# What type of math background would I need to learn PDEs?

I’d like to learn about the Black Scholes equation which is a partial differential equation. What type of math is this? What type of math background would I need to understand this equation?

You’d probably just need a course in multi-variable calculus to gain a pretty good understanding of the equation; specifically, you’d need to understand the concept of partial differentiation. I don’t think that an actual course in PDEs would be necessary to gain a good understanding of this particular equation, especially if you’re more concerned with its applications than with its inner mathematical properties.

“Multi-variable calculus” generally means the third course it the university Calculus sequence. Depending on exactly what you’re trying to understand about it, you may or may not even need that much, or far more. I’m not an economics major (1 class), and while I have a minor degree in mathematics, I can’t even pretend to really know anything about this equation. You can certainly follow the derivation and review the methodology to do that; if your intent is to actually build a parametric model from the principle itself rather than just plugging in the formula, then that’s complicated, probably too much so for incidental interest. You’d really need to know more about both PDEs and numerical methods to do something useful with that.

Someone who has a real math degree might come along and correct me, but I don’t think even your introductory Partial Differential Equations course is really going to get you too far with this; that class is kind of a potpourri of different methods for solving some of the most basic PDEs, usually simple one- and two-dimensional systems with linear or periodic behavior. If you’re interested in this equation alone, you’re better off studying a specific derivation and methods associated with it.

Stranger

To really understand Black-Scholes, you need a background in stochastic calculus, which is pretty advanced stuff. You’ll need a measure-theoretic background in probability, which is not something that the typical math student is going to run into on the way to a master’s degree.

It’d be helpful to know exactly what your math background is like, but you might look for a copy of McDonald’s “Derivatives Markets”, 2e. There’s a relatively informal discussion in there.

To really understand Black Scholes, you’re going the wrong way to study PDEs. Many, probably most, of the people who use Black-Scholes could not solve the equation. To undetsand it, you want the financail intuition. You can get this from the binomial model. If you want the math of Black-Scholes equation and solution, you can then get those from the limit of the binomial model, and you need nothing more than an introductino to calculus for this.

If you want to go further and and study exotic options, you can do it pretty much the same way.

If you want a mix of math and finance, I’d check out Paul Willmott’s books. They’re expensive so you probably want to try to borrow them to look at first.

You need to understand continuous time stochastic processes. I’d suggest picking up a book on options and derivatives that shows mathematical derivations. As I recall, Hull’s book derived Black-Scholes in a couple of pages, but that was after some preliminary work. The text also has a nice reading list that will guide you in digging into more details. I found the book quite readable and would recommend it.

The last math class I took was a calculus class for business majors. I also had an elementary and intermediate class in statistics.

Ideally, I’d like to be able to understand Options, Futures, and other Derivatives by John Hull, which is considered to be one of the most important books on option pricing.

Ph.D. in math here, and I don’t really grok PDEs. I can understand what they’re saying, but a real intuitive grap… I don’t get it.

Really? It was pretty much the fourth math class for the science/math/engineering majors at my college. It would go Calc I, Calc II, Differential Equations, Multi-Variable Calculus, and then on to the upper classes that mostly math majors take (hell, most science and engineering majors just stopped at Diff Eq.)

Which school was this? ODEs at most I’ve seen come after Calc3, and usually after linear algebra as well.

Come to think of it, how would you teach ODEs at all sensible without multivariable calculus and linear algebra?

I’m going off McDonald’s text here, which is similar to Hull but slightly less rigorous. At a bare minimum, you’ll need two semesters of calculus, some linear algebra, and some probability theory. It certainly wouldn’t hurt you to have that third semester of calculus and differential equations.