Text fromGlobal Investor, July 1, 2000.

Nonlinear Maths:

Handmaiden of Post-Modern Finance

Nonlinearmathematics, a discipline that arose in the first half of the 20th century, butwhich failed to achieve the profile it deserved in the financial world, is nowcapturing the imagination of modern portfolio theory adherents.

There are two reasonsfor this. One is that most observers are coming to accept that traditionalmarket models can't describe how markets behave. The second is the growingaccessibility of nonlinear applications as a result of vastly increasedcomputing power. (The Big Picture)

Nonlinear mathematics techniques analyzeprices trade by trade. This continuous price stream is known as tick data.Adapting algorithms that meteorologists have used since the 1940s, nonlinearmaths adherents claim that they can predict future short-term price trends fromtick data with probabilities of several percentage points above the 51% to 54%that linear techniques can now predict. However, the calculations used inanalyzing tick data are labor intensive which is why the rise of nonlinearmaths in finance has only accompanied the increase of computer power describedby Moore's Law.

In 1965, Gordon Moore, co-founder of Intel,observed that the number of transistors per square inch on integrated circuitboards had doubled every year since they were invented. He predicted thisdoubling every 12 months would continue into the foreseeable future. It didn'texactly, but what has continued is the doubling of data density on computerchips every 18 months. Experts, including Moore, believe this law that has beenoperating for the last four decades will remain valid for the next two, atleast.

The Hurst exponent

The idea that nonlinear maths techniques canmodel a chaotic universe was pioneered by the British hydrologist and dambuilder H.E. Hurst (1900-1978) who created an 'exponent'. That model was usedto predict rainfall in watersheds feeding the River Nile so that Hurst couldtell Nile dam builders what height their dams should be. Rainfall patterns are,like securities prices, seemingly random, but the Hurst exponent was able tomeasure how long aperiodic trends would persist. No wonder it caught theattention of traders.

At the beginning of the 1990s, the mysteriousapplication of nonlinear statistical techniques hitherto almost unknown infinance promised untold riches for those who could apply them.

One of the very few who caught this trendearly was Richard Olsen who, in 1985, set up Olsen & Associates in Zurichwhich has devoted itself to forecasting foreign currency market movements usingrigorous mathematical exercises well-known to physicists and meteorologists.

Anyone who looks at a price series seesrepeated patterns. For almost a century, this observation has invited technicalanalysts the world over to try to identify, describe, codify and predict thefrequency of repetition of these patterns.

"I clearly remember, when I was atOxford, the first lecture [in a course given] who later wonthe Nobel Prize for Economics," Olsen recalls. "He explained whycurrent financial models, which are still in wide use today, were wrong."

Olsen spent "a lot of time" justthinking about the foundations of economics. It became apparent to him thateconomic thinking was based on building blocks that when compared to economiclife he saw around him just couldn't be true. Economic models were static, buteconomic activity is dynamic.

To Olsen, the challenge was to build a kind ofdynamic economic model for markets. Olsen didn't want to start with somethingstatic as a static model is just one instance of the dynamic market continuum.

After university, Olsen worked as a foreignexchange trader. "I saw that some of the secret recipes successful foreignexchange traders were using were exactly what I had predictedtheoretically," Olsen recalls. They were based on direct market experiencebecause virtually none of the traders knew advanced mathematics or computermodeling and testing.

It was obvious to Olsen that his models wouldhave to be based on tick data. "This is how people respond to markets, so,it follows that you had better do the same thing if you want your models towork," he says. Like the models of the traders he studied, Olsen sawrepetitive patterns. He and others also noticed that there was a degree ofdivergence between the predictions of linear pricing tools and these actualprices.

For smaller positions, this divergence betweenmodels and reality was tolerable, but as world wealth grew in 1980s,institutional assets also grew. As they did, large institutional traders beganto see this divergence was becoming more significant, but few knew what to doabout it. Since Olsen's work was proprietary, applications of nonlinear mathsto finance were simply not on most people's radar screens.

The catalyst for the popularization ofnonlinear maths in finance was James Gleick's Chaos (1986) which inspiredthoughtful financial people to 'find a better way.' One example was ChristopherMay, who says that in 1989, he stayed up all night to read it. In the showergetting ready for work the next day (trading warrants for Baring's in Tokyo),he had what he describes as an "epiphany", though given his locationa 'watershed' seems a neater description. May quit and later started TLBPartners, a hedge fund, before writing 'Nonlinear Pricing' (published byWiley) last year. "The beauty of nonlinear maths is that you can modelthese price series more accurately than with linear methods," reports May,"the same way you can see an ant better if you use a microscope."

"It became clear to me pretty early in the 1990s that usingnonlinear statistics was a good way to go," adds. John Moody who isDirector of the Computational Finance program at the Oregon GraduateInstitute in Beaverton, Oregon, and head of Nonlinear PredictionSystems, a financial investment firm.

What Moodyuses are nonlinear algorithms to extract 'weak' signals from 'noisy' prices; toidentify 'pockets of predictability'; to quantify high probability tradingopportunities; to adapt to changing market conditions; to avoid dataoverfitting; and, to manage risk. (See--Bidding US T-bill auctions). Using thesame techniques, Moody can also estimate how reliable his predictionmight be.

Bidding US T-Billauctions

Every Monday at 1:00 in the afternoon, US

primary bond dealers must bid on three-month

and six-month Treasury bills.

Because US Treasury bond dealers are

accustomed to financing everything they

buy and selling it as quickly as possible, it

would be ideal if all the inventory were sold

at a profit by the following Thursday

morning when they are required to pay for

the T-bills they bought Monday.

Bond dealers need to know where T-bills

will come so that can prepare their

customers to accept the price levels of

Monday's auctions. If they can anticipate

that the market is going down, they can

short old bills and replace them with new

bills bought at lower prices for a profit. If

they know the market is going up, they buy

old bills now, and swap them for new bills at

higher prices.

Can nonlinear statistical techniques

help them? John Moody, Director of the

Computational Financeprogram at the

Oregon Graduate Institutein Beaverton,

Oregon, and head ofNonlinear Prediction

Systems, a financialinvestment firm

explains how it canbe done using nonlinear

maths techniques.

"If you haveprice movements of T-bills or

other relatedsecurities during the current

day, the currentweek, and the most recent

couple of months, thesimplest forecast

would be predictingprobability of direction

relative to currentprices. Under the

standard assumptionof a random walk or an

efficient market, youhave an unbiased

coin. But, by makinguse of information

from recent marketprice behavior, you may

be able to identifypatterns, perhaps even

weak trends whichmeans you now have a

biased coin to toss.

"You might beable to get, says, 60%

heads applied to aprobability range of bill

prices. You are notrestricted to simply

using recent pricebehavior in the bill

market. You can useprice movements in

multiple markets tosearch for relevant

inter-marketrelationships. At first, the

probability thatbills might come at 4.52%

might be X% and at4.51%, X plus some

percent, and so on.As you get closer to the

auction, expectedranges can be refined.

"We have lookedat different kinds of

price series indifferent market sectors;

stocks, foreignexchange, commodities

prices, financialfutures in price data series

ranging from monthlydata to tick-by-tick. If

you have aninter-market relationship you

think is valid, wecan test that relationship

with certainnonlinear techniques to

determine whether ornot it is a real

relationship in advanceof waiting for

events to prove thatit is or isn't. There is a

lot of work in doingany of this. In all of what

we have found, we'veseen that there is no

one magic bullet,just better probabilities."

Olsen says the beauty of using high frequencyor tick data means there is no debate about market theory; no second guessing.Results using nonlinear techniques are immediate and obvious. Managers have abetter probability of making a series of reasonable predictions. But, whenusing it, Olsen has two caveats. People assume that using nonlinear maths iseither impossible, or very easy. In fact, it is neither, but using it does meana lot of hard work.

The second point is how to use what onelearns. "It's one thing to know how a petrol explosion works," Olsencautions. "It is quite another to build a combustion engine that canharness this explosion."

The need for nonlinear

Because of the increase in data crunchingspeed explained by Moore's Law, coupled with the enormous growth of availablecapital, big pension funds now have a problem believes Ronald Layard-Liesching,a partner at Pareto Partners' London office.

"Once you get north of $ 20 billion, yourealize that you can hardly give even a really good manager 10% of yourfund," says Layard-Liesching. It is now much harder for active managers toadd value, especially when they manage large pools of assets. The onlymanagerial style big funds can now use in hopes of 'beating the market' areglobal macro or long/short strategies. Both involve a higher number of tradesthan pension funds have been accustomed to in the past, but both can beimproved with nonlinear estimation techniques.

Pareto's Layard-Liesching has an unusualapproach to managing assets. "Seven years ago, we began working withHughes Aerospace because, like military defense establishments, we areprotecting assets in a hostile environment," he says. Pareto turned toHughes because, reports Layard-Liesching: "Defense people are pragmatic.They will use whatever technique it takes to optimize their PK [Probability ofKill] ratio."

Pareto, under Layard-Liesching's guidance, has often consulted with JohnMoody to apply neural networks to Pareto's method. The result is a neuralnet modelled on defense missile requirements that Layard-Liesching says hasbeen performing well for the last six years.

Real results

What nonlinear maths can do in finance isquantified , chief investment officer for Deutsche Asset Managementin North America. In 1993, Barr founded LBSCapital, one of the early users of nonlinear mathematical techniques forinvestments. Reformed in 1996 as Advanced Technology Investment, Barr and hiscompany have recently joined Deutsche Bank

In 1994, Barr applied for a patent that waslater granted to transform, analyze and process data to create expected returnsfor some 3,000 securities using nonlinear estimation techniques called backpropagation neural networks. After estimating returns, the process selectssecurities to build optimal portfolios based on these anticipated returns.

"Nonlinear techniques try to learn fromthe data itself to create a model without first describing the problem,"Barr says. The important distinction is the contextual relationship of factors.Linear maths examines a one-to-one relationship. Nonlinear maths tracks theinterdependency of variables with each other. It offers a statement ofunconditionality, meaning one does not know the forms and substance of whatneeds to be defined. "This is an area of advanced mathematics that triesto derive the form and substance of a model of price behavior which isn't knownor assumed previously," says Barr.

"We are trying to capture subtle littlepatterns to help us forecast in a more robust fashion the expected returnbehavior or expected return of a particular asset class or security," heexplains. Barr has several strategies including long, long/short and large-cap."All of them have outperformed the S&P by more than 100 basis pointson an annualized basis," he reports.

Software for laymen

If this is encouraging, even better news forasset managers is that Barr built his models using a forerunner of NeuroShellTrader from Ward Systems in Frederick, Maryland. That system, in itscomplete form, costs a trifle less than $ 2,500 and can be run on almost anypersonal computer.

Steve Ward founded Ward Systems in 1988, andhas since sold thousands to nonlinear programs to customers as diverse as theUS Post Office, the US Departments of the Army and the Navy, the MassachusettsFish Hatchery Department (for estimated future food and game fish populations)and scores of corporations and private traders.

Through Ward Systems, nonlinear mathstechniques are available even to users who have only a layman's understandingof how nonlinear algorithms actually work. NeuroShell Trader takes raw data,such as moving averages, from which it derives sets of predictions. Tradingrules are then derived from these predictions. Buy/sell signals are generatedfrom these trading rules. On the surface it sounds simple, but what thisprogram can do is discover multidimensional patterns in price time series thatare too complex to be seen in a standard chart.

The process obviously requires backtesting. Itcan also do walk-forward testing of predictions it has made, modifying itsparameters as it does by 'learning' to adapt to changes conditions as it doesby 'retraining' itself.

Later this year, Olsen & Associates plansto launch a trading platform that will make markets in foreign exchange on a24/7 basis. "At the core of this platform is a market-making engine thatuses non-linear predictive models for which we have applied for patents,"Olsen explains. This is a new approach to foreign exchange market making whichin the past was largely driven by dealers who routinely laid their exposure offon customers through their sales and advisory networks.

Thanks to Moore's Law, there appears to be no end todeveloping nonlinear math applications for finance. "When I compare thecomputing power that was available 15 years ago to what is available now, it'shard to anticipate what we will all be doing 15 years from now," Olsenmuses.

However, Olsen says there are two importantthings to remember when contemplating the future of finance. "One is thatfinancial markets are a non-zero sum game," he says.

This observation leads to the secondconclusion that because markets are a non-zero sum game, it is possible for thecareful practitioner to add value by outperforming market. Whether or not thiswill turn out to be true remains to be seen, but win, lose or draw, nonlinearmaths will be an integral part of this process.

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