University of Ljubljana
Faculty of Mathematics and Physics
Department of Physics
.
Borut Polajnar
Seminar
Philosophy of Financial
Markets Behavior
Advisor: prof. dr. Rudolf Podgornik
Ljubljana, March 2008
Abstract
This seminar paper deals with historical development of the way
we view the hustle and bustle of events that constitute a system of
financial market. The story starts in the year of 1900 when, then
young mathematician Louis Bachelier used mathematics of what is
now known as Brownian motion to describe movements of price of
assets traded on Paris Bourse. His work, after being over-looked for
more than half a century, was then adopted by some of the promi-
nent economists of 20th century, of whom Harry M. Markowitz and
William F. Sharpe and their contribution to the Modern theory of
finance (MTF) will be dealt with specifically. The concepts devel-
oped were part of every financial adviser’s or analyst’s toolkit in form
of frame of thinking and software support, but came under major
scrutiny after some catastrophic market breaks occurring throughout
the otherwise very prosperous 90’s. Better late than never, would ar-
gue Benoit B. Mandelbrot, the man to whom already in the 60’s were
known the deficits of using independent Gaussian variables as the
platform on which to build good quantitative description of financial
markets. Mandelbrot has certainly revolutionized the world of science
with his work, the field of economics being one of the most subtle of
his interests that set him on the path of his greatest discovery, that is
fractal geometry.
1
1 Introduction - the dichotomy of the
world we live in
1.1 Deterministic versus probabilistic view
The book Essai philosophique sur les probabilités by Pierre-Simon
Laplace, although discussing probability issues, starts with the fol-
lowing words [1]:
We may regard the present state of the universe as the effect
of its past and the cause of its future. An intellect which
at a certain moment would know all forces that set nature
in motion, and all positions of all items of which nature is
composed, if this intellect were also vast enough to submit
these data to analysis, it would embrace in a single for-
mula the movements of the greatest bodies of the universe
and those of the tiniest atom; for such an intellect nothing
would be uncertain and the future just like the past would
be present before its eyes.
Einstein, being famously succinct in his sayings, expressed a similar
notion by his renowned words [2]:
God does not play dice.
Both men, although not avoiding dealing with probability theory
or using it as a tool to solving problems, still held determinism in
higher esteem over probabilistic view of nature. One could even say
they believed that probabilistic explanation is not understanding the
problem, but merely describing it.
Mandelbrot, on the other side, argues that “being able to model
reality is a form of understanding” [3]. His firm belief is that the world
of finance cannot be tamed by some deterministic rules that would
describe its past and its future, both at the same time through its
present state1. Among other numerous factors that influence market
behavior he pinpoints anticipation, as a factor that could never fit
under deterministic umbrella of cause-and-effect [4]:
And there is the most confounding factor of all, anticipa-
tion. A stock price rises not because of good news from the
company, but because the brightening outlook for the stock
means investors anticipate it will rise further, and so they
buy. Anticipation is a feature unique to economics. It is
psychology, individual and mass - even harder to fathom
1One can discuss determinism of financial markets only ex post facto, whilst ex ante
description probability theory offers proves to be the only manageable one.
2
than the paradoxes of quantum mechanics. Anticipation is
the stuff of dreams and vapors.
Still, his belief in the form of ‘probabilistic understanding’ is not
of kind allowing for any kind of model that comes close to describing
reality. He can be labeled reductionist, in that he is demanding prob-
abilistic model being as parsimonious as possible. In his view, there
is no knowledge in elaborate algorithmic structures that grow out of
patching the noted discrepancies of models with reality. Rather he
demands generalization to the extent when all the characteristics of
observed reality follow from a self-contained, albeit random generator.
1.2 ‘Mild’ versus ‘wild’ view
Historically speaking, one of the most important implementations of
probabilistic description was in handling of error terms in measure-
ments. It was inferred that aggregated errors should preserve distri-
bution of every which aggregating element. It was for the stability
argument, and also for having well-defined and quickly convergent
mean and variance – and all other moments, as a matter of fact – that
the Gaussian or Normal distribution was chosen to serve as represent-
ing distribution of error terms. It was also noted that the Gaussian
served as a limit for many of the other distributions under aggregation
– the fact known today as Central limit theorem (CLT).
Implications of CLT have a great influence on our life, in that
it is commonly believed that every process will sooner or later re-
sult in a aggregation that is Gaussian distributed and therefore fully
defined by two simple, self-explanatory parameters, mean µ and stan-
dard deviation σ. Such belief has as a direct corollary that the world
is perceived as being made of tiny pieces. After all, practically every
element (95%) falls quite safely under the sway of 2σ from the mean.
Gaussian distribution does not leave much space for extraordinary ex-
ceptions, therefore all the constituent pieces fall together to form a
greater picture, with not much harm done had any of the aggregating
elements been taken out or added to the bunch.
But what if the creator of the world were not a coin flipper whose
cumulative wins or loses sooner or later fit the familiar bell-shaped
curve with well-defined averages, but rather a blindfolded archer, who
chose the size of the building blocks of world by the amount he missed
the designated target on an infinite wall. For fictitious example of
such archer being 1 unit from wall that he randomly shoots at, the
distribution of length of misses from the center can be calculated as
dP
dx
=
dP
dϕ
∣
∣
∣
∣
dϕ
dx
∣
∣
∣
∣
=
1
π(1 + tan2 ϕ)
=
1
π(1 + x2)
,
(1)
3
since x = tanϕ. Cauchy probability density, defined by (1), has so
called ‘fat tails’, since dP/dx ∼ x−2 as x → ∞. Following power-law
for large x as opposed to exponential Gaussian decay, has as a direct
corollary in this particular case an infinite variance. Aggregation of
Cauchy variables 2 could not and would not be perceived as a compo-
sition of timid grains, but of elements varying rapidly in magnitude,
some of them having size comparable to the size of the aggregation
as a whole. Looking at composite structure made of ‘Cauchy bricks’
could therefore be described as revealing rough, discontinuous, ‘wild’
fluctuations, as opposed to timid, continuous, ‘mild’ fluctuations in
the case of ‘Gaussian bricks’.
Figure 1: Sample averages ∑
N
i Xq/N for q = 1 (lower graphs) and q = 2
(upper graphs) for Cauchy-like probability density (Pr{X>x} ∼ x−1, for
large values of x) with varying sample-size N for three distinct sample groups.
A kind of erratic behavior is easily perceived that is both sample-size and
sample dependent. Averages are clearly not easily calculated when dealing
with wild variability. Graphics reproduced from [3].
The mild and the wild are terms both chosen by Benoit Mandel-
brot to describe what he calls states of randomness, with intentional
allusion to the states of matter. Mild reflects solid state, having low
energies and well-defined structure and volume, whilst wild reflects
gaseous state with high energies and non-existent structure. There
2It should be noted that Cauchy distribution, like Gaussian, is a stable one. They are
both special instances of general class of L-stable distributions to be introduced in section
3.1.1.
4
is a third state to reflect liquid state, which Mandelbrot calls ‘slow’.
Further division of states may be obtained by various more specialized
criteria, but the subject is beyond the scope of this seminar. The list
of all states of randomness can be found in [3] (p. 140-141), with more
detailed explanation in accompanying chapter.
It was long believed that the principle way in which the world
was built was mild. Famous economist Alfred Marshall asserted such
belief by saying: “natura non facit saltum” (quotation can be found as
illustration of traditional views on nature both in [3] and [4]), which
translated means nature does not undergo jumps. Nature is often
assumed to proceed in smooth continuous fashion. However, as it
will be shown in subsequent chapters, the ‘normality’ of Gaussian
world comes under severe questioning when the financial markets are
concerned, since their behavior is not tamed, but rough and wild. But
let us first describe how the classical view of financial world, which
incorporates the mild randomness as its key part, came to exist and
how does it function in principle.
2 The classical view
2.1 What is the nature of financial markets?
Louis Bachelier in his doctoral thesis had no doubt to whether or not
the description of market should be a probabilistic one. After listing
some of the probabilistic pros, he firmly stated [5]:
Although the market does not predict the movements, it does
consider them as being more or less likely, and this proba-
bility can be evaluated mathematically.
For the successive movements of price Bachelier predicated Gaus-
sian distribution, also assuming that such movements are independent.
The choice was motivated by the fact that, apart from stability under
aggregation, he demanded the mathematical expectation of speculator
to be zero. Such assertion is description of what is now called Effi-
cient market hypothesis. According to [6] it was not properly mathe-
matically formulated until 1965 when Paul Samuelson stated through
equation
E(Zt+1|Zt,Zt−1,...,Z0) = Zt,
(2)
that the expected value of future price Zt+1, knowing all the previous
prices Zt,Zt−1,...,Z0, is the current price Zt itself. Price variation
process is therefore a martingale3, Gaussian stable process, or more
3More about martingales can be found, for instance, in [7].
5
commonly Brownian motion, being the simplest stable process of that
form.
Bachelier perceived the connection between price change moves
and the fundamental solution of heat transfer equation, the most im-
portant result of this parallel being relation
σ ∼
√
t.
(3)
Therefore, as might come as some surprise to some believers in primacy
of physics over all other sciences, he made the connection between
Brownian motion and its continuous time limit in form of diffusion
or heat transfer process 5 years before Einstein independently did the
same in his famous paper on molecular motion.
Last but not least, Bachelier can also be credited with being self-
critical as to notice that some jumps of price simply did not fit un-
der the reach of bell-shaped Gaussian curve. Still, he dismissed such
jumps, treating them as ‘contaminators’ or ‘outliers’.
2.2 What is risk?
When trying to determine what efficient investing would be, Harry
Markowitz came up with the idea that rational investor’s utility func-
tion U4 should only depend on two parameters, namely expected re-
turn rate E(R) and risk σ. Marking risk as σ is no coincidence, since
Markowitz, like Bachelier before him, adopted the idea of Gaussian
movement of prices, predicating standard deviation a good measure
of risk. Such thinking lead him to formulation of efficient portfolio as
being one having [8]:
• maximal expected return rate at a given risk level
or equivalently
• minimal level of risk at a given expected return rate.
With the assumption of mild Gaussian price movements, the ex-
pected return rate E(RP ) and risk σP of portfolio P, compound of
individual assets or sub portfolios i, with weights wi, and mutual cor-
relation factors ρij, are [9]:
E(RP ) =
∑
i
wiE(Ri)
(4)
σ2
P
=
∑
i
w2
i σ2
i +
∑
i
∑
j
wiwjσiσjρij.
(5)
4Utility function is a measure of individual’s preferability. Rational investor therefore
thrives to maximize his utility function.
6
One can easily see, that given assets with similar expected return,
it is preferable to combine ones that are uncorrelated or even anti-
correlated in order to minimize risk - the concept well described with
the single word of diversification. In a general case of N available
assets or sub portfolios, 2N possible combinations fill up the portion
of return-risk space whose boundary is the hyperbola-like curve, the
upper half of which is called Efficient portfolio frontier or Markowitz
frontier. On it lay portfolios satisfying stated condition of efficiency.
Markowitz’s theories, extended with the works of others, including
Sharpe, whose contribution is discussed in the following section, are
now collectively known as Modern portfolio theory (MPT).
Figure 2: The risk-return space filled with sample portfolios. The ones lying
on the upper half of hyperbola-like curve that borders the space that sample
portfolios fill, are the so called efficient portfolios. The tangency portfolio
marked by darker spot is more commonly known as market portfolio and
is characterized by largest Sharpe measure of all efficient portfolios. Sharpe
measure is exactly the slope of Capital allocation line (CAL). Further details
are explained in section 2.3. Graphics reproduced from [9].
2.3 What is the value of an asset?
As Markowitz’s doctoral student, William Sharpe was confronted with
a task of simplifying calculations of MPT. Apart from initial demand
to calculate N(N −1)/2 correlation factors, 2N iterations needed to be
done to perform the full calculation for N elements in form of assets or
sub portfolios. While pondering on it, Sharpe realized that among ef-
ficient portfolios, one with maximum reward-to-risk ratio could be sin-
gled out. In the context of MPT reward-to-risk ratio is called Sharpe
7
measure and is defined as [10]:
S =
E(RP ) − Rf
σP
,
(6)
where Rf is the risk free rate that can be emulated by a rate of
short-term government-issued bonds. Portfolio with maximal Sharpe
measure came to be known as market portfolio, since market is by
implications of Efficient market hypothesis the most optimal trader
that cannot be beaten in its craft. Such logic marked the dawn of
stock-index funds that make possible investing in shares in the same
proportion as the real market does.
But while reducing the task of creating efficient portfolio by simply
letting the market itself do the math, he also devised a cleaver way
to value an individual asset. Market being the almighty arbiter, the
optimal risk premium E(Ri)−Rf of an asset should equal that of the
market itself, multiplied by the so called systematic or market risk
factor of investment i on a given market m, labeled usually as βim.
To summarize [11]:
E(Ri) − Rf = βim(Rm − Rf ),
(7)
where βim in turn is defined as [11]:
βim =
Cov(Ri,Rm)
Var(Rm)
= ρim
σi
σm
(8)
After estimating E(Ri), net present value (NPV) 5 of an asset can
readily be calculated, where future cash flows are determined by some
sort of fundamental analysis. Obtained value should be an optimal
price of an asset, the most appealing feature being the classification
of diverse asset values comprising real market by a single variable β.
Sharpe’s asset valuing model came to be known as Capital asset
pricing model (CAPM), which is itself an integral part of MPT.
2.4 The misfits of classical view
Considering price charts alone, Brownian motion passes as a respecta-
ble model (see figure 3). However, when price moves are considered,
5The NPV formula emulates the notion of ‘time value of money’. NPV is calculated
as the sum of future cash flows, each of them discounted by interests factor of form
(1 + E(Ri))ti [12], where E(Ri) is the expected rate of return (or some form of interests),
relevant at time i, and ti is portion of time from present to i in fraction-of-year units. In
case of one-time present evaluation of E(R) the value of the asset could be determined as
∑
i CFi/(1 + E(R))ti .
8
Brownian motion induced white noise clearly stands out as a very un-
realistic option (see figure 4). Being direct heirs of Bachelier’s model,
Markowitz’s and Sharpe’s theories inherited bad assumptions of mild
Gaussian variation, especially critical being presumed continuity of
price change. Out of this assumption grew very intuitive but flawed
measure for risk in the form of standard deviation of price changes σ
and a similar notion of variance against the market in form of Sharpe’s
β. Both have the misfortune of being critically unstable in the case
of wild variability (see figure 1). Further shaky assumptions are those
connected to rationality of investors, which, taken to the extreme, ren-
ders them all equal, where real-case situation proves to be quite the
opposite. People can be quite ‘irrational’, the ‘friction’ between many
behavioral groups they form being one of the main generators of wild
variability.
Figure 3: The so called fever charts of price. It is hard to distinguish between
two model and two real charts. From the top the charts are: IBM stock,
Brownian motion, USD/DEM exchange rate and multifractal model of price
change. Graphics reproduced from [4].
3 New deal
3.1 What is the nature of financial markets?
Like Louis Bachelier before him, Benoit Mandelbrot is himself a true
believer in overwhelming power of probabilistic approach. Yet he was
9
Figure 4: The charts of relative price changes. Here the inadequacy of Brow-
nian motion model quickly becomes apparent. Its white noise signal lacks
three main characteristics of real charts: discontinuity, long-term depen-
dence and clustering of volatility. The multifractal model, however, remains
indistinguishable from real charts. As in previous figure, from the top the
charts are: IBM stock, Brownian motion, USD/DEM exchange rate and
multifractal model of price change. Graphics reproduced from [4].
the one to do a crucial step away from Gaussian view of the world,
becoming fascinated with power-law distributions, instead. These are
often referred to as scaling, because of their invariance under condi-
tioning W defined as 6
P(u) = Pr{U>u}
W
−→ PW (u) = Pr{U>u|U>w} =
P(u)
P(w)
. (9)
Now, for power-law probability distribution P(u)=(u/˜u)−α, condi-
tional probability distribution is according to (9):
PW (u) =
(u/˜u)−α
(w/˜u)−α
=
( u
w
)−α
,
(10)
thus preserving functional form, that is except for the change in scale.
6Result (9) can easily be understood. Consider the probability density p(x). Knowing
U>w does not effect relative probability of any of the possible values to occur, it only
narrows the interval on which probability is spread. To calculate conditioned probability
density pW (x), one only multiplies (rescales) unconditioned probability density p(x) with
a constant A so that ∫
∞
w
Ap(x)dx = 1. Thus pW (x) = p(x)/P(w).
10
There is another typical infinite range distribution that also pre-
serves functional form under conditioning, namely exponential distri-
bution P(u) = exp(−λ(u − u0)):
PW (u) =
e−λ(u−u0)
e−λw
= e
−λ(u−˜u)
,
(11)
but the change is in location rather than scale. To illustrate the point,
one can observe that the moments of power-law distribution become
dependent on conditioning W7:
E(U
q
W
) = −
∫ ∞
w
uqdP(u) =
α
α − q
wq,for q < α,
(12)
whilst in the exponential case they clearly stay the same, since trans-
lation cannot change the surface under the functional curve by it-
self. Scaling distribution is therefore term used exclusively to refer to
power-law distributions.
Of course, the fact that exponential does not change under condi-
tioning comes as no surprise to a physicist, to whom it is perfectly clear
that radioactive decay at a certain moment in time has nothing to do
with the length of life of particular nuclei. But what about when it
comes to people, and the social structures and systems we have build?
It turns out that power-laws are everywhere. The first of them was
empirically discovered by Italian economists Vilfredo Pareto, who in
1909 observed that the wealth distribution had a power-law tail, with
Pareto’s estimation for α to be around 3/2.8 Another striking thing
was that the same α could be obtained for various countries and his-
torical eras. Pareto himself was so astounded by his discovery, that
he claimed this intriguing fact to be the consequence of “something
(some fundamental law) in the nature of men” [4].
Another man believing in the power of power-laws was George
Kingsley Zipf.9
In his book Human Behavior and the Principle of
Least Effort he accounted for almost every social phenomena he could
think of with a power-law. Perhaps one of the most fascinating is
scaling of word frequencies in a given text or speech10. Paying his
attention to James Joyce’s Ulysses specifically, he estimated the α ex-
ponent to be around 1. Unfortunately, universality of that particular
7Since P(u) = ∫
∞
u
p(x)dx, probability density can be obtained by differentiating prob-
ability distribution, that is p(u) = −dP(u)/du.
8References on Pareto can be found in [3], [4] and [6].
9References on Zipf can be found in [3] and [4].
10Zipf granted every word a rank according to its frequency – the most frequent word
got rank 1, the second most frequent rank 2 etc. – and noted that the frequency is a
power-law function of the granted rank.
11
law is not as far reaching as in Pareto’s case, since people as eloquent
as James Joyce are not what one would call a ‘representative sample’.
It should also be noted that in the title of Zipf’s book the notion
of ‘the principle of least effort’ appears. By using it, Zipf was referring
to the fact that scaling distributions fit well the folklore and common
wisdom type of ‘knowledge’, such as that luck or wealth produces
even more luck and wealth. Suppose now not only that the wealth
distribution is scaling, but also the amount of wealth accumulated
in one’s lifetime – which is not so different notion, after all. Then
according to (12), for α = 2, having accumulated w of wealth, one is
expected to accumulate at least as much until his or hers life’s work is
brought to an irreversible halt. Being a ‘fundamental law’ of human
nature that a path of least resistance is a preferable one, Zipf assumed
that our nature in some way induces the distributions that enable one’s
following such path. Once a certain amount of wealth is accumulated,
not much more need to be done in the world of scaling, since what is
expected is that the wealth will multiply ‘by itself’.
Zipf was also convinced that power-laws were something intrinsic
to human nature and therefore social sciences, but as it later turned
out they are quite common in our physical world as well. In physics
a whole new field dealing with critical phenomena emerged in which
scaling found its part to play.
3.1.1 The meaning of cotton - Noah effect
So, what about the changes of price? Certainly the Gaussian does
not fit, as Benoit Mandelbrot found out when exploring the case of
historical cotton price moves in the 60’s. Adding to the population, of
say sample monthly data, one price datum after another, the sample
volatility11 was violently changing with constant occurrence of ‘pollut-
ing events’ and ‘outliers’, rendering it impossible to conclude that price
changes followed a ‘simple’ Brownian motion. However, exploring the
distribution of difference of logarithms of price12 L(t, T), defined as
L(t, T) = log Z(t + T) − log Z(t), Z(t) being the spot price of cotton
at time t, for different time-spans T, it turned out that the relation
log P
±
(l) = −αlog(±l)
(13)
11Volatility is a term used to describe variability of price changes and is often used as a
synonym for standard deviation.
12Change in logarithm of price is often dealt with instead of the absolute price change
as such. The obvious motive is, of course, to render proportional price changes equal.
Apart from that, the use of logarithm preserves additive nature of price change, whereas
it would be turned into multiplicative process, had relative price changes been used.
12
holds, where P±(l) are Pr{L(t, T) > l} for positive values of l and
Pr{L(t, T) < l} for negative values of l, respectively, and α ≈ 1.7.
Relation (13) is a clear-cut expression of scaling property underlying
the price change process.
The most appealing aspect of the matter is that the data show
coefficient α being independent of time t and time-span T. Thus
scaling principle does not change with historical time nor it is affected
by aggregation. The only thing that changes being the scale of price
changes as such, result is very pleasing in that it is expressing a sort
of universal principle behind the process, avoiding at the same time
the need for Gaussian assumption.
Figure 5: Graphical representation of scaling property of cotton price change
process. Positive (1) and negative (2) price changes are dealt with separately.
Apart from that, three cases are presented: a – daily price changes from 1900-
1905, b – daily price changes from 1944-1958 and c – monthly price changes
from 1880-1940. All lines exhibit scaling with α ≈ 1.7. Graphics reproduced
from [3].
To classify the distribution he found, Mandelbrot turned his atten-
tion to the work of his professor from Paris, Paul Lévy, who solved for
general problem of stable probability density’s characteristic function
ϕ(q), finding that it should be of the form [6]:
lnϕ(q) =
{
iµq − γ|q|α[1 − iβ q
|q|
tan(π
2
α)], forα = 1
iµq − γ|q|[1 + iβ
q
|q|
2
π
ln|q|], forα = 1
,
(14)
where the four parameters are alphabetically:
• α ∈ (0,2] – kurtosis factor
13
• β ∈ [−1,1] – skewness factor
• γ ∈ (0,∞) – scale factor
• µ – location factor.
Figure 6: Samples of L-stable probability densities for varying α (left case)
and β (right case). With decreasing kurtosis factor α the curves are be-
coming more and more leptokurtic, that is gaining sharper and narrower
peaks and fat-tails. Having skewness factor β = 0 has as a affect asymmetry
in probability distribution. Varying scale factor γ, or c as it is labeled on
graphs, would stretch or compress the curve the way the changing σ effects
the shape of a Gaussian curve. Varying location factor µ would only shift
the peak around. Graphics reproduced from [13].
Only three of the functions in the entire class have their respective
analytical form, namely Cauchy (α = 1 and β = 0), Gaussian (α = 2
and β = 0) and Lévy-Smirnov (α = 1/2 and β = 1) probability density
functions. For large values of u, with honorable exception of Gaussian,
it holds that
P(u) =
∫ ∞
u
F
−1
(ϕ(q))dx ∼ u
−α
,
(15)
where F−1 is reverse Fourier transform respectively. Distribution
Mandelbrot found in the case of cotton price changes therefore belongs
to the class of L(évy)-stable distributions. The stochastic process itself
is often referred to as the L(évy)-stable flight.
Being a description of extreme events driven process, Mandelbrot
calls L-stable variability Noah effect, after the tale of great floods from
the Old Testament.
3.1.2 The meaning of Nile - Joseph effect
One now justifiably wonders what exactly has the river Nile to do with
price changes and financial markets in general. Well, as fate would
have it, Nile came to be a source of influence for Benoit Mandelbrot,
14
specifically through the work of Harold E. Hurst, who in his study of
Nile noticed that [4]:
Although many natural phenomena have a nearly normal
frequency distribution this is only the case when their order
of occurrence is ignored. When records of natural phenom-
ena extend over long periods there are considerable varia-
tions both of means and standard deviations from one pe-
riod to another. The tendency to occur in groups makes
both the mean and the standard deviation computed from
short period of years more variable than is the case in ran-
dom distributions.
Hurst summed his thinking and facts provided by experimental data
into the result that the range R of an optimal dam that would perfectly
dampen variability in the river Nile discharges over the period of N
years, is related to the average value of yearly standard deviation σy
by [4]:
log
( R
σy
)
= K log
(N
2
)
,
(16)
where he measured K to be approximately 0,7. Thus, even though
that shuffled yearly changes fit a perfect Gaussian, the persistence
of the process causes the variability of aggregated process to change
differently than in the case of perfectly uncorrelated changes, where
K = 1/2 would have been measured.
Mandelbrot formalized Hurst’s empirical laws in defining the pro-
cess of Fractional Brownian motion, that is characterized by properties
[3]
E(BH(t + T) − BH(t)) = 0
(17)
and
E((BH(t + T) − BH(t))2) = T2H,
(18)
with constant H ∈ [0,1], now know as Hurst-Hölder exponent. For
such a process the correlation C between past and future average,
defined as (B(t+T)−B(t))/T and (B(t)−B(t−T))/T, respectively,
is [3]
C =
1
2
(2T)2H − T2H − T2H
T2H
= 22H−1 − 1.
(19)
With the exception of ‘usual’ Brownian motion with H = 1/2, for
which it is zero, the correlation does not vanish for any value of time-
span T, thus implying long-term, even infinite dependence. Exponents
H = 1/2 are found in many price series, one notable example being
exchange rates for currencies.
15
Again the biblical example has given alternative name for the long-
term dependence driven variability. After Joseph, son of Jacob, who
interpreted Pharaoh’s dream of seven fat cows eaten by seven lean
cows in terms of 7 good and prosperous and 7 years of famine, it is
called Joseph effect.
3.1.3 Noah and Joseph joining hands
Figure 7: The distortion of price change through trading time. The distor-
tion through line at 45◦ to any of the axis defining θ-t plane would have no
consequence, whilst breaking this line into parts of various slopes has as an
effect packaging of many units of trading time into a single unit of physical
time, or reversely stretching a single unit of trading time over many units
of physical time. The former case being emulation of quick-running market
with many big jumps over short periods of physical time, the latter case
being its opposite in the form of slow-running market with moderate change
over lengthy periods of physical time. Multifractal model is capable of ac-
counting for all three major discrepancies between original Brownian motion
and reality, namely discontinuity of price change, long-term dependence and
clustering of volatility. Graphics reproduced from [4].
The Noah and Joseph effect are instances of fractal models of price
change. The essence of fractality being repeatability on all scales, it
is the scaling exponent α and dependence exponent H that measure
fractality for Noah and Joseph effect, respectively, since both are scale-
independent constants embodying essence of a ‘greater truth’ about
16
the market behavior, repeatedly seen on all scales of observation. Of
course, the question arises whether both concepts can be combined
to form an even better model of the way the markets behave. The
answer is that such combination can be made. And it is through
the ‘distortion’ of physical time with the intention to mold it into
the concept of trading time that this is achieved. Such distortion is
reasonable in the sense that it emulates the fact that markets can
move ‘faster’ or ‘slower’. There are times when a lot of information
amasses in a mere hour of trading and there are times when nothing
happens for nearly a week. Surely such intervals cannot be treated
on equal footing. The technique itself is a well-known concept of
compounding 13. Market time θα is called directing function, whilst
price moving function BH is called a compounding process14. The
result is a multifractal model of price change, namely
BαH(t) = BH(θα(t)).
(20)
Compound process BαH(T) gives rise to countless new options of vari-
ability in modeling price changes. It is worth mentioning – as a sort of
satisfying token of internal consistency of the theory –, that Brownian
motion of properly chosen fractal time B1/2(θα(t)) reproduces exactly
L-stable flight with exponent α. But it is, of course, the general case
of fractional Brownian motion in fractal time, giving rise to genuine
multifractality, that is most interesting. Such is the model capable
of describing to a very satisfying level most price records of various
assets - including those with scaling exponent α > 2, since stability is
guaranteed by compounding process and therefore its restrictions need
not be imposed on directing function itself. For illustrative example
of how characteristics of compounding process can be altered through
subordination to directing function see figure 7.
3.2 What is risk and what is an asset worth?
- Conclusion
Not explicitly recognized by the section 2, the questions of risk and
value of a particular asset are essentially the same thing, since in
valuing an asset it all comes down to evaluating its risk. Unfortunately,
the risk yard stick in the case of wild variability is not as apparent and
13More about compounding can be found, for instance, in [7], where the technique is
referred to as subordination.
14The choice of indexes show that the underlying concept is to induce extreme price
jumps through occasional packaging of loads of trading time into a unit of physical time,
whilst long-term memory remains property of subordinated price changing motion.
17
intuitive as standard deviation is in the Gaussian case, since variance
is not defined for the general case of α < 2. And even when it is
defined, its convergence is not tamed, but can vary greatly depending
on sample size or the sample as such, and can therefore be misleading.
The α and H are, of course, by themselves a measure of risk, telling
us the story of markets that are far riskier on general than in α = 2
and H = 1/2, that is mild Gaussian case. Still, neatly packed theories
and cookbook recipes like those of MPT are yet to be developed in
the case of wild variability.
Figure 8: To analyse ruin problems is one possible approach to better assess-
ing risk under wild variability conditions. Here, the simplified model of the
ruin problem in insurance business is presented. Linear trend in the growth
of capital is due to collecting premiums, whilst drops are due to paying
variable claims. The dangerous world of probabilistic wildness resulting in
real-life bankruptcies reveals itself immediately to contrast the steady growth
Gaussian prediction. Graphics reproduced from [4].
18
References
[1] http://en.wikipedia.org/wiki/Laplace (3/2008)
[2] http://en.wikiquote.org/wiki/Albert Einstein (3/2008)
[3] Benoit B. Mandelbrot, Fractals and Scaling in Finance: Disconti-
nuity, Concentration, Risk. New York: Springer (1997)
[4] Benoit B. Mandelbrot and R. L. Hudson, The Mis(behavior) of
Markets: A Fractal View of Risk, Ruin, and Reward. New York:
Basic Books (2004)
[5] Mark Davis and Alison Etheridge, Louis Bachelier’s Theory of
Speculation: The Origins of Modern Finance, Princeton University
Press (2006)
[6] Rosario N. Mantegna and Eugene H. Stanley, An Introduction
to Econophysics: Correlations and Complexity in Finance. Cam-
bridge: Cambridge University Press (2000)
[7] William Feller, An Introduction to Probability Theory and Its Ap-
plications, New York: Wiley (1970)
[8] http://www.investorwords.com/1673/efficient portfolio.html
(3/2008)
[9] http://en.wikipedia.org/wiki/Modern portfolio theory (3/2008)
[10] http://en.wikipedia.org/wiki/Sharpe ratio (3/2008)
[11] http://en.wikipedia.org/wiki/Capital Asset Pricing Model
(3/2008)
[12] http://en.wikipedia.org/wiki/Net present value (3/2008)
[13] http://en.wikipedia.org/wiki/Levy skew alpha-stable distribution
(3/2008)
