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Lesson 10: Calibration

Note: This Lesson has been censored to protect the innocent.

'A cynic is a man who knows the price of everything and the value of nothing,' said Oscar Wilde. At least a cynic may know that there is a difference between price and value. A typical quant thinks that these two are the same. To fully appreciate finance theory and practice you need to distinguish between these two and also to understand the concept of worth. Go back to page 200 and remind yourself (or maybe you remember the go-cart example?)

One of the most damaging effects of confusing price and value is the ubiquity of unthinking calibration. By assuming that there is perfect information about the future contained in option prices we are abdicating responsibility for performing any analysis, estimation or forecasting for that future. We are in effect saying that 'the market knows best,' which has to be one of the stupidest knee-jerk statements that people make in quantitative finance. I know that markets do not know best; I have two pieces of information for this. First, I have been involved in successful forecasting, making money because of the difference between forecasts of the future and the market's view of the future. Second, I speak to more people in this business than almost anyone else, and I know that the vast, vast majority of them are using models that make ridiculous assumptions about the market and its knowledge (and aren't even properly tested). If almost everyone is basing their models around the prices of some coked-up 23-year-old trader and not on any statistical analysis then there are going to be plenty of (statistical) arbitrage opportunities for a long time to come.

I hold three people responsible for the popularity of calibration. Three very nice, very intelligent people who all published similar work at about the same time. These three are _______,15_______and__________16

Professor ________ is a very experienced practitioner, now academic in the city of N_Y_. He has a great enthusiasm for understanding and modelling the markets. I get the impression that whenever he finds (yet) another violation of a model's assumptions he is first disturbed, then intellectually stimulated, and finally turns his mind to improving and quantifying. Dr ___________ is a practitioner. He takes a very European, laid-back, approach to this business, almost as if he knows he will never come up with perfection. But he keeps trying! I do not know Professor ____________very well, only having met him once and exchanged some emails. Clearly he is very talented, and seems quite charming, with the looks of a 1930's swashbuckling matinee idol! Had his book with__, published in 19_, been printed in paperback and cheaper[1]then I have no doubt that quantitative finance would be a completely different, and better, subject from what it is now.

All of these researchers are men I admire and respect, and they have all produced brilliant work over the years. But by some once-in-a-lifetime alignment of the heavenly stars they all in 19_/_ published independently the same idea of __, the model that_ is a function of _________price and_. And this is the 15_'s work was joint with_. However, I don't know this latter gentleman so won't comment on his niceness or intelligence! 16I don't really blame them. Their research was interesting and clever, just in my view misguided. It's the sheep who implemented the ideas without testing that I blame. idea I hate more than any other in quantitative finance![2]

The function in question is found by looking at the market prices of exchange-traded contracts. In other words, rather than estimating or forecasting ______and then finding the prices of options, we work backwards from the prices to the ___________model. This is calibration.

To be fair to these three, this wasn't the invention of calibration. Ho & Lee (1986) published a fixed-income paper which did something similar. But in fixed income this is not so bad - interest rate volatility is lower than in equities and there are many contracts with which to hedge.

First of all how does calibration work? I'll explain everything using this __________model.

When you look at the market prices of exchange-traded options you see how there are strike and term structures for implied volatility. Implied volatility is not constant. The strike structure goes by the names of skews and smiles. If volatility is constant and the Black-Scholes model is correct, then these prices cannot be correct otherwise there'd be arbitrage opportunities. The existence of arbitrage opportunities seems to make some people uncomfortable.[3] What is the simplest way to make our model output theoretical prices that exactly match market prices? In equities the simplest such model is the _________________________model.

Implied volatility is a function of two variables, strike and expiration, so let's make actual volatility a function of asset price and calendar time, two variables again. There is enough freedom here that if we choose this actual volatility function carefully enough then the solution of the Black-Scholes equation will match market prices of vanilla options of all strikes and expirations. That's calibration.

And why do people do it? Here's another story. Your task is to price some exotic equity option. It just so happens that you've been doing a part-time PhD in volatility forecasting, so naturally you'll want to use your new volatility model. So you price up the exotic and show the results to your boss. 'Fantastic,' he says, 'I'm sure you've done a great job. But there's one small problem.' Here it comes. 'I think your model is wonderful, I know how clever you are. But those @*&%$ in risk management are going to need some convincing before they'll let you use your new model. Tell you what, if you can value some vanillas using your model for comparison with market prices then I'm sure that'd go a long way to persuading them.' So you do as you are told. You value the vanillas using your model. However your model only happens to be a fantastic volatility-forecasting model, it is not a model that calibrates to market prices. Therefore your values and the market prices of the vanillas disagree. Your boss tells you to dump your model and use the__ model like everyone else. Your bank has just missed a fantastic opportunity to make a killing, and will probably now lose millions instead.

This model was adopted with little thought simply because of quants obsession with no arbitrage! Ask a quant for an estimate of volatility and he will look to option prices. Ask him to estimate volatility using only the stock price, or for an asset on which there are no traded options, and watch the look of confusion that comes over his face.

Advocates of calibrated models will defend it by saying two things. First, how do you know that your volatility model is so great? Second, in practice we hedge with vanillas to minimize risk that the model is wrong.

The first defence is silly. I know very few walks of life in which we know anything with any certainty. Or any walks of life in which we know the probabilities of events, outside of a few casino games. Does this stop people going about their day-to-day business? Getting drunk, chatting up women, starting families, having an affair, nothing has either a certain outcome or has known consequences with known probabilities. We are spoiled in quantitative finance that we seem, according to theories, to need little knowledge of asset price distributions to value derivatives. We don't need to know asset growth rates for example. And this seems to have caused our ability to attempt to forecast or to analyse historical data to have completely atrophied. And worse, not only have people become lazy in this respect, but they also seem to think that somehow it is a fundamental law of finance that we be given everything we need in order to correctly value an option. This is utter nonsense. Here's an arbitrage that some people believe exists: do a Masters in Financial Engineering, get a job in a bank, do a little bit of programming, take home a big pay packet, and retire at 35. If you are terrified of the thought of the existence of arbitrage, why is this one any different?[4] No, I'm afraid that sometimes a little bit of effort and originality of thought is required.

The second defence is really saying, well we know that calibrated models are wrong but we solve this by hedging with other options so as to reduce model risk. This is a big, fat fudge. At least these people accept that calibration isn't as good as they've been led to believe. My problem with this is that this fudge is completely ad hoc in nature, and no one really tests whether the fudge actually works. And the reason for this is that risk management don't really know enough about the markets, the mathematics or the research literature.

The above strongly hints that the _ model and calibrated models generally are rubbish. They are. I will now tell you how to prove it for yourself, and then explain why.

Today you take price data for exchange-traded vanillas. You work backwards from these prices to find the actual volatility using the ______________model (details in

Wilmott, 2006a, even though I don't believe in it) or any other calibrated model. You have now found the local volatility surface or model parameters. Imagine making the local volatility surface out of modelling clay, which is then baked in the oven. Once found, this surface, or model parameters, are not allowed to change. It is set in stone. If there is a change then the model was wrong. You come back a week later when the data has changed. You recalculate (recalibrate) the model. Now look at the local volatility surface or parameters and compare with what you found last week. Are they the same? If they are then the model may be right. If they are not, and they never, ever are (just ask anyone who has ever calibrated a model) then that model is wrong. If our subject were a science (which it could be if people wanted it to be) then the model would be immediately thrown away. As it is, people continue to use the model (with the sort of ad hoc fudges mentioned above) and pretend all is well. See Schoutens et al. (2004) for nice examples of this.

A simple analogy is that one day you go to a fortune teller. She tells you that in August next year you will win the lottery. But then you go back a week later, she doesn't recognize you, and now she says that sadly you will be run over by a bus and killed... in June next year. These two pieces of information cannot both be right, so you conclude that fortune telling is nonsense. Similarly you should conclude that calibration is nonsense.

Mathematically we have here what is called an 'inverse problem,' meaning that we work backwards from answers/values to find parameters. Such problems are notoriously unstable in the sense that a small change to initial conditions (implied volatility) can have a huge effect on the results (actual volatility). Diffusion equations, such as the Black-Scholes equation, cannot be 'run backwards in time.' But you get inverse problems in many branches of applied mathematics. In an episode of CSI Miami there was a murder on a yacht. I think one of the big-breasted and bronzed babes was the vic and H. had to find the murderer. The clue as to the perp was a piece of fabric on which there was some writing. Unfortunately the fabric had got wet and the ink had diffused. H. takes the fabric back to the lab, runs it through one of their hi-tech gizmos and before our eyes the lettering appears. What they had done was to reverse time in a diffusion process, an inverse problem not unlike calibration. As every applied mathematician knows this is a no-no, and at this point I lost all respect for CSI! Actually it's not as bad as that, as long as the diffusion has not acted for too long and as long as we know that the image on the fabric was made up of letters then we can with a certain degree of confidence figure out what the writing might have been. With inverse problems in finance such as the _____________model such reverse engineering would only make sense if the assumed model, that volatility is a function of asset price and time, were a good one. It just is not. And no amount of fancy mathematics or numerical analysis is going to make this bad model any good.

So that's how you prove that calibrated models are rubbish: Calibrate now and recalibrate a week later. Then take your results to the most senior risk managers you can find. Tell them that what they want you to do is dangerous and insist on implementing a decent model instead!

Here are a few problems with calibrated models.

• Over fitting: You lose important predictive information if your model fits perfectly. The more instruments you calibrate to the less use that model is.

• Fudging hides model errors: Perfect calibration makes you think you have no model risk, when in fact you probably have more than if you hadn't calibrated at all.

• Always unstable: The parameters or surfaces always change when you recalibrate.

• Confusion between actual parameter values and those seen via derivatives: For example there are two types of credit risk, the actual risk of default and the market's perceived risk of default. If you hold an instrument to maturity then you may not care about perceived risk of default, if you sell quickly then all you care about is market perception since there is little risk of actual default in that time.

Why is calibration unstable?

In a recent piece of research (Ahmad & Wilmott, 2007) we showed how in the fixed-income markets one can back out the 'market price of interest rate risk.' For every random quantity there is a market price of risk, measuring how much expected return in excess of the risk-free rate is needed for every unit of risk. Economists and believers that humans are rational will probably say that this quantity should be, oh I don't know, 3, say. It should be a nice stable parameter representing the nice stable behaviour of investors. Of course, sensible people know better and indeed when we look at this parameter, shown in Figure 5.12, we see exactly what you would expect, a lot of variation. (For technical reasons this parameter ought to be negative.) In some periods people need more return for taking risk, sometimes less, sometimes they'll even pay to take risk! And this is why calibration is

Market price of interest rate risk versus time.

Figure 5.12: Market price of interest rate risk versus time.

doomed. When you calibrate you are saying that whatever the market sentiment is today, as seen in option prices, is going to pertain forever. So if the market is panicking today it will always panic. But Figure 5.12 shows that such extremes of emotion are shortlived. And so if you come back a week later you will now be calibrating to a market that has ceased panicking and is perhaps now greedy![5]

Calibration assumes a structure for the future that is inconsistent with experience, inconsistent with commonsense, and that fails all tests.

And finally, can we do better? Yes, much better! And to get you started go back to Lesson 9 and go through that reading list!

Note: If your goal is to fool risk managers into believing that you are taking little risk while actually taking enormous risks then, yes, use any of these calibrated models, the model, Heston, etc. I can even tell you which model is best for hiding risk. Not hedging risk, note, I mean hiding it from anyone who might want to restrict the size of your trades and hence your bonus. But, for obvious topical reasons, I think we've gone beyond that now!

  • [1] It is still one of the most expensive books I have ever bought!
  • [2] Heston (1993) is a very close second.
  • [3] As I've said elsewhere, life and everything in it is about arbitrage opportunities and their exploitation. I don't see this as a bit problem. It's an ... opportunity. Evolution is statistical arbitrage in action! Ok, now I see why maybe Americans don't like arbitrage, they don't believe in evolution!
  • [4] Note that I'm not saying I think this arbitrage exists, only that many people do.
  • [5] So I would advocate models with stochastic market price of risk, as being sensible and not too far removed from classical models.
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