PDE Books for Black–Scholes and Option Pricing
Summary
This discussion recommends ways to learn PDE techniques used in derivatives pricing. It suggests starting with an options text that introduces Itô’s lemma, binomial pricing, and explicit and implicit finite-difference methods, then moving to more specialized treatments of computational finance and PDE-based option pricing. It also points to work on the alternating direction explicit method and to books covering path-dependent options and dimension reduction.
The recommendations span accessible practitioner material and more technical references on parabolic PDEs, Green’s functions, and stochastic differential equations. The evidence is the respondents’ descriptions of the books and their own experience, rather than a systematic comparison or a worked Black–Scholes derivation. Readers should choose based on their mathematical background and desired depth; some recommendations are described as abstract or technically demanding. The discussion offers a reading path, not a self-contained PDE course or an evaluation of the methods’ performance.
Key ideas
- An options textbook can connect Itô’s lemma and binomial pricing to finite-difference solutions of pricing PDEs.
- Implementing explicit and implicit finite-difference schemes can help build practical understanding.
- Specialist texts extend the material to methods such as operator splitting and path-dependent option pricing.
- Green’s function and parabolic PDE references may suit readers seeking more theoretical depth.
- The recommendations are practitioner opinions rather than a structured assessment of textbooks.
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# Any book which is intro to PDEs but prioritises techniques useful for solving Black-Scholes? # Any book which is intro to PDEs but prioritises techniques useful for solving Black-Scholes? Summary: Can you recommend any book which is: - Intro/first course in PDEs - Covers solution methods useful for Black-Scholes model? Background I have just started learning about PDEs (after studying the ODEs) with the aim to understand in depth the Black-Scholes PDE solution (an then move to more complex models). As a start I'm following a course which is solving the heat equation using the method of Separation of Variables and we end up with a Fourier series solution. Ok, flows nicely from the ODEs but it's not anywhere near the Black-Scholes formulae! I need an introduction to PDEs but have a strong preference to learn techniques used in the Black-Scholes derivation i.e. Fundamental Solution and Green's Function over using other methods not suitable for Black-Scholes. Is that possible? Can you recommend any books with that angle? ## Answer by James Spencer-Lavan (score 6, accepted) https://quant.stackexchange.com/a/68987 Ok I'll answer this from a practioner's perspective rather than a purist's (which I am solidly not). Below are the books that influence my understanding of this space, listed in chronological order of when I bought them. TL;DR - buy Hull and Wilmott's books & read in that order. But also search for papers by Duffy after having read Hull. - Options, Futures, and Other Derivatives, John Hull I'd consider this essential reading for anyone interested in derivatives. It covers tonnes of ground you need in general but specific to PDE pricing it has: Ito's Lemma in sufficient detail for you to be dangerous, binomial trees in sufficient detail for you to get the idea but also see their limitations and, finally, finite difference methods / illustrations for pricing European and American options in both explicit and implicit FDM. Hugely useful for the junior practitioner - but make sure to implement explicit & implicit FDM in code yourself before you move on! You won't know it otherwise. - Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach, Daniel Duffy This book is for the serious purist - method of lines, operator splitting, etc. Powerful techniques heavily influenced by some of the greats of early PDE study (lots of Soviets) but as a practitioner text I found it too dry / abstract to be of strong utility in my day job. Duffy is a very good author (and a very nice guy, having conversed with him over the years) with a number of papers I like (specific to PDEs you need to read The Alternating Direction Explicit (ADE) Method for One-Factor Problems - this is my go-to method for general PDE option pricing). - Paul Wilmott on Quantitative Finance 2nd Edition: 3 Volume Set This is where you can really explore PDE option pricing as he extends into path-dependent options, auxiliary variables, dimension reduction techniques etc. Much like Hull, this book is direct and accessible with sufficient mathematical precision to avoid ambiguity. Huge practical value and a lovely book to read. Lots of exciting ideas to try out. I hope this list is helpful for you. JSL ## Answer by pierrot (score 1) https://quant.stackexchange.com/a/77851 I like the book "Computational Methods for Quantitative Finance: Finite Element Methods for Derivative Pricing (Springer Verlag)" from Norbert Hilber. Very easy to read and exhaustive. The book "Tools for Computational Finance (Universitext) 6th ed. 2017 Edition" from Rudriger Seydel is a good introduction too and takes you through the implementation steps of Black Scholes PDE. Last, for more advanced readers, the book "Computational Methods for Option Pricing" from Olivier Pironneau is a must. ## Answer by Daneel Olivaw (score 1) https://quant.stackexchange.com/a/77853 It depends on the degree of technicality you are seeking. Here are two books that might be relevant for an analysis based on Green’s function, although they are quite technical (the exposition of the Feynman-Kac theorem in Karatzas & Shreve (1991) is based on these): > Friedman, Avner (1964). Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ. Friedman, Avner (1975). Stochastic Differential Equations and Applications, Academic Press, New York.
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