Ought you study Complex Analysis and/or Variables in university?

Question

I deliberately haven't stipulated the kind of Quant job, as I'm asking in general. This r/quant comment answers "what is complex analysis used for in quant finance?":

Some pricing models, and some analysis of distributions. E.g. Option valuation using the fast Fourier transform by Peter Carr and Dilip B. Madan has 2207 citations according to Google.

But can you learn Complex Analysis and/or Variables on your own? Or ought you study them in university? I'm assuming Complex Analysis and Variables differ like how regular and honors multivariable calculus. Physics Forums:

The impression I have is that the Complex Variables class is more concerned with computation and calculus using complex numbers (something that as a physicist you may have to do a lot). And the Complex Analysis class will be more about developing the theory of complex numbers and their use in calculus and whatnot. A complex analysis course will mostly be concerned with proving things, while I imagine the complex variables class will be all about using complex numbers to help with computation.

In my grad complex analysis class, we reviewed the entire complex variables course in a day and a half. In other words, the variables course is sort of a prereq for the analysis course. Depending on your familiarity with the complex plane, some topology, and calculus, you could probably go into the analysis class directly. It's certainly more enjoyable.

Dr Transport wrote

Unless you do a PhD in Mathematical Physics, a theorem-proof class in my opinion would not be that helpful. I'm a theoretician and have not had a need for that level of mathematical rigor.

Mathwonk wrote

I am not a physicist, I am a mathematician, but I have taught both those courses. I would imagine that for you the applied course is more useful. I.e. you would probably rather understand how to use complex analysis than how to prove the theorems.

Andy Nguyen discourages it:

Complex analysis has little to no use in FE program. You would better off spend the summer working on your C++.

Vic_Siqiao:

complex analysis has no direct use in FE, but it helps sometimes do calculations invloving complex variables. and i think some topics such as residual theorem are important, which i was asked in a fin math program interview.

macroeconomicus:

I think stochastic calculus will give you better benefit/cost at this stage. Stochastic calculus is used a lot in asset pricing and mathematical finance, and I assume in some other subjects in economics (macro perhaps?). Complex analysis is used a little in advanced probability to work with characteristic functions and such, and also for some things in time series, but you probably don't need to take a whole analysis course to follow. I heard you can just pick it up along the way.

Answer 1

In the context of mathematical finance and financial economics, complex analysis naturally arise in derivative pricing. Specifically, some models impose that the conditional characteristic function of the underlying will be affine in all state variables. In those cases, you can generally obtain a quasi-analytical formula for pricing European options where you evaluate an integral whose integrand is a function of the conditional characteristic function of the underlying. It looks something like this: \begin{equation} \int_0^\infty \text{Imag}\left( g \circ \psi(\phi - i) \right) d\phi \end{equation} Because of the conditional characteristic function $\psi(.)$, $g \circ \psi(\phi - i)$ is going to be complex-valued, so it spits out numbers of the form $a+bi$ where $i^2 = -1$. You're really just working with a grid of $\phi'$s and a corresponding grid of $b'$s when you seek to numerically approximate this integral... So, you don't need a whole course in complex analysis to understand this.

Another place where you will find complex analysis is in time series econometrics. The reason is that you can think of a time series in the time space, just as in the frequency space. I have seen a lot of people trying to get papers on this subject off the ground, but it's the sort of paper almost no one reads -- and even less uses in practice.

My advice: if you're going to put time on something, put time on stochastic calculus and computer programming. Why? Stochastic calculus is the lingua franca of deritative pricing, so almost no matter what you do, it will be useful. As for computer programming, you need to be able to solve problems numerically, as well as to implement analytical solutions. There's nothing like getting your hands dirty, trying to do everything from the theory to the calibration to the data to understand how models work (and sometimes don't work).

Answer 2

I really enjoyed the harmonic analysis course that I had in graduate school decades ago. But is it useful for finance? I looked at the paper Silke Prohl. Harmonic Analysis for Mathematical Finance. and it sounds like fun (did not read it in detail, hope to do it some other time) but I don't see immediate finance relevance.