Mathematicaltools from quantum theory are being used to develop increasingly complex
financial products that exploit the uncertainties of the markets to make a profit
Paul Darbyshire
-
EVERYDAY,trillions of dollars are tra- I
whereby decisions to buy and sell are
ded on the world's financial markets. I
based on the counterintuitive rules of
The bulk of these transactions are carsuperpositionand entanglement.
ried out by traders working in banks,
who stakeinvestors' money on financial
The path to profit
"instruments" such as the FTSE-IOO
Path integrals are essentially a way to
share index or the exchange rate beadd together probabilities, and date
tween two currencies. However, as inback to the work of Norbert Wiener in
vestorshavedemandedgreaterpotential
the early 1920s.However, they are probreturns on their investments,this basic
ably better known among physicistsvia
trading activity hasbecomeincreasingly
Richard Feynman, who in 1948 used
more sophisticated.
path-integral methods to reformulate
Modern trading strategies are based
the rules of quantum physics.
on financial "derivatives" - financial
Feynman suggestedthat when conassetswith values that are determined
sidering the quantum mechanics of a
by, or derived from, the price of some
moving particle, every conceivablepath
underlying asset.This underlying asset
could be assigned a certain complex
could be the shareprice of a company;
number called the probability amplian exchange rate, or the price of a
tude for that path. The probability amcommodity such as sugar or oil. Deplitude for any event,suchasan electron
rivatives can basically be structured
passing through a slit in a screen, can
around anything that has a value that
then be obtained by summing the probcan vary in an unpredictable way: In- Rocketscience-thedecades-oldpracticeofabilities for every possiblepath between
deed, it is even possible to structure a ~,nputting
a"se~
o,fmarket
vari~bles
into.abasic
the initial and final position of the par.black.box~rlcl~g~~deltof~ndthe ~rlceofa
ticle (figure 1). Furthermore heshowed
derivative around .a quantity .that does financial
derivativeISIncreasinglybeingreplaced
..'
not have an ObVIOUS
financIal value, bymorerealistic
models,
someofwhichexploit
that the resultIng mtegral can be solved
such as the average temperature or techniques
fromquantum
physics.
to produce a neat mathematical packlevel of rainfall in a particular country.
agecalled the "propagator", which conThe need for quantitative methods to determine the value tains all the dynamical information about the system as it
of the many different types of derivatives has led to an in- evolvesover time.
creasing number of physicistsand mathematicians joining
Many of the techniques developedby Feynman can easily
the lucrative world of investment banking. Most large fi- be adapted to apply to random; or stochastic, rather than
nancial institutions now employ a small number of "quants" quantum processes.The only major differenceis that the con(or "rocket scientists" asthey are more affectionately known) tribution from each path will be a real number rather than a
to develop,.among other things, new pricing models to guide complex one, and that the total from all contributions will
traders and help generate bigger profits.
representa probability rather than a probability amplitude.
One recent trend in the growing field of quantitative finance
The first person to apply Feynman path integrals to finanis to apply techniques borrowed from quantum physics to cial modelling was Jan Dash - a particle physicist at the
price derivatives.Chief among theseare path integrals,which University of California in Berkeley who had moved into
were originally developedto describethe interactions of ele- banking - in the mid-1980s. At that time, little notice was
mentary particles. Furthermore, trading activity in the more taken of his radical endeavour, but later many academics
distant future may even take place on a "quantum board", began to investigate ways in which path integrals could be
PHYSICS WORLO MAY 2005
physicsweb.org
25
applied to the financial markets. The
reasonwas simple: the value of a financial derivative depends on the "path"
followed by the underlying asset.
One of the best ways to illustrate this
is to consider a type of derivative called
~n option, of which there are two types. A
B
~ "call" option is a financial cont~act
$
ISsuedby one party to another that gIves
2
the buyer the right, but not the obligation, to buy the underlying assetfor Inthe1940sRichard
Feynman
pioneered
anew
a specified "strike" price at a future path-integral
approach
tosolving
problems
in
"maturity" date. The seller or under- quant,um
mechanics.
Consider
~double-slit
I
.
.
wnter, normally working for a bank,
charges the buyer a premium for the
opdon up front. If the value of the unym
. g assetis higher than the strike
derl
.
.
prIce when the optIon matures, the
buyer will presumably exercise their
right to buy the asset at the lower price
.
.
and. the.n sell!t at Its present value, resultmg m an mstant profit less the premium paid to buy the option in the first
place (seebox on page 27).
A " put " optIon
. IS
. a Slffi
. il ar contract
certainty and is therefore free from risk.
As a result, it should earn the risk-free
rate of interest that is applicable to safe
investmentssuch asgovernment bonds,
at leastfor a very short period in time.
In banking terms, this meansthere are
"no arbitrage" opportunities. Arbitrage
is based on the fact that two identi~al
assetsshould sell for the same pnce
acrossmarkets:if the prices differ, an opportunity arisesto buy the underpriced
asset and then sell the identical overpriced asset for a profit. To maintain
..
experiment:
anelectronfrompolntAreaches
.
.
pointBbytakingoneoftwopossible
paths:
AS1B the Black-Scholes nsk-free posItIon m
orAS2B.
Although
weknowwhere
theelectron
practice, the stock amounts must theresta.rted
~~dwhere
itfinished.
wehavenoi~ea
fore be constantlychangedby frequently
WhIChsiltltcamethrough.Bytreatln~theflnal.
Purchasin
actualpathasa quantumsuperposition,
the
. g. or selling. the amount of
probability
amplitudes
canbesummedoverall
stock Wlthm the portfolIo (figure 2).
possiblepathsthatoccurbetween
AandBto
From this analysis, Black, Scholes and
determineatotalamplitude.If weletthe number Merton were able to derive a general
of slits get very large,then the sum approaches
anintegral-i.e.thepathintegralfromAto B.
Sucha path-integral
formulation
showsvery
pr?mising
r~sultswhenanalysing
derivatives
th$
pricesof whicharedep~ndent
onthe path
followed by the underlYing asset.
. .J~£r'
.
partial Ullleren.tIal equ~tIon for the value
of a stock optIon, whIch turned out to
look very similar to the heat-diffusion
equation from thermodynamics (seebox
on page.29) Th e so1utIon
. 0f thIS
o equa-
that givesthe buyer the right to sell the
underlying assetat maturity for an agreed strike price. If the
price of the assetis lower than the strike price, the investorwill
buy the asseton the open market at the lower price before
immediately selling it at the higher strike price to the company that sold the option. Again, this will result in an instant
profit lessthe initial premium.
Pricing an option is a complex mathematical problerI!,~t
involves diffusion processessuch as Brownian motion. This
random movement, first observed by the botanist Robert
Brown with pollen grains suspendedin a liquid, is frequently
observedin nature. Due to the unpredictable behaviour of
the underlying assets,the derivativesmarkets are similar.
tion led to analytic formulas for the price
of standard call and put stock options. This meant that., for
the first time, it was possibleto find the fair value of any call
or put stock option by simply knowing the price of the stock,
the strike price of the option, the risk-free interest rate, the
volatility of the underlying stockprice, and the time to maturity of the option. This value was the price of the option that
was quoted by financial institutions and subsequentlytraded
around the globe.
Wall Streetwas ecstatic;traders could simply punch in a set
of market variables and out popped the value of the option.
However,practitioners and academicssoonbegan to uncover
the shortfalls and holeswithin the Black-Scholes model due
to its restrictive assumptions.For example, both the volatility
Option pricing
and the risk-freeinterest rate are assumedto be constant over
The idea of developing a mathematical model for pricing time, which is clearly implausible in today'sfinancial markets.
an option datesback to 1900,when Louis Bachelier proposed
When the markets, in particular the foreign-exchange and
a stochasticprocessto depict the evolution of a stock price. equity markets, act outside the Black-Scholes world, traders
Much later, in 1973,Fischer Black and Myron Scholesin col- are often guided by past experience and rely on "gut feellaboration with Robert Merton, while all at the Massachusetts ing". While suchinstincts are useful, they can never compete
Institute of Technology (MIT), revolutionized derivatives with a trader equipped with a robust pricing mechanism.
pricing by developing a pioneering formula for evaluating Without doubt, such models can make a great deal of
non-dividend-paying stockoptions.
money, which is why so many researchers are looking for
In order to simplify the analysis,the MIT team devised a new ways to price derivatives.
specialportfolio comprising two financial assets:a "short" (or
seller)position in an option and a "long" (or buyer)position in Quantum pricing
the assetunderlying this option. The researchersnoted that A path-integral description of the Black-Scholes model was
both the stockprice and option value are affectedby the same recently developed by Belal Baaquie of the National Unisource of uncertainty; namely the movement of the stock versity of Singapore and co-workers.From this, Baaquie and
price. As a result, over a very short period of time the price of co-workerswent on to devisea quantum-mechanical version
a call option is perfectly correlated with the price of the un- of the Black-Scholes equation to describethe price of a simderlying stock, while the price of a put option is negatively pIe, non-dividend-paying option.
correlated with the underlying stock.
In quantum mechanics, the state of a physical systemcan
This meansthat by making appropriate adjustmentsto the be representedby a wavefunction, I'll>, and the expectation
portfolio, the profit or lossfrom the option position is com- value of an observablethat is describedby an operator A and
pletely offsetby the profit or lossin the stockposition. In other is given by the "inner product" < 'II IA I'll>, where <'II I is the
words, the overall value of this specialportfolio is known with Hermitian conjugate of the wavefunction. In the financial
26
physicsweb.org
PHYSICS WORLD
MAY 2005
be $500. Now move forward three months. If the stock price is lower
than $100, the trader will clearly choose not to exercise the option
and accept the $500 loss on their initial investment. However, if the
stock price has risen to, say, $120, the trader would obviously
exercise their rightto buy the stock at $100 and could then sell it
immediately at $120, making a profit of $2000, less commission
and brokerage fees and the $500 cost of the option.
stock
price S
This diagram shows the relationship
between the price of a call option, c, and
the price of the asset or stock underlying the option, S, in a special portfolio in
which the holder has been sold one call option and has bought a certain
amount of the underlying asset. Ata particular point in time, a small change
the stock price, i1S, and the resultantsmall
option, i1c, might be such thati1c=0.4i1S
in
change in the price of the call
- i.e. the slope or gradient of the
relationship between c and S is 0.4. In this case, the risk-free portfolio would
consist of a "long" position of 0.4 of the stock and a "short" position in one call
world the value of an option at a certain time, t, can then be
interpreted asthe inner product <II x), wheref is the option
price and x is the price of the underlying asset.
The evolution of the option value with time,f(t), can be
written as If(t)= exp(tH) If(O), where His the appropriate
differential operator or Hamiltonian and f(O) is the value
of the option at t= O.The path integral for the option then
models the stochastic process followed by the price of the
underlying asset,in the sameway that the Feynman path integral for, say;an electron takes into account all its possible
trajectories. Using simple boundary conditions for the value
of the option at certain times, a self-consistentquantum~ystern for the price of an option can be determined.
Baaquie and co-workers used Monte Carlo techpiques in which random numbers and probability distributions are
used to simulate real physical systems- to solve the path
integral. This involves fIXing an initial point for the underlying assetprice on the path, x, while allowing the final point,
x', to evolve over time. Then, by incorporating each price x'
with the option's "pay-off" function - i.e. how far the
underlying assetis aboye (for a call option) or below (for a
put option) the strike price - an averageprice for the option
can be determined.
This technique is then repeated for several values of x
until the value of the option converges,giving its price at the
maturity date (this is analogous to summing all the possible
paths of a particle in the double-slit experiment to determine
which slit the particle went through). Baaquie and co-workers
found that their Monte Carlo method agreed well with the
Black-Scholes analytic formulas for simple non-dividendpaying options. Moreover, unlike the classicalapproach, their
quantum description can easily be extended to so-called
exotic options.
A good example of an exotic option is a barrier option, the
price of which depends on the underlying assetreaching a
certain value during the lifetime of the option: a "knock-out"
barrier option is "killed off" when the underlying assetprice
hits a certain barrier, while a "knock-in" barrier option
comes "alive" when the underlying assetreachesthe barrier.
Although barrier options can be very lucrative, they alsohave
their disadvantages.Movements in the price of the underP.v.",.
WnR,n
Moy
7nn~
option. However, this position
is attained
only for a very short period of time, so
to remain risk-free the portfolio must be adjusted using a concept called
dynamic hedging. For example, in five days' time, the relationship
between c
and S might change such that i1c = 0.5i1S, in which case 0.5 of the stock must
be bought for each call option sold. Nevertheless,
in a very short, or
instantaneous,
period of time, the return from the risk-less portfolio is the
risk-free rate of interest.
This argument
was central to the derivation
Black-Scholes
pricing formulas
for non-dividend-payingstock
analytic
of the
options.
lying assetcan be very unpredictable, and a sudden spikecan
lead to a barrier being hit and an investor losing all their
money; This introduces a discontinuity in the dynamics of
the systemthat can causemajor problems for the trader and
risk manager.
Baaquie and co-workers have modelled barrier options by
incorporating a potential, V(x),into their quantum description, which constrainsthe stochasticprocess.The team is also
studying more complex path-dependent derivatives, such as
double-barrier options (the values of which reduce to zero if
one of two barriers is crossed).Initial results show that this
pricing problem is similar to a quantum-mechanical particle
in an infinite potential well.
Other researchers,suchasKirillllinski of Birmingham University in the UK, havetaken a slightly different path-integral
approach, based on quantum electrodynamics(QED). This
theory treats the force between two electronsas being due to
the exchange of a "virtual" photon, rather than the electric
field produced by eachelectronasdescribedby Coulomb's law;
llinski has shown how QED can be used to replicate the
Black-Scholes"no-arbitrage" argumentsto obtain the special
portfolio suggestedin the original MIT analysis.
In llinski's theory; the financial market is mapped onto
the QED model. Particleswith positive and negativecharges,
corresponding to securities and debts, respectively;interact
quantum mechanicallywith eachother through "electromagnetic" fields.llinski makesthe analogybetweenthe virtual particlesin QED that damp the force between a pair of electrons
and thosetraderswho try to eliminate arbitrage opportunities
within the financial markets.He goeson to showthat there are
opportunities to make a profit within a "virtual arbitrage"
world that were not envisagedin the original Black-Scholes
analysis.Nevertheless,he is quick to point out that quantum
financial methods are not the final word
on modelling the markets,but merely a
step to gaining a deeper understanding
and a better theo~
-
-
-
As well as the path-integral approach
to pricing derivatives,some researchers
are trying to take advantage of quantum phenomena by viewing the financial markets as a "quantum game".
Game theory describescompetitive scenarios betweena number of individuals
or groups who try to maximize their
own profit, or minimize the gains made
by their opponents, via co-operation
or conflict (see PhysicsWorld October
2002 pp25-29). However, by adopting
quantum trading strategies,rather than
classicalones,it seemsthat players can
make more informed decisions, which may lead to better
profit opportunities.
Last year,Edward Piotrowski of the University of Bialystok
and Jan Sladkowski at the University of Silesia,both in Poland, studied trading strategiesbased on superpositions of
different trading decisions in an abstract vector space.Any
trading activity or strategy is performed via "unitary" transformations on the statesin the vector space,which describe
the evolution of the systemover time.
Recent advances in quantum entanglement and cryptography could make these futuristic-sounding quantum trading systemsa reality. If two particles such as photons are
produced in such a way that they are entangled, any dist~bance of the state of one photon will instandy disturb the
other - no matter how far the particles are apart. Any attempt to intercept an entangled photon in transit would
therefore be immediately obvious to those monitoring the
state of the other photon in the pair, allowing ultra-secure
communication. Indeed, only last year a quantum financial
transaction was performed between two buildings using
entangled photons (figure 3).
Entanglement is also an essentialingredient in quantum
computation and information processing,which are necessary if quantum trading strategiesare to be put into practice.
Piotrowski and colleagues argue that such irading activity
would take place on a "quantum board" that contained the
setsof all possible statesof the trading game. However, the
researchersare careful to point out that to actually play sucha
game would require major advancesin technology;
To illustrate this, consider the classical "prisoner's dilemma", in which you and a friend are picked up by the police
and interrogated in separatecells without a chance to communicate with each other. You are both told the same thing:
if you both confess,you will both get five years in prison; if
neither of you confesses,the police will be able to pin part
of the crime on both of you and you will both get three years;
if one of you confessesbut the other doesnot, the confessor
will make a deal with the police and will go free while the
other one goesto jail for six years.
Unable to confer and lacking faith in the other's trustworthiness, each prisoner concludes that the best strategy is to
confess,because,no matter what your friend does,you will be
28
better of[ To your disadvantage,your friend alsorealizesdlis,
so you both end up getting five years. Ironically; if you had
both "co-operated" (i.e. refused to confess),you would both
havebeen much better off with only a three-year sentence.
So what if there were quantum rules?In the classicalprisoner's dilemma, there is only a singlechoice to either co-operate or defect.JensEisert and Martin Wilkens at the University
of Postdam,along with Maciej Lewenstein at the University
of Hanover, have recendy shown that in a quantum version
there is a third option: a superposition of confessingand staying quiet. Moreover, the prisoners' choicescan be entangled,
sothat one can influence the other.
Eisert and colleaguesdescribea physical model of the prisoner's dilemma in which both prisoners have secret access
to quantum particles and can manipulate their state.That is,
the prisoners become quantum players. It turns out that the
best strategy for both players is initially not to confess or
remain silent, but to "feel each other out" through strange
quantum combinations of the possibleoutcomes and, in the
end, make the choice that best rewards them. It is hard to
give a classicaldescription of this strategy;other than to say
that when both players use it they both come off as well as
they possibly can. In other words, this quantum strategy is
not only the most rational but alsothe most profitable.
Clearly th~ extra possibilities offered by quantum game
strategiescan lead to more successfuloutcomes than purely
classicalones.This hasfar-reaching consequencesfor trading
behaviour and could lead to fascinating developments in
quantum-designedfinancial marketsand rule-basedinvesting.
Outlook
This article givesonly a flavour of the potential applicationsof
quantum physicsin finance.However,many difficult problems
remain to be solvedbefore "rocket scientists" can take early
retirement. For example, we need to somehow include additional stochasticvariablessuchasvolatility and interestratesin
the quantum pricing systemto make the modelsmore realistic,
which posesan extremelydifficult mathematical challenge.
There are also sociological hurdles to be overcome. Developing and discussingpath integrals and quantum markets
with hard-nosedtraders, whosebottom line is to simply make
a profit, is not always easy:Indeed, most traders have litde or~
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PHYSICS WORLO MAY 2005
Tears of the Tree
TheStoryof Rubber
john Loadman
The Black-Scholes partial differential equation is written as
~5t
~
~
+ .:!c 0"2 S2
+ rS
= rf
2
as2
OS
where tis the value of the derivative, t is the time to maturity, 0"is the
volatility of the underlying asset, S is the underlying asset price, and
ris the risk-free interest rate. In order to determine the analytical
formulas for the price of call and put options, it is necessary to solve
this differential equation using various boundary conditions and a
final condition known as the option's pay-offfunction. By
considering a relevant change of variable, the Black-Scholes
equation can be written in terms of differential operator (or
Hamiltonian)HBs,such that
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at
where
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~ - r ) ~ax + r
H =:::-i!-.£ + (
BS 2 ax2
2
and x is the value of the underlyingasset. Fromthis Hamiltonian
descriptionit is fairly straightforwardto derivethe path-integral
representationfor the stochasticprocessfollowed bythe value of
the asset.It is then possibleto derivea pricing"kernel" forthe stock
optionthat is solvableusingfamiliar methodsfrom computational
physics.such as MonteCarlotechniques.
no scientificbackgroundand they often wonderwhat PhD
physicistsaredoingon thetradingfloor in thefirst place. ~
Nevertheless,there has been rapid progress in quantum
cryptography and quantum computation over the past decade, so there is reason to be optimistic about quantum financial methods as well. With the rapid advancement in
technology and the constant desire for bigger profits, it is
doubtful that we will have to wait too long.
Further reading
B Baaquie et a/. 2004 Hamiltonian and potentials in derivative pricing models:
exact results and lattice simulations Physica A 334 531-557
F Black and M Scholes 1973 The pricing of corporate liabilitiesJ. Political
Economy 8j. 637-654
Z Chen 2002 Quantum finance: the finite dimensional case arXiv.org/absl
quant-ph/0112158
J W Dash 2004 Quantitative Finance and Risk Management: A Physicist's
Approach (Singapore, World Scientific)
E Derman 2004 My Ufe as a Quant: Reflections on Physics and Finance
(New York, Wiley)
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BorisS.Bokstein.Mikhail I. Mendelev.
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William Barford
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Yvesjean and Colin Marsden
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clearteacher [...] This
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[...] It corresponds ideally to what is needed.'
R Feynman and A Hibbs 1965 QuantumMechanics and Path Integrals
O. Eisenstein University of Montpellier
(New York, McGraw-Hili)
March 2005 I 304 pages
0-19-853093-5 I Hardback £39.95
J Eisert et a/. 1999 Quantum games and quantum strategies Phys. Rev. Lett.
K lIinski 2001 Physics of Finance: Gauge Modeling in Non-Equilibrium Pricing
(New York, Wiley)
E Piotrowski and J Sladkowski 2002 Quantum game theory in finance
a rXiv.org/ a bsl q uant-ph/0406129
Paul Darbyshire
~~-is a visiting professor
in the UK, and an independent
e-mail [email protected]
PHYSICS WORLD MAY 200!
at the Nottingham
consultant
and investment
Business
broker.
School