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by Paul Wilmott

I'm not sure readers of this magazine are going to be pleased with the following statement,
but here goes: "There is too much mathematics in finance". Ooh, er!

Recently the Financial Times had a short article on quantitative finance and the current crisis, taking the position that what was needed was more mathematics. In any formal debate one side takes one position and the other side takes the opposite. Hence my statement above. In truth it's not so much about quantity of mathematics as quality. The Financial Times article simply seemed to think that more is better, without any thought for the type of mathematics involved. My position is therefore better stated as:

"The quality of mathematical modelling in finance is poor." And "the quality has been decreasing for at least the last decade."

Twenty years ago people from all backgrounds, whether mathematics, science, economics, physics, etc. were welcomed into finance and were encouraged to contribute. There was a great deal of attempted technology transfer. Yes, most of it fell flat, that's inevitable, but the point is that their opinions and ideas were listened to with respect. Somewhere in the mid nineties the subject of quantitative finance became axiomatized, and very rapidly there became just the one way to do quant finance, and every other approach was dismissed. Even the physicists joining in had to adopt the 'accepted' way of thinking. Groupthink began to dominate.

"I think I know how this happened, and I certainly know why it is bad."

How? I blame the ubiquitous and overrated Masters programmes. Sadly such courses prey on the young and impressionable, usually those straight out of a first degree, leaving them heavily in debt and having to be re-educated once/if they get a job. These courses are almost invariably taught by academics with no practical experience. The courses are made abstract and over-complicated because that's what the professors know. Final word on this: even Madoff only preyed on the rich and didn't exploit poor young students.

Bad? It is extremely bad because now there is an entire generation of quants and wannabe quants who really do believe that knowing mathematics is all that is needed. But they are incapable of questioning assumptions, or building their own models unless they are but a minor, trivial tweaking of somebody else's model. They cannot think outside any box. They have no awareness of the link between human behaviour and the markets, but that is what drives everything! I know all this because I am one of those people having to do the re-education.

I launched the Certificate in Quantitative Finance (CQF), see, in 2003, it has become an antidote to the Masters programs. The CQF is fundamentally about the practice of quantitative finance. This still means that delegates must know about the mathematical methods and models, but they also learn how to criticize models, and build their own. This program has now grown to be the largest quant program in the world at that high level. Most people taking the CQF come from banks, and are people who appreciate the uneasy relationship between theory and practice in this subject.

Recently I was giving a talk about the CQF to an audience of people from all sorts of backgrounds. Afterwards one of them told me that he'd like to take the CQF but wanted to skip the first three modules on the grounds that he knew it all, having taken a Masters. And do you know what he meant by 'knowing it all'? Apparently he could 'derive the Black-Scholes equation' …and that was it! Well, sunshine, that's the trivial bit! What about when it breaks down, about its robustness, what about the popular 'improvements' that make it worse? No, he didn't know any of this. And I said to him "You can be a Nobel Prize winner and you still start the CQF at module one, because even in the first lecture there are things Nobel laureates don't know!"

Having set the tone I am going to spend the rest of this paper describing, in brief, some of the main problems with current quantitative finance modelling and suggest areas for research. In my view these are where people should concentrate their attention, not on yet another closed-form solution of a stochastic volatility model:

Real challenges of real quantitative finance

Lack of diversification
Despite the benefits of diversification being well known among the general public, and despite there being elegant mathematical and financial formulations of this principle, it is still commonly ignored in banking and fund management. This is partly due to the compensation structure within the industry whereby bonuses are linked simply to profit. The end result is that people will try to maximize the amount they invest or bet without any concern for risk.

Supply and demand
Prices are ultimately dictated by supply and demand. But 'price' is different from 'value'. Valuation is what quants do, to come up with a theoretical value of a product based on various assumptions. The role of the salesperson is then to sell the contract for as much above that theoretical value as possible, the difference being the profit margin. But then the quant, who rarely distinguishes price and value often gets confused. He sees a contract selling for $10, its price, and by confusing this price with the theoretical value of the contract deduces something about the mathematical model for that contract.

Jensen's Inequality arbitrage
The expectation of a non-linear function of a random variable and the function of the expectation are not the same. If the function is convex then the former is greater. But options are just non-linear functions of something random, and the difference between the expectation of the function and the function of the expectation is where the value is in an option. The moral is do not assume things are deterministic when they are random, otherwise you will be missing value and mispricing.

Sensitivity to parameters
No one really knows what volatility will be in the future. So risk managers and quants do sensitivity analyses in which they assume a number for volatility, see what is then the value of a contract, change the volatility and see how the contract's value changes. This gives them a range of possible values for the contract. Wrong. They assume a quantity is a constant and then vary that constant! That does not give a true sensitivity, and is completely different from assuming at the start that volatility is a variable. Such analyses can easily give a false sense of security. Because the sensitivity of a value to another quantity is called a greek, I have christened sensitivities to parameters 'bastard greeks' because they are illegitimate.

There are two problems with correlation. First it is not quite what people think it is and second it doesn't do a good job of modeling the relationship between two financial quantities. It's not what people think it is because when asked they talk about two highly correlated stocks trending in the same direction. But more often than not correlation to the quant is about the relationship between two random numbers, returns, for example, over the shortest timescale, and nothing to do with trending. And correlation is a very unsubtle tool for modeling the very subtle relationship between quantities.

Reliance on continuous hedging (arguments)
Derivatives theory is based almost entirely on the idea of continuous hedging eliminating risk. But in practice people don't and can't hedge continuously. They hedge every day for example. Such discrete hedging means that risk has not been eliminated and so all the theories based on risk elimination cannot necessarily be used. That's fine, as long as they openly admit that there is far more risk than they've been leading people to believe.

For every buyer there's a seller. Correct. Therefore derivatives are a zero-sum game. Incorrect. One side, usually the option seller if it's an exotic, is also hedging the options by trading in the underlying asset. And this trading can move the markets. Such feedback is currently not adequately addressed in derivatives modelling or in trading, even though it has been blamed for the 1987 crash.

Reliance on closed-form solutions
There are good models and there are bad models. You'd think that people would choose models based on comparison with statistics, or maybe on some insight gained from fundamental economic considerations. No. Most popular models are chosen because they have simple closed-form solutions for basic contracts. This is seen especially in fixed-income models. Despite the ease of number crunching of most finance models quants still like to work with elegant models when they should be working with the most accurate models. Typical closed-form models used are the Vasicek- or Hull & White-model.

Valuation is not linear
Despite people's everyday experience of going to the shops, popular valuation models are almost all linear. The value of a portfolio of derivatives is the same as the sum of the values of the individual components. The relevance of the visit to the shops is that we have all experienced nonlinearity in terms of either discount for bulk, economies of scale, buy two get one free. Or the opposite, only one item per customer. There are several great non-linear quant models, with some fantastic properties.

Calibration means making sure that the output of a model matches data in the market. It has been likened to the calibration of a spring. That is a completely misleading analogy. Springs behave the same way time and time again. Hooke's law is a very good model relating the tension in a spring and its extension, as long as that extension is not too great. Finance is not like this. Stock prices do not behave the same way, even probabilistically, time and time again because they are driven by human beings. For this reason calibration is fundamentally flawed.

Too much precision
Why bother with too much accuracy in your numerical analysis? First, you don’t know whether volatility is going to be 10% or 20%. Second, you don't ever monitor individual contracts to expiration, together with cash flows and hedging instruments, to see whether you made a profit. Third, you lump all contracts together into one portfolio for hedging, and it's the portfolio that makes a profit (or doesn't). You make money from the Central Limit Theorem, so there's no point in pretending that you know the contract's value more precisely than you do.

Too much complexity
One can no longer see the wood for the trees in quantitative finance. Most researchers are so caught up with their seven-factor stochastic correlation model, with umpteen different parameters, calibrated to countless snapshots of traded prices, that they cannot see that their model is wrong. If the inventors and users of the model cannot see what is going on, then what hope has the senior management, and those who have to sign off on valuations and risk-management practices? Quant finance needs to get back to basics, to work with more straightforward models, to accept right from the get-go that financial models will never be perfect. Robustness is all.

Quantitative finance is one of the most fascinating branches of mathematics for research. But it is not fascinating because of the sophistication of the mathematics. Classical quantitative finance, unfortunately, as taught in universities only encompasses a relatively small amount of mathematical tools and techniques. No, quantitative finance is fascinating because of its potential to embrace mathematics from many different fields, a potential that has mainly not been realized. And fascinating because we are trying to model that which is extremely difficult to model, human interactions.

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Articles on these research areas and others may be found in Wilmott magazine and Wilmott journal, see

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Paul Wilmott

Next issue: July 2024
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