Monday, January 25, 2010

Actuarial theology? Wait a second...

It's funny how different subjects can overlap. This point was forcibly brought home to me by a paper I was reading which compares different "stochastic asset models" (SAMs).

Would you expect a topic like this to tie directly into the philosophy of religion? I didn't. But guess what I found...

Meet SAM

SAMs are bits of mathematics and/or computer code that are used to predict roughly what possible values an asset could hold in future. Actuaries use these to guesstimate how much money you need to put into a pension scheme to get a given payout, and how that money should be invested in the meantime.

As a very simple example, we generally stick money in high-interest (high-risk) assets to start with, and then move it into low-risk (low-interest) assets later on. That's because, as someone approaches retirement, there's less time to recover from disasters - if the market crashes two days before they hit 65, we want to ensure that the piggy-bank is still reasonably full.

Normally the asset ratio is taken to be a simple linear function of time. So for someone retiring at 65, you might say: put min(1,(65-age)/10)*100% of their money into a high-risk fund (e.g. Asian shares) and max(0,(age-55)/10)*100% into a low-risk fund (e.g. government bonds). This is a nice simple linear relationship.

However, it is also a very rough rule of thumb pulled straight out of our actuarial buttocks. The problem is, we don't know how the assets are likely to behave over time, so we can't come up with a more realistic strategy. SAMs are intended to fill that hole.

A SAM will normally be a set of difference equations (or differential equations) with a bit of randomness ("innovations") thrown in for good measure. The equations are intended to codify the relevant bits of economic theory - for example we expect that retail prices will be affected by inflation. The model is then initialised with numbers estimated from the last few years' worth of real-world economic data.


The problem* is, these models can sometimes have "interesting" side-effects. For example, the paper I'm reading refers to a type of model which it calls a "random walk variant with alpha-stable innovations". I think I know what that means, but anyway the interesting bit is the following quote:

"The alpha-stable distribution [...] is in some ways the most intellectually satisfying possibility. [...However, they are] not satisfactory for the pricing of derivatives because the prices of simple options are theoretically infinite [...]"

Yeah, that'd be a problem.

An option is a simple kind of contract - basically a form of insurance. If I run a sawmill, I want to ensure that I can get some logs cheaply no matter what. One approach would be to arrange a "future" contract - a contract which sets a price and time at which I can and must buy the wood.

However, this isn't perfect: what if the price of wood drops massively? In that case I'll be paying good money for something I could buy cheaper elsewhere. Instead of a future, I could arrange an "option" contract, which sets a price and a time at which I can - but don't have to - buy the wood**. This gets me the best of both worlds - no risk of excessive gain or loss. I'll normally have to pay a bit of a premium for the privilege, of course.

These options are vital tools of business, and to be told that they should be infinitely expensive is disturbing. How can that be the case?


As far as I can tell, the logic seems to be: extreme market events are more likely than you'd think. Let's consider a simple case, an option to buy a tonne of wood for £100. We need to work out how much loss the option seller expects to suffer. This will depend on the probability distribution of the lumber's cost C at the option's expiration date.

If we knew for sure that the cost would be £150, we could say P(C=150) = 1. The option seller would then have to buy wood at £150, and sell it to me for only £100. Her loss would be £50, so she'd presumably charge me at least £50 for this (rather useless) option***.

If we guessed that the option price would either go up or down by £30, and that the odds were the same, we could say P(C=50) = P(C=150) = 0.5. If the price went down, I'd be daft to exercise the option, so the seller's expected loss is 0. If the price went up (as it would 50% of the time), the seller would lose $50 as before. £50 × 50% = £25, so the option would probably cost a bit more than £25.

Let's consider a more complicated case:
P(C<=100) = 1/2
P(C=200) = 1/4
P(C=300) = 1/8
P(C=500) = 1/16
P(C=900) = 1/32
P(C = 100*(1+2^n)) = 1/4 × 1/2^n for n between 0 and infinity

It's easy to prove that 1/2 + 1/4 + 1/8 + 1/16 + ... = 1, so this is a valid probability distribution.

In this case, what is the option seller's expected loss? It will be:
0×P(C<=100) + (200-100)×P(C=200) + (300-100)×P(C=300) + (500-100)×P(C=500) + ...

This can be rewritten as:
Sum[n = 0 to infinity] { 100 × 2^n × 1/4 × 1/2^n }
= 25 * Sum[n = 0 to inf] { 1 }

But if you add ones together an infinite number of times, you get infinity! This is what's known as a "fat-tail" effect - the "tail" of the distribution is so long and wide that it completely overwhelms the rest of the calculation.

Wager 2.0

From a practical economic perspective, this is a disaster for the model. It's implausible that the value of anything can be infinite - there isn't that much money on the planet.

But I'd like to draw your attention to something. Here we have a model - a belief system of sorts - which, if true, can grant us amazing rewards. All we need to do is act on this belief (e.g. buy an option) and our expected gain goes up to infinity.

Does that remind you of anything?

Pascal's wager is one of the oldest arguments for God. It states that, if you believe in God, you could get to go to Heaven - an infinitely valuable reward - and at worst you'll lose a finite amount of time spent praying etc. If you don't believe, you definitely won't go to Heaven. The "fat tail" of Heaven's infinite goodness is supposed to overwhelm the mere finite rewards you can get from living a happy Godless life.

By the same logic, the infinite return we can expect from buying an option under the alpha-stable model should overwhelm our mere finite uncertainty about whether the model is actually valid. There is no way we can be infinitely sure that the alpha-stable model is wrong - statistics just doesn't work like that. So (in theory) the logic will always hold.

I've given some thought to Pascal's wager, and I think I've found a fair number of holes in it. I haven't thought much about option pricing yet; the same holes may be present there. But in the meantime, a single thought is buzzing round and round in my head:

Should I buy an option?

* OK, so there are all sorts of other problems. But I'll save them for another post.

** Actually this is only one type of option (a European call, to be precise). For more information, see the Wikipedia page.

*** I'm ignoring interest here as it's not really relevant to the point.

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