Text from
Global Investor, July 1, 2000.
Nonlinear Maths:
Nonlinear
mathematics, a discipline that arose in the first half of the 20th century, but
which failed to achieve the profile it deserved in the financial world, is now
capturing the imagination of modern portfolio theory adherents.
There are two reasons
for this. One is that most observers are coming to accept that traditional
market models can't describe how markets behave. The second is the growing
accessibility of nonlinear applications as a result of vastly increased
computing power. (The Big Picture)
Nonlinear mathematics techniques analyze
prices trade by trade. This continuous price stream is known as tick data.
Adapting algorithms that meteorologists have used since the 1940s, nonlinear
maths adherents claim that they can predict future short-term price trends from
tick data with probabilities of several percentage points above the 51% to 54%
that linear techniques can now predict. However, the calculations used in
analyzing tick data are labor intensive which is why the rise of nonlinear
maths in finance has only accompanied the increase of computer power described
by Moore's Law.
In 1965, Gordon Moore, co-founder of Intel,
observed that the number of transistors per square inch on integrated circuit
boards had doubled every year since they were invented. He predicted this
doubling every 12 months would continue into the foreseeable future. It didn't
exactly, but what has continued is the doubling of data density on computer
chips every 18 months. Experts, including Moore, believe this law that has been
operating for the last four decades will remain valid for the next two, at
least.
The Hurst exponent
The idea that nonlinear maths techniques can
model a chaotic universe was pioneered by the British hydrologist and dam
builder H.E. Hurst (1900-1978) who created an 'exponent'. That model was used
to predict rainfall in watersheds feeding the River Nile so that Hurst could
tell Nile dam builders what height their dams should be. Rainfall patterns are,
like securities prices, seemingly random, but the Hurst exponent was able to
measure how long aperiodic trends would persist. No wonder it caught the
attention of traders.
At the beginning of the 1990s, the mysterious
application of nonlinear statistical techniques hitherto almost unknown in
finance promised untold riches for those who could apply them.
One of the very few who caught this trend
early was Richard Olsen who, in 1985, set up Olsen & Associates in Zurich
which has devoted itself to forecasting foreign currency market movements using
rigorous mathematical exercises well-known to physicists and meteorologists.
Anyone who looks at a price series sees
repeated patterns. For almost a century, this observation has invited technical
analysts the world over to try to identify, describe, codify and predict the
frequency of repetition of these patterns.
"I clearly remember, when I was at
Oxford, the first lecture [in a course given] by James Mirriees who later won
the Nobel Prize for Economics," Olsen recalls. "He explained why
current financial models, which are still in wide use today, were wrong."
Olsen spent "a lot of time" just
thinking about the foundations of economics. It became apparent to him that
economic thinking was based on building blocks that when compared to economic
life he saw around him just couldn't be true. Economic models were static, but
economic activity is dynamic.
To Olsen, the challenge was to build a kind of
dynamic economic model for markets. Olsen didn't want to start with something
static as a static model is just one instance of the dynamic market continuum.
After university, Olsen worked as a foreign
exchange trader. "I saw that some of the secret recipes successful foreign
exchange traders were using were exactly what I had predicted
theoretically," Olsen recalls. They were based on direct market experience
because virtually none of the traders knew advanced mathematics or computer
modeling and testing.
It was obvious to Olsen that his models would
have 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 to
work," he says. Like the models of the traders he studied, Olsen saw
repetitive patterns. He and others also noticed that there was a degree of
divergence between the predictions of linear pricing tools and these actual
prices.
For smaller positions, this divergence between
models and reality was tolerable, but as world wealth grew in 1980s,
institutional assets also grew. As they did, large institutional traders began
to see this divergence was becoming more significant, but few knew what to do
about it. Since Olsen's work was proprietary, applications of nonlinear maths
to finance were simply not on most people's radar screens.
The catalyst for the popularization of
nonlinear maths in finance was James Gleick's Chaos (1986) which inspired
thoughtful financial people to 'find a better way.' One example was Christopher
May, who says that in 1989, he stayed up all night to read it. In the shower
getting ready for work the next day (trading warrants for Baring's in Tokyo),
he had what he describes as an "epiphany", though given his location
a 'watershed' seems a neater description. May quit and later started TLB
Partners, a hedge fund, before writing 'Nonlinear Pricing' (published by
Wiley) last year. "The beauty of nonlinear maths is that you can model
these 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 using
nonlinear statistics was a good way to go," adds. John Moody who is
Director of the Computational Finance program at the Oregon Graduate
Institute in Beaverton, Oregon, and head of Nonlinear Prediction
Systems, a financial investment firm.
What Moody
uses are nonlinear algorithms to extract 'weak' signals from 'noisy' prices; to
identify 'pockets of predictability'; to quantify high probability trading
opportunities; to adapt to changing market conditions; to avoid data
overfitting; and, to manage risk. (See--Bidding US T-bill auctions). Using the
same techniques, Moody can also estimate how reliable his prediction
might be.
Bidding US T-Bill
auctions
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 Finance
program at the
Oregon Graduate Institute
in
Systems, a financial
investment firm
explains how it can
be done using nonlinear
maths techniques.
"If you have
price movements of T-bills or
other related
securities during the current
day, the current
week, and the most recent
couple of months, the
simplest forecast
would be predicting
probability of direction
relative to current
prices. Under the
standard assumption
of a random walk or an
efficient market, you
have an unbiased
coin. But, by making
use of information
from recent market
price behavior, you may
be able to identify
patterns, perhaps even
weak trends which
means you now have a
biased coin to toss.
"You might be
able to get, says, 60%
heads applied to a
probability range of bill
prices. You are not
restricted to simply
using recent price
behavior in the bill
market. You can use
price movements in
multiple markets to
search for relevant
inter-market
relationships. At first, the
probability that
bills might come at 4.52%
might be X% and at
4.51%, X plus some
percent, and so on.
As you get closer to the
auction, expected
ranges can be refined.
"We have looked
at different kinds of
price series in
different market sectors;
stocks, foreign
exchange, commodities
prices, financial
futures in price data series
ranging from monthly
data to tick-by-tick. If
you have an
inter-market relationship you
think is valid, we
can test that relationship
with certain
nonlinear techniques to
determine whether or
not it is a real
relationship in advance
of waiting for
events to prove that
it is or isn't. There is a
lot of work in doing
any of this. In all of what
we have found, we've
seen that there is no
one magic bullet,
just better probabilities."
Olsen says the beauty of using high frequency
or tick data means there is no debate about market theory; no second guessing.
Results using nonlinear techniques are immediate and obvious. Managers have a
better probability of making a series of reasonable predictions. But, when
using it, Olsen has two caveats. People assume that using nonlinear maths is
either impossible, or very easy. In fact, it is neither, but using it does mean
a lot of hard work.
The second point is how to use what one
learns. "It's one thing to know how a petrol explosion works," Olsen
cautions. "It is quite another to build a combustion engine that can
harness this explosion."
The need for nonlinear
Because of the increase in data crunching
speed explained by Moore's Law, coupled with the enormous growth of available
capital, 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, you
realize that you can hardly give even a really good manager 10% of your
fund," says Layard-Liesching. It is now much harder for active managers to
add value, especially when they manage large pools of assets. The only
managerial style big funds can now use in hopes of 'beating the market' are
global macro or long/short strategies. Both involve a higher number of trades
than pension funds have been accustomed to in the past, but both can be
improved with nonlinear estimation techniques.
Pareto's Layard-Liesching has an unusual
approach to managing assets. "Seven years ago, we began working with
Hughes Aerospace because, like military defense establishments, we are
protecting assets in a hostile environment," he says. Pareto turned to
Hughes because, reports Layard-Liesching: "Defense people are pragmatic.
They will use whatever technique it takes to optimize their PK [Probability of
Kill] ratio."
Pareto, under Layard-Liesching's guidance, has often consulted with John
Moody to apply neural networks to Pareto's method. The result is a neural
net modelled on defense missile requirements that Layard-Liesching says has
been performing well for the last six years.
Real results
What nonlinear maths can do in finance is
quantified by Dean Barr, chief investment officer for Deutsche Asset Management
in
In 1994, Barr applied for a patent that was
later granted to transform, analyze and process data to create expected returns
for some 3,000 securities using nonlinear estimation techniques called back
propagation neural networks. After estimating returns, the process selects
securities to build optimal portfolios based on these anticipated returns.
"Nonlinear techniques try to learn from
the 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 the
interdependency of variables with each other. It offers a statement of
unconditionality, meaning one does not know the forms and substance of what
needs to be defined. "This is an area of advanced mathematics that tries
to derive the form and substance of a model of price behavior which isn't known
or assumed previously," says Barr.
"We are trying to capture subtle little
patterns to help us forecast in a more robust fashion the expected return
behavior or expected return of a particular asset class or security," he
explains. Barr has several strategies including long, long/short and large-cap.
"All of them have outperformed the S&P by more than 100 basis points
on an annualized basis," he reports.
Software for laymen
If this is encouraging, even better news for
asset managers is that Barr built his models using a forerunner of NeuroShell
Trader from Ward Systems in
Steve Ward founded Ward Systems in 1988, and
has since sold thousands to nonlinear programs to customers as diverse as the
US Post Office, the US Departments of the Army and the Navy, the Massachusetts
Fish Hatchery Department (for estimated future food and game fish populations)
and scores of corporations and private traders.
Through Ward Systems, nonlinear maths
techniques are available even to users who have only a layman's understanding
of how nonlinear algorithms actually work. NeuroShell Trader takes raw data,
such as moving averages, from which it derives sets of predictions. Trading
rules are then derived from these predictions. Buy/sell signals are generated
from these trading rules. On the surface it sounds simple, but what this
program can do is discover multidimensional patterns in price time series that
are too complex to be seen in a standard chart.
The process obviously requires backtesting. It
can also do walk-forward testing of predictions it has made, modifying its
parameters as it does by 'learning' to adapt to changes conditions as it does
by 'retraining' itself.
Later this year, Olsen & Associates plans
to launch a trading platform that will make markets in foreign exchange on a
24/7 basis. "At the core of this platform is a market-making engine that
uses non-linear predictive models for which we have applied for patents,"
Olsen explains. This is a new approach to foreign exchange market making which
in the past was largely driven by dealers who routinely laid their exposure off
on customers through their sales and advisory networks.
Thanks to
However, Olsen says there are two important
things to remember when contemplating the future of finance. "One is that
financial markets are a non-zero sum game," he says.
This observation leads to the second
conclusion that because markets are a non-zero sum game, it is possible for the
careful practitioner to add value by outperforming market. Whether or not this
will turn out to be true remains to be seen, but win, lose or draw, nonlinear
maths will be an integral part of this process.