Lesson 12: Too Much Complexity
'Four stochastic parameters good, two stochastic parameters bad.' (Thanks to George Orwell.)
Maths is fun. I've had many jobs and careers in the last three decades, and started various businesses, but the one thing that I keep coming back to is mathematics. There's something peaceful and relaxing about an interesting maths problem that means you can forget all your troubles, just get totally absorbed in either the detail of a formulation, calculation or solution, or lie back and think of deep concepts. I wonder if one of the reasons quantitative finance is in such a mess is that people treat the subject as if it's of academic interest only, to be made as abstract and complicated, and therefore as respectable, as possible.
I'm going to let you in on the big secret of quantitative finance, and you must keep this secret because if word got out then that would be the end of all masters in financial engineering programs. And universities make a lot of money from those.
Ok, the big secret... Quantitative finance is one of the easiest branches of mathematics.
Sure you can make it as complicated as you like, and plenty of authors and universities have a vested interest in so doing. But, approached correctly and responsibly, quant finance is easy.
Let's talk about the different levels of maths you see in quant finance.
As I said earlier, some people try to dumb the subject down. There are plenty of textbooks that kid you into thinking that there is almost no mathematics in the subject at all. These books may dabble in the binomial model but go no deeper. Now anyone with a second-year undergraduate knowledge of numerical methods will recognize the binomial model for the inadequate and cumbersome dinosaur that it is. I like the binomial method as a teaching tool to explain delta hedging, no arbitrage and risk neutrality. But as a way of pricing derivatives for real? No way! Watching the contortions people go through on the wilmott.com Forums in order to make their binomial code work is an illuminating experience. Dumbing the subject down is not good. You cannot price sophisticated contracts unless you have a decent mathematical toolbox, and the understanding of how to use those tools. Now let's look at the opposite extreme.
Some people try to make the subject as complicated as they can. It may be an academic author who, far from wanting to pass on knowledge to younger generations, instead wants to impress the professor down the corridor. He hopes that one day he will get to be the professor down the corridor who everyone is trying to impress. Or maybe it's a university seeing the lucrative QF bandwagon. Perhaps they don't have any faculty with knowledge of finance, certainly no practical knowledge, but they sure do have plenty of people with a deep knowledge of measure theory. Hey presto, they've just launched a masters in financial engineering! Making this subject too complicated is worse than dumbing it down. At least if you only work with the binomial method you can't do much harm, simply because you can't do much of anything. But with all those abstract math tools at your command you can kid yourself into believing you are a derivatives genius. Never mind that you don't understand the markets, never mind that the people using your models haven't a clue what they are doing. I believe that the obscenely over-complicated models and mathematics that some people use are a great danger. This sort of maths is wonderful, if you want to do it on your own time, fine. Or become a finance professor. Or move into a field where the maths is hard and the models are good, such as aeronautics. But please don't bring this nonsense into an important subject like finance where even the best models are rubbish. Every chain has its weakest link. In QF the weakest links are the models, not the maths, and not the numerical methods. So spend more time thinking about your models and their robustness and less on numerical inversion of a transform in the complex plane.
Here's a true story that illustrates my point quite nicely. Not long ago I was approached by someone wanting to show me a paper they hoped to get published. The paper was about 30 pages long, all maths, quite abstractly presented, no graphs. When I'd read the paper I said to the author that I thought this was a good piece of work. And I told him that the reason I thought it was good was because, unfortunately for him, I'd done exactly the same piece of research myself with Hyung-sok Ahn a few years earlier. What I didn't tell him was that Hyungsok and I only took four pages to do what he'd needed 30 for. The reason for the huge difference in derivations was simply that we'd used the right kind of maths for the job in hand, we didn't need to couch everything in the most complicated framework. We used straightforward maths to present a straightforward problem. Actually, what he had done was worse than just unnecessarily obscure the workings of the model. There was a point in the paper where he trotted out the old replacement-of-drift-with-the-risk-free-rate business. He did this because he'd seen it done a thousand times before in similarly abstract papers. But, because the paper was about incomplete markets, the whole point of the model was that you were not allowed to make this substitution! He didn't understand the subtle arguments behind risk-neutral valuation. That was the place where his paper and ours diverged, ours started to get interesting, his then followed a well-worn, and in this case incorrect, path.
Rule 1 of quant finance seems to be 'Make this as difficult as we can.' It's going to be years before the tendency for people to make quantitative finance as difficult as they possibly can is eradicated. And that'll be years while money is lost because of lack of transparency and lack of robustness in pricing and risk-management models. (But on the other hand, there'll be lots of research papers. So not all bad news then!)
There've been a couple of recent Forum threads that perfectly illustrate this unnecessary complexity. One thread was a brainteaser and the other on numerical methods.
The brainteaser concerned a random walk and the probability of hitting a boundary. Several methods were proposed for solving this problem involving Girsanov, Doleans-Dade martingales, and optimal stopping. It must have been a really difficult problem to need all that heavyweight machinery, no? Well, no, actually. The problem they were trying to solve was a linear, homogeneous, second-order, constant-coefficient, ordinary differential equation! (Really only first order because there weren't even any non-derivative terms!) The problem was utterly trivial. Talk about sledgehammers and nuts.
The other thread was on using non-recombining trees to price a simple vanilla option. People were really helpful to the person asking for advice on this topic. But no one, except for me, of course, asked the obvious question, 'Why on Earth are you doing such a silly thing?' I can hardly imagine a more cumbersome, slow, and generally insane way to solve a simple problem.
It disturbs me when people have been educated to such a level of complexity that they can throw about references to obscure theorems while at the same time being unable to think for themselves. To me, mathematics is about creativity in the use of tools not about being able to quote 'results.' Even knowledge of the names of mathematicians and what they are famous for is something I find a bit suspect. If you know the names of all the theorems but don't know when to use them then you are a historian not a mathematician. Perhaps maths is an art, and I'm not impressed with painting by numbers.
If you look through the various Forums on wilmott.com you will see that we have some areas for people to talk about mathematics, research papers, etc., and then there are areas to talk about trading, general finance, etc. You will notice that the majority of people are comfortable in only either the maths areas or the trading areas. Not many people are comfortable in both. That should tell you something, the overlap of skills is far less than one would expect or hope. Who would you trust your money to? A mathematician who doesn't know the markets or a trader who doesn't know maths? Ideally, find someone who is capable in both areas.
And so to the middle ground, not too dumb, not too clever for its own good. Let's start with the diffusion equation. As every mathematician knows there are three important classes of partial differential equation: elliptic, hyperbolic, parabolic. There are various standard techniques for solving these equations, some of them numerical. The diffusion equations that we see so often in QF are of parabolic type. Rather conveniently for us working in QF, parabolic equations are by far the simplest of these different types to solve numerically. By far the simplest. And our equations are almost always linear. Boy, are we spoiled! (I've thought of publishing the 'Wilmott Ratio of salary to mathematical complexity for various industries. Finance would blow all others out of the water!)
Or take the example of some fancy exotic/OTC contract. You start with a set of model assumptions, then you do the maths, and then the numerics. Most of the time the maths can be 100% correct, i.e. no approximations, etc. Given the assumptions, the pricing model will follow as night follows day. Then you have to crunch the numbers. Now the numerics can be as accurate as you like. Let s say you want the value and greeks to be 99% accurate. That's easy! It may take a few seconds, but it can usually be done. So where s the problem? Not the maths, not the numerics. The problem is in the model, the assumptions. Maybe you get 70% accuracy if you are lucky. It seems odd therefore that so many people worry about the maths and the numerics, when it is very obvious where the main errors lie!
There is a maths sweet spot, not too dumb, not too smart, where quants should focus. In this sweet spot we have basic tools of probability theory, a decent grasp of calculus, and the important tools of numerical analysis. The models are advanced enough to be able to be creative with new instruments, and robust enough not to fall over all the time. They are transparent so that the quant and the trader and the salesperson can understand them, at least in their assumptions and use.
Because the models are necessarily far, far from perfect, one must be suspicious of any analytical technique or numerical method that is too fiddly or detailed. As I said above, the weakest link in the chain is not the maths or the numerics but the model assumptions. Being blinded by mathematical science and consequently believing your models is all too common in quantitative finance.
-  And we weren't afraid of the measure-theory police, so we did things that mathematicians have been doing for a century and we did them without any fuss.