SUPPOSE that money, as well as being at the root of all evil, is the
鈥渆lementary stuff鈥� of the economic realm. That it is, in its own setting, every
bit as fundamental as the subatomic particles of the physical world. As a
metaphor, you might have no trouble swallowing that one.
But what if the relationship between physics and finance goes beyond
metaphor? Suppose a deep and real mathematical link runs between some of most
abstract theories of quantum physics and the trillions running round that most
risky of businesses鈥攖he derivatives markets? It would mean that these
markets behave in the same way as the fundamental particles and fields of the
Universe.
This notion may take some extra chewing. Yet two Russian physicists believe
it to be true. Kirill Ilinski and Gleb Kalinin of the University of St
Petersburg came up with the idea last year. And working with equations that
resemble those of the quantum theory of electrons and photons, they have now
created a theory that reproduces the complex rules that traders have used for
years to work the markets. A strange mathematical fluke? Perhaps. But stranger
still is the possibility that the laws of physics might contain the seeds of a
unified theory of finance.
Advertisement
Ilinski never intended to start thinking about finance. After finishing a PhD
in mathematical physics at the Steklov Mathematical Institute in Leningrad, he
came to Britain as a postdoc at the University of Birmingham. There he met a
colleague who was playing the Prague stock exchange, and Ilinski became hooked
on the mathematical complexities of the market. 鈥淚 was sure that physics
methodology would be useful,鈥� he says. After returning to Russia, and studying a
pile of books on financial management and financial economics, he hit on an idea
that triggered memories of his PhD work on gauge theory鈥攖he framework for
all modern theories of particle physics.
Gauge theory is a set of mathematical tricks that lets physicists describe
fundamental particles and the fields through which they interact. Modern gauge
theory has its roots in the 1920s, when physicists noticed that the quantum
description of a particle鈥攊ts wave function鈥攈as a strange
mathematical redundancy. It always contains an unimportant number called a
鈥减丑补蝉别鈥�.
Fields come into the picture because quantum particles have wave-like
properties, and tend to smear themselves out in space. So a particle doesn鈥檛
have just one phase, but many鈥攐ne for every point in space. Changing all
these phases in exactly the same way, known as making a gauge transformation,
has no effect on the basic equations of quantum theory. This seems like a
trivial matter until you realise that physicists believe that such 鈥済auge
invariance鈥� is a fundamental property of the laws of nature.
But if different points in space receive different changes, then the
equations do change. Roughly speaking, gauge theory says that the fundamental
fields of the Universe exist in order to maintain the status quo of gauge
invariance, and to prevent changes in these unimportant phases from having any
effect. If the fields also change, and in just the right way, they can
compensate for any phase changes that are different.
The simplest gauge theory, quantum electrodynamics (QED), deals with the way
electrons and positrons interact via the electromagnetic field, and it is one of
the most accurate theories ever devised. In QED, the close connection between
the fields and the phases has some peculiar consequences. Like most things
quantum, the phases of particles fluctuate randomly. To stop these quantum
fluctuations causing brief violations of gauge invariance, the field has to
fluctuate in a corresponding way. It does this by stirring up 鈥渧irtual鈥� photons
which transmit forces between charges, causing them to move and restore the
balance. On large scales, the virtual photons become unobservable, and classical
electromagnetism emerges.
Electric cash
Ilinski, working with his colleague Kalinin at St Petersburg, has translated
this arcane conceptual framework into the financial world. Instead of electric
charge, think about cash sitting in trading accounts. If the account is in
surplus, it鈥檚 a positive charge, while if it鈥檚 in debit, it鈥檚 a negative charge
(see Diagram).
Cash can be invested in dollars, yen, pounds, whatever,
because at any given time, the exchange rates are fixed, so the choice is
unimportant.
This, Ilinski and Kalinin claim, is the financial equivalent of gauge
invariance, with the choice of currency corresponding to a choice in phase for
an electron鈥檚 wave function. If every currency suddenly doubled in value, it
shouldn鈥檛 affect the value of our account because the exchange rates stay the
same. However, if just one rate (say dollar to yen) suddenly changes randomly,
it creates an opportunity for what the financial world knows as 鈥渁rbitrage鈥�. In
other words, someone, somewhere sees the imbalance and can profit by making a
clever exchange of currencies.
Gambling on prices
The imbalance stirs up an 鈥渁rbitrage field鈥� that acts to restore the balance,
just like the electromagnetic field in QED (see Diagram). This field serves as a
force between trading accounts, causing cash to flow between them as traders
eliminate the arbitrage opportunity by taking advantage of it (see 鈥淎rbitrage鈥�).
Sounds nice. But does this way of looking at the markets pay any theoretical
or practical dividends? Well, Ilinski and Kalinin have shown that the behaviour
of some of the more complex financial instruments, derivatives such as options
and futures, quite simply tumble out of the equations of their theory.
Derivatives play an important role in protecting investors against the
uncertainty鈥攐r volatility鈥攐f the markets. Suppose, for example, that
you are head of a giant bakery and you are worried about El Ni帽o driving
up wheat prices next year. There are two ways in which derivatives can help. You
can buy a 鈥渇uture鈥�, where you enter a contract to pay today鈥檚 price for wheat in
a year鈥檚 time. However, since El Ni帽o is unpredictable, it might actually
lead to a glut in wheat, drive down prices, and you would be stuck paying a
higher price than you need to. The solution is to buy an 鈥渙ption鈥� on wheat,
which gives you the right鈥攂ut not the obligation鈥攖o buy wheat at
today鈥檚 price in a year.
Until 1973, options were a financial sideshow because anyone who sold one had
no way to tell how the market might change by the time the option
matured鈥攊n short, they seemed to be taking an unquantifiable risk. But
then independent financial theorist Myron Scholes, working with Fischer Black of
the University of Chicago and Robert Merton of the Massachusetts Institute of
Technology, came up with an epoch-making theory of how to handle them
(鈥淐alculated gambling鈥�, New 杏吧原创, 9 August 1997, p 36).
They took for granted that the variations in the price of things鈥攕uch
as wheat鈥攁re unpredictable. To work out the worth of, say, our wheat
option, they followed two key ideas. First, they realised that the effect of an
option鈥攖he movement of goods and funds that it engenders鈥攃an be
replicated, or mimicked, by buying and selling various amounts of wheat and cash
each day. That is, an option can be 鈥渂uilt鈥� out of other more familiar products,
the values of which are closer to hand.
Second, they assumed that arbitrageurs in the market are extraordinarily
effective. This supposition of 鈥減erfect鈥� arbitrage implies that the value of an
option has to match that of its replicating portfolio. For arbitrageurs would
quickly exploit any imbalance, buying and selling for profit, and would cause
the forces of supply and demand to bring the prices back into balance.
From these ideas follows an equation which allows an option on wheat to be
valued without knowing anything about the future price of wheat. The
Black-Scholes-Merton theory (which earned its inventors a Nobel prize last year) acts
like a set of Newton鈥檚 laws for the derivatives markets. Emboldened by the risk
management ideas contained within it, financial institutions now use derivatives
to offer cheaper, more efficient products such as mortgages and savings
accounts. The BSM equation also instructs traders as to how they should juggle
the proportions of options, cash and assets in their portfolios, so as to
鈥渉edge鈥� their risks, immunise themselves against movements in the markets, and
earn profits from the charges they put on trades.
Ilinski and Kalinin鈥檚 first triumph is to derive the entire BSM theory from
the heart of their gauge theory. That in itself is quite an achievement. But
there鈥檚 more: just as quantum theory 鈥渃orrects鈥� Newton鈥檚 laws of classical
physics, so Ilinski and Kalinin鈥檚 calculations correct the BSM theory. They
offer a more powerful version of the traditional theory of derivatives鈥攐ne
that takes into account the effects of arbitrage fluctuations.
This is important, because the BSM assumption of 鈥減erfect鈥� arbitrage isn鈥檛
really true. First, arbitrageurs are fallible. And second, when an arbitrage
opportunity occurs in the market, it persists for a short time before being
washed out. Just as in QED electrons interact with vacuum fluctuations of the
electromagnetic field, which causes them to skitter to and fro, so cash flows
through markets in response to momentary imbalances that produce arbitrage
opportunities. This is an effect the BSM model doesn鈥檛 include.
`It ain鈥檛 physics鈥�
Ilinski and Kalinin talk of the 鈥渞ationality鈥� of traders. This 鈥渞ationality鈥�
has little to do with everyday reason, but is a number鈥攁nalogous to the
charge of the electron鈥攚hich measures how quickly and strongly traders
respond to exploit arbitrage opportunities. Volatility鈥攁 measure of how
much market prices fluctuate鈥攑lays a role similar to Planck鈥檚 constant in
QED. Because Planck鈥檚 constant h is exceedingly small, quantum
fluctuations normally only reveal themselves in the atomic world. If it were
much larger, then our world would be a much stranger place. Similarly, as the
volatility in a market becomes larger, the quantum-like fluctuations in it grow
as well. On the other hand, and in line with QED, the traditional BSM theory of
derivatives emerges as a 鈥渃lassical limit鈥� when volatility is small.
Ilinski and Kalinin hope that the corrections offered by their theory will
provide some practical tools for traders working in real markets. As each new
generation of derivatives hits the market, competition between banks forces
traders to charge less for their services and drives the creation of new, more
sophisticated products. This puts derivative traders in a tough spot. Unlike
their clients, they try to avoid betting directly on the markets. Nor do they
want to become victims of arbitrage by underpricing their derivatives.
Will Ilinski and Kalinin鈥檚 theory help? Finance theorists have spent many
years already trying to tackle such problems. And seasoned traders are often
cynical about the contributions of academics. 鈥淲hat these people forget,鈥� says
Nassim Taleb, an adviser at the investment bank Paribas, 鈥渋s that every real
market price is not a point on a random walk, but a contractual agreement
between two rational adults. It ain鈥檛 physics.鈥�
Making money
Robert Campbell of Lehman Brothers is also blunt: 鈥淚鈥檇 like to see these
academics come in to Lehman at 7.30 in the morning and do some real pricing and
hedging. It鈥檚 a big mistake to rely too much on models鈥攃ommon sense is
very important.鈥� Glancing at Ilinski and Kalinin鈥檚 paper, he adds: 鈥淭his may be
aesthetically pleasing, but it has little relevance to what we do at
尝别丑尘补苍.鈥�
Over at Goldman Sachs, however, Emanuel Derman, head of quantitative
strategies, is more forgiving: 鈥淭heir work sounds interesting, and there are
good reasons for trying to estimate the effects of imperfect hedging.鈥� Derman is
an ex-physicist himself, which perhaps explains his warmer attitude. 鈥淲hat they
need to do,鈥� he adds, 鈥渋s explain to people in finance why they should switch to
their formalism.鈥�
And yet even Derman acknowledges a profound difference between the worlds of
physics and finance. 鈥淚n physics, you鈥檙e playing against God,鈥� he says, 鈥渨ho
doesn鈥檛 change his mind very often. But in finance, you鈥檙e playing against God鈥檚
creatures, whose feelings are ephemeral, at best unstable, and the news on which
they are based keeps streaming in.鈥�
So whether traders will soon switch to the Ilinski and Kalinin approach is
unclear. As is whether it would make them any money. Efficient arbitrage should
quickly render useless any model that gives one market player an advantage over
the others. Unlike the natural world with its fixed and immutable laws, the
financial universe seems to evolve mischievously to confound new ideas that help
us understand it. If there ever is a Grand Unified Theory of Finance, it may
remain someone鈥檚 closely guarded secret.
THE exchange rates between dollars, pounds and yen fluctuate constantly,
buffeted by supply and demand and other economic factors. You might think the
dollar-pound rate will go up tomorrow, and exchange your pounds for dollars now
in anticipation. But you have no way of knowing this鈥攁ll you can do is bet
on the market, and your hunch might be wrong.
If, however, you have the capacity to trade dollars, pounds and yen all at
the same time, there is a way of making a risk-free profit. It鈥檚 called
arbitrage. Suppose 拢1 = $2, 拢1 = 200 yen, and $1 = 100
yen, then by converting a pound into dollars, then into yen, then back to
pounds, you wouldn鈥檛 make any profit. But if the dollar to yen rate suddenly
changed so that $1 was the equivalent of 125 yen, then the same round
trip transaction would yield a profit of 50 yen, or 25p. If this could be done
instantaneously, it would amount to risk-free arbitrage.
In the language of Ilinski and Kalinin, the blip in rates would create an
arbitrage field which would create trading that would restore the balance
between currencies, by changing the dollar-pound and pound-yen rates, for
example.