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Home arrow Business & Finance arrow Frequently Asked Questions in Quantitative Finance


(Answers on page 378)

Question 1: What are the advantages of diversification among products, or even among mathematical models?

Question 2: If you add risk and curvature what do you get? Question 3: If you increase volatility what happens to the value of an option?

Question 4: If you use ten different volatility models to value an option and they all give you very similar values, what can you say about volatility risk?

Question 5: One apple costs 50p, how much will 100 apples cost you?

Lesson 1: Lack of Diversification

One of the first lessons in any course on quantitative finance will be about portfolio construction and the benefits of diversification, how to maximize expected return for a given level of risk. If assets are not correlated then as you add more and more of them to your portfolio you can maintain a decent expected return and reduce your risk asymptotically to zero. (Risk falls off like the inverse square root of the number of different uncorrelated assets.) Colloquially, we say don t put all your eggs into one basket.

Of course, that s only theory. In practice there are many reasons why things don t work out so nicely. Correlations never behave as they should, the relationship between two assets can never be captured by a single scalar quantity. We'll save detailed discussion of correlation for later in the chapter! For the moment I m more worried about people or banks not even attempting to diversify.

Part of the problem with current mechanisms for compensation is that people are encouraged to not diversify. I don t mean 'not encouraged to, I do mean 'encouraged to not.

Example It's your first day as a trader in a bank. You're fresh out of an Ivy League Masters programme. You're keen and eager, you want to do the best you can in your new job, you want to impress your employer and make your family proud. So, what do you trade? What strategies should you adopt? Having been well educated in theoretical finance you know that it s important to diversify, that by diversifying you can increase expected return and decrease your risk. Let s put that into practice.

Let s suppose that you have the freedom to trade whatever you like. Would you make the same trades as others around you? You look around and see that a certain trade is popular, should you copy this or do something 'uncorrelated ? Suppose that you choose the uncorrelated trade. Several things can happen, let s look at a couple of possibilities. First, you lose while all around are raking it in. Not necessarily your fault, maybe just a stat-arb strategy that was unlucky, next month will probably be better. Tough, you don t have a month, you are taking up valuable desk space, space better given to those people making the money. You're fired. But what if the opposite happens? You make money while all the others with their popular trade are losing it. Couldn't be better, right? Sadly, the others have lost so much money that there isn't going to be a bonus this year. Your relatively tiny profit hardly begins to fill in the hole made by the others. You certainly won t be getting any bonus, no one will. If you are lucky you get to keep your job this time. As Keynes said, 'It is better to fail conventionally than to succeed unconventionally.

There is no incentive to diversify while you are playing with

OPM (Other People s Money).

Example Exactly the same as above but replace 'trades' with 'models. There is also no incentive to use different models from everyone else, even if yours are better.

The problem is in the way that bank employees are rewarded by gambling with OPM, the compensation system that rewards excessive risk taking and punishes diversification. Diversification is not rewarded.

Now, I m all in favour of using the 'best possible models, but I can see that there is an argument for different people to use different models, again on the grounds of diversification because ultimately there is no 'perfect model and perhaps even seeking the 'best model is asking too much. Model error wouldn't matter so much if there was more diversification. But sadly how good the models are is of secondary or even tertiary consideration within banks. How can one take as much risk as possible while appeasing risk management? That s number one (and two).

If everyone else is doing similar trades then, yes, you should do the same. Let s do some very simple mathematical modelling of the current system of compensation, nothing complicated, and very easily understood by non-mathematicians. To do a thorough job of modelling compensation we ought to look at:

• Probabilities of each trader making money, distributions, etc.

• How skilful you think you are versus how skilful you think other traders are. (Note that reality may be irrelevant!)

• Number of traders

• Details of the compensation scheme

But we are just going to take a very simple example. Suppose you think you are clever, but you think all your colleagues are fools. And these fools are all stupidly doing pretty much the same trades as each other. To get any bonus two criteria must be satisfied, first that you make a profit and second that between you all you make a profit. Now if your colleagues are idiots you might reckon that it's 50-50 whether they make a profit between them, think of tossing a coin. And they are all betting on the same toss of the same coin. You, on the other hand, are much cleverer, the probability of you making money is 75%, say. Now there are far more of them than you so the probability of it being a profitable year for the group is about 50%. For you to get a bonus first the group as a whole must make a profit, that's 50%, and then you must also make a profit, that's 75%. If you and they are independent then there is a 37.5% chance of you getting a bonus. Now who s stupid? If you copy their trades the probability of you getting a bonus is a significantly bigger 50%!

So what is the logical reaction? All do the same! This is just classical game theory.

Add to this the natural (for most people) response to seeing a person making a profit (even on a coin toss) and you can easily see how everyone gravitates towards following a trading strategy that perhaps made money once upon a time.

And that is the logical thing to do given the nature of the compensation.

Even if everyone starts off by following independent strategies, if you or anyone thinks that one of your colleagues is really clever, with a great strategy, then the logical thing to do is drop your strategy and copy his.

It is easy to see that the tendency is for everyone to converge to the same strategy, perhaps the one that has performed well initially, but this does not guarantee 'evolution to the best strategy or even one that works.

The same happens between banks as well. They copy each other with little thought to whether that is the right thing to do. But this has other consequences as well. The banks compete with each other and if they trade the same contracts then this inevitably reduces profit margins. But profit margins are also margins for error. Reduction of profit margin increases the chance of large losses, and such losses will then happen simultaneously across all banks doing the same trade.

Finally, there's a timescale issue here as well. Anyone can sell deep OTM puts for far less than any 'theoretical' value, not hedge them, and make a fortune for a bank, which then turns into a big bonus for the individual trader. You just need to be able to justify this using some convincing model. Eventually things will go pear shaped and you ll blow up. However, in the meantime everyone jumps onto the same (temporarily) profitable bandwagon, and everyone is getting a tidy bonus. The moving away from unprofitable trades and models seems to be happening slower than the speed at which people are accumulating bonuses from said trades and models!

Unless the compensation system changes then as long as you want a bonus you must do the same trade as everyone else and use the same models. It doesn't matter whether the trade or the models are good. No wonder the man in the street thinks that bankers are crooks. And this does rather make the remaining 11 lessons rather a waste of time!

My scientist within would prefer each bank/hedge fund to have 'one' model, with each bank/hedge fund having a different model from its neighbor. Gives Darwin a fighting chance! I see so many banks using the same model as each other, and rarely are they properly tested, the models are just taken on trust. (And as we know from everyone s problems with calibration, when they are tested they are usually shown not to work but the banks still keep using them. Again, to be discussed later.)

There are fashions within investing. New contracts become popular, profits margins are big, everyone piles in. Not wanting to miss out when all around are reaping huge rewards, it is human nature to jump onto any passing bandwagon. Again this is the exact opposite of diversification, often made even worse because many of those jumping on the bandwagon (especially after it s been rolling along for a while) don t really have a clue what they are doing. To mix metaphors, many of those on the bandwagon are in over their heads.

The key point to remember is something that every successful gambler knows (a phrase I use often, but shouldn't have to), no single trade should be allowed to make or break you. If you trade like it is then you are doomed.

We all know of behavioural finance experiments such as the following two questions.

First question, people are asked to choose which world they would like to be in, all other things being equal, World A or World B where

A. You have 2 weeks vacation, everyone else has 1 week

B. You have 4 weeks' vacation, everyone else has 8 weeks

The large majority of people choose to inhabit World B. They prefer more holiday to less in an absolute sense, they do not suffer from vacation envy. But then the second question is to choose between World A and World B in which

A. You earn $50,000 per year, others earn $25,000 on average

B. You earn $100,000 per year, others earn $200,000 on average

Goods have the same values in the two worlds. Now most people choose World A, even though you won t be able to buy as much 'stuff' as in World B. But at least you'll have more 'stuff' than your neighbours. People suffer a great deal from financial envy.

In banking the consequences are that people feel the need to do the same as everyone else, for fear of being left behind. Again, diversification is just not in human nature.

Now none of this matters as long as there is no impact on the man in the street or the economy. (Although the meaning of 'growth' and its 'benefits' are long due a critical analysis.) And this has to be a high priority for the regulators, banks clearly need more regulatory encouragement to diversify. Meanwhile, some final quick lessons. Trade small and trade often. Don't try to make your retirement money from one deal. And work on that envy!

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