## Thoughts On Mathematical Finance (3F/MFE Exam)

I moved over to the actuarial side of the business a few years ago from our capital markets team. I’m not an actuary, but I took an interest in modeling and followed the advice of my first boss in the business: “want to get ahead? Be more useful to us”. We were starting an actuarial department and I joined in.

I didn’t start taking the exams seriously until my wife got pregnant. It’s been a breakneck pace since and I just wrote my fourth exam, the 3F/MFE, subject of which is the math of derivative securities. I’m fairly familiar with the area from the CFA exams, but what does this have to do with valuing P&C liabilities, the job I’m training to do? And anyway, isn’t all this math exactly what is supposed to be wrong with finance?

There are two big ideas in the syllabus that are worth discussing. The first has some notoriety, which is that we assume asset returns are distributed normally and so prices are distributed lognormally. The immensity of this assumption cannot be overstated. First because just about everything in mathematical finance requires it to be true; second, it’s just about complete garbage. In fact, there is absolutely no attempt to justrify this assumption in any of the required readings.

The other idea is a bit more esoteric and is called risk neutrality. Risk neutrality is a way of dealing with the problem that people are risk averse, in that we’re unlikely risk \$1 of loss for \$1 of gain on a 50/50 bet. So we’d only take a 50/50 bet if we could win \$1.5 and lose \$1.

In a strange twist, the math skips over the ‘utility’ of a dollar of profit vs dollar of loss. Instead, we reweight the probabilities so that the return is the risk free rate. In a risk neutral world, it’s not a 50/50 bet.

Don’t worry, I don’t really understand it, either. One of my complaints is that, once I’ve passed the test (8 weeks ’till they tell me!), it doesn’t matter whether I understand it or not, I’m never going to use this knowledge. The understanding is literally worthless.

So why did I have to learn about it? In the the released sample problems there’s this really interesting remark (number iii after the solution to problem 71 in this pdf document) in respect of risk-neutral pricing:

Arguably, the most important result in the entire MFE/3F syllabus is that securities can be priced by the method of risk neutral pricing… Some authors call the following result the fundamental theorem of asset pricing: in a frictionless market, the absence of arbitrage is “essentially equivalent” to the existence of a risk-neutral probability measure with respect to which the price of a payoff is the expected discounted value.

I didn’t understand what that meant until I really got stuck into the Black Scholes derivation. The story of the that derivation goes like this: borrow some money and buy a derivative. Now hedge away the risk with some shares. All that’s left is the risk free rate. The big trick is figuring out how many shares you need to buy/sell to hedge he risk of the derivative. That’s where normally distributed returns come in and that’s where everything falls apart.

The CAS/SOA don’t make similar comments about any other part of syllabus, so it’s interesting that they decided to play Hitchcock and jump in for a cameo here.*

Why do they think this here is such a big deal? And, more importantly, why do they think actuaries should know this stuff?

There’s a bit of history to this exam. Originally it was paired with a more statistics-heavy exam for the complete Exam 3 (now they’re 3F/MFE and 3L). The split exams were half-exams, too, but the MFE has crept back up to a full-on 3-hour test. Why?

And think of the social context. During the 90s and early 00s, financial mathematics was a pretty big deal. Physicists were pouring out of science departments and onto trading floors, probably amazed that all this abstract math was a valuable skill. It’s no surprise that the CAS/SOA tried to hop onto that bandwagon.

But ultimately the ideas aren’t terribly compelling. I’ve studied these models intimately now (the fixed income ones are laughably useless and WAY more complicated), and I’d never give someone my money to trade with them.

*Get ready for a tangent:

There’s an essay at the beginning of my copy of Moby Dick that for no good reason pops into my head all the time. Melville is describing the most essential characteristic of a whale, the spout, and the essayist claims he has sheds the character of Ishmael and starts writing as himself:

the spout is nothing but mist. And besides other reasons, to this conclusion I am impelled, by considerations touching the great inherent dignity and sublimity of the sperm whale… He is both ponderous and profound. And I am convinced that from the heads of all ponderous profound beings, such as Plato, Pyrrho, the Devil, Jupiter, Dante and so on, there alwasy goes up a certain semi-visible steam while in the act of thinking deep thoughts. While composing a little treatise on Eternity, I had the curiosity to place a mirror before me and ere long saw reflected there a curious involved worming an undulation in the atmosphere over my head. The invariable moisture of my hair, while plunged in deep thought, after six cups of hot tea in my thin shingled attic of an August afternoon, this seems an additional argument for the above supposition.

Incidentally, I don’t buy it; I think he’s still Ishmael here (treatise on Eternity? Puh-leeze). But I think the sentiment remains. Melville is clearly in awe of the whale and pays it a kind of honor by comparing it to his greatest heroes.

Maybe the SOA/CAS is just so impressed with the elegance of the math that they HAD to break in for a chat.