Mathematical Preparation for Interest Rate and Local Volatility Models
Summary
The discussion considers what mathematical background can help a reader understand and implement models in Brigo and Mercurio, including HJM, LMM, and local volatility models. The questioner has studied introductory stochastic calculus and is working through Shreve, and asks whether further preparation is needed.
The answer says Shreve’s standard material can provide a starting point, while warning that the book’s treatment may not fully prepare readers for the semimartingale perspective used in more advanced finance. In particular, it recommends understanding quadratic variation and covariation as distinct concepts from the heuristic rule that squared Brownian increments behave like a time increment. It points to more advanced texts on Brownian motion, stochastic calculus, mathematical finance, and semimartingales as possible study or reference material. These are recommendations rather than a formal prerequisite sequence; the answer also suggests that Brigo and Mercurio can be read alongside references once the introductory Shreve material is covered.
Key ideas
- Introductory stochastic calculus can provide an entry point to Brigo and Mercurio.
- Semimartingales and quadratic covariation may require study beyond the standard Shreve treatment.
- Advanced texts on Brownian motion, stochastic calculus, and mathematical finance can deepen preparation.
- The recommended books are optional references rather than a required sequence.
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Full text
# Background required for the book by Brigo and Mercurio # Background required for the book by Brigo and Mercurio My aim is to be able to read and understand almost all of the book by Brigo and Mercurio including HJM, LMM and the Local Vol models. So that I am able to implement these models on my own. My question is what background do I need to be able to do this? I have read Rannie and Baxter as an introduction to Stochastic Calculus and am now reading the book by Shreve. Do I need to read anything else after Shreve as a preparation or will it be sufficient? I am happy to read and put in the hours, as long it helps me understand and master Brigo and Mercurio and then more advanced books on this topic as well. Your guidance is much appreciated. ## Answer by user25064 (score 4) https://quant.stackexchange.com/a/10099 You can start to understand Brigo and Mercurio from the standard Shreve material but it does not look at things from the perspective of semimartingales which will possibly be confusing at some point. You're probably going to want to understand $d[X,Y]_t$ quadratic variation notion vs just the whole "$(dW(t))^2 = dt$" concept from the Shreve book that I'm assuming OP is referring to. Shreve wrote another excellent book Karatzsas and Shreve, Brownian Motion and Stochastic Calculus with the follow up Methods of Mathematical Finance both of which I can absolutely recommend as being challenging but amazing to learn from, something like "blue rudin" at times. There is another book which I have never read personally but have heard great things about regarding stochastic calculus with respect to semimartingales Diffusions, Markov Processes, and Martingales: Volume 1, Foundations and Diffusions, Markov Processes and Martingales: Volume 2, Itô Calculus The books mentioned above would certainly make Brigo and Mercurio easier to understand but there is no reason that they can't be used for reference, once you're through Shreve I and II.
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