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Summary:

Can you recommend any book which is:

  1. Intro/first course in PDEs
  2. 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?

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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.

  1. 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.

  1. 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).

  1. 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

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  • $\begingroup$ Hm... so your advice to go for purely quant finance approach and skip the mathematical / engineering take on these equations... Are these suitable for those without PDE background (I enjoy Hull but it doesn't have enough PDE material. Will need to check Wilmott)? My plan was to study the basics of PDEs first and then move into finance applications whilst I build intuition around the relevant PDEs and some theory... $\endgroup$ Dec 4 '21 at 10:58
  • $\begingroup$ A copy of Hull is sufficient material for you to be able to derive the Black Scholes pricing equation and implement the numerical solutions needed to price options under that economy. Am sure there is much grounding you can benefit from in non-quant finance books on PDEs - but my understanding of your ask was to focus on Black Scholes - which is specifically a quant finance topic. $\endgroup$ Dec 4 '21 at 11:07
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    $\begingroup$ Black Scholes PDE is a Parabolic PDE. In general PDE books you will also learn much interesting material about Hyperbolic and Elliptic PDEs, which AFAIK has not much application in Finance. Mr. Duffy seems to have done a good job of ransacking PDE books to find material applicable to derivatives pricing. You might want look at his work. $\endgroup$
    – noob2
    Dec 4 '21 at 11:11
  • $\begingroup$ @noob2 good idea to narrow my focus to Parabolic PDEs. Do you have any recommendations of Duffy's books in addition to the one mentioned in James' answer? -> "Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach" $\endgroup$ Dec 4 '21 at 17:57

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