended for rigorous Brownian motion and stochastic calculus.
- Shreve’s finance volumes and Musiela and Rutkowski are suggested for formal mathematical finance study.
- Harrison and Pliska papers are cited for foundational results on risk-neutral measures, no-arbitrage, and completeness.
- Delbaen and Schachermayer are recommended for general treatments of the Fundamental Theorem of Asset Pricing.
- The recommendations reflect individual reading experience rather than a systematic evaluation of the references.
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Full text
# Reference request for arbitrage pricing with martingale theory # Reference request for arbitrage pricing with martingale theory I am a mathematician. What's the go-to reference for a proper math-based introduction to martingale theory and arbitrage pricing? The books I am being referred to deal mostly either with the discrete case, or, if its continuous, then it does not contain all the proofs and there's a lot of hand-waving (for example, Bjork's Arbitrage theory in Continuous time, which does not even contain proper proofs of, say, the fundamental theorem of finance or the all-important Ito formula). ## Answer by Brian B (score 3) https://quant.stackexchange.com/a/33649 As a mathematician I have preferred Karatzas and Shreve, "Brownian Motion and Stochastic Calculus" (ISBN 978-0387976556). It has all the theorems and proofs, and is well-written. Shreve noted the popularity of his book and later wrote the 2-volume set "Stochastic Calculus for Finance". The second volume may also be satisfactory to you, but I have not read it. ## Answer by FKaria (score 2) https://quant.stackexchange.com/a/33655 I don't think there many books that proof the fundamental theorem of asset pricing as is quite technical and not very interesting for the usual audience studying quantitative finance. Also, Ito formula is stochastic calculus subject, is a requisite for many mathematical finance books. That said, I like these books for a more formal approach: - Musiela, Rutkoswki - S.E. Shreve I for the binomial model - S.E. Shreve II for continuous time pricing. For a proof of the fundamental theorems of asset pricing see: - Harrison, Pliska 1980 (Risk neutral probability measure implies no arbitrage) - Harrison, Pliska 1981 (Unique risk neutral measure if and only if market is complete). ## Answer by Daneel Olivaw (score 2) https://quant.stackexchange.com/a/33663 Regarding the proof of the Fundamental Theorems of Asset Pricing (FTAP), as explained by @FKaria not many books present exhaustively and rigorously the proof as it is quite long and technical while not that useful in practice. However, you might want to look at the paper that shows the result is the most general framework: > Delbaen, Freddy; Schachermayer, Walter (1994). "A General Version of the Fundamental Theorem of Asset Pricing". Mathematische Annalen. 300 (1): 463–520. ## Answer by red_trumpet (score 0) https://quant.stackexchange.com/a/85641 There is The Mathematics of Arbitrage by Delbaen and Schachermayer. I can't really comment on the later parts of the book, but I read the proof of the finite version of the Fundamental Theorem of Asset Pricing and found it quite accessible.
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