Math Meth-Financial
Price Analys, Lecture 2
Mathematics GR5360 2
Is There an Order in Constant π?
• π is a mathematical constant equal to the ratio of a circle’s
circumference to its diameter, approximately (as a double) equal to
3.141592653589793.
• π is an irrational number: cannot be expressed exactly as a common
fraction. Therefore, π decimal representation is infinitely long and never
settles into a permanent repeating pattern.
• “Feynman point” is a sequence of decimal digits of π that starts at spot
762 and contains six 9’s in a row.
Math Meth-Financial
Price Analys, Lecture 2
Mathematics GR5360 3
Is There an Order in Constant π?
• π digits seem to be randomly distributed, although up until now no
rigorous proof of this was discovered.
• π is a transcendental number, that is not a root of any non-zero
polynomial having rational coefficients – it is impossible to square the
circle with a compass and straight-edge.
• Currently over 1013 digits of π were computed. For all practical scientific
applications 40 digits of π or less are sufficient. Humans were able to
memorize up to 67,000 digits.
• Digits of π do not seem to have any apparent order or pattern. A
number of infinite length is called normal when all possible sequences
of digits of any given length appear equally often.
• Multiple series and integral representations of π are known.
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Math Meth-Financial
Price Analys, Lecture 2
Mathematics GR5360 4
Is There an Order in Constant π?
• Using the file “Pi1.25Million.txt” of the first 1.25 Million digits of π we will
count sequences of same digits of different lengths (1,2,3,4,5) and
compare those frequencies to those generate totally randomly.
• At first look, we seem to see signs of “order” in π.
Math Meth-Financial
Price Analys, Lecture 2
Mathematics GR5360 5
Is There an Order in Constant π?
• A more careful analysis shows that this seeming “order” is nothing more
than statistical sample noise.
Math Meth-Financial
Price Analys, Lecture 2
Mathematics GR5360 6
Elements of Statistics Relevant to Price Analysis*
• Two-point Probability Density Function
* - with some changes from “Statistical Hydrodynamics” by Monin and Yaglom, ref. B6.
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Math Meth-Financial
Price Analys, Lecture 2
Mathematics GR5360 7
Elements of Statistics Relevant to Price Analysis
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Math Meth-Financial
Price Analys, Lecture 2
Mathematics GR5360 8
Elements of Statistics Relevant to Price Analysis
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Math Meth-Financial
Price Analys, Lecture 2
Mathematics GR5360 9
Elements of Statistics Relevant to Price Analysis
( )? R :functions response lconditiona theare What 4.
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Math Meth-Financial
Price Analys, Lecture 2
Mathematics GR5360 10
Push-Response Functions( )
( ) .)()(
response"" average lconditiona a findcan we),( series-time
financial agiven ,)()( xpush""every for example),for 1,(
shift timeof choice particular aFor function. Response-Push a
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Math Meth-Financial
Price Analys, Lecture 2
Mathematics GR5360 11
More on Push-Response Functions*
.)()(
:responsemean lconditiona for the have Then we
.
2
),(),(),( and
2
),(),(),(
: where,),(),(),( :parts
asymmetric and symmetric a into PDF variate-bi compose-decan One
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minutes), say, in, (measured of case particular aWith
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21
212
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a
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as
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* - from papers by V. Trainin et. al., refs. A43-46.
Math Meth-Financial
Price Analys, Lecture 2
Mathematics GR5360 12
More on Push-Response Functions*
Walk).(Random litiespredictabi no havemay
series-price theas wellas possible, are shapescomplex more Naturally,
reversion-mean term-long and following- trendterm-short 4.
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* - from papers by V. Trainin et. al., refs. A43-46.
Math Meth-Financial
Price Analys, Lecture 2
Mathematics GR5360 13
Fourier Expansion and Fourier Transform*
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* - from Monin & Yaglom, vol. II, ref. B6.
Math Meth-Financial
Price Analys, Lecture 2
Mathematics GR5360 14
Fourier Expansion and Fourier Transform*
.conditions general very someunder possible is tscoefficien eduncorrelat
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* - from Monin & Yaglom, vol. II, ref. B6.
Math Meth-Financial
Price Analys, Lecture 2
Mathematics GR5360 15
Fourier Expansion and Fourier Transform*
( ) ( )( )
. ,2 ),( ,
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* - from Monin & Yaglom, vol. II, ref. B6.
Math Meth-Financial
Price Analys, Lecture 2
Mathematics GR5360 16
Fourier Expansion and Fourier Transform*
. phases of sticscharacteri lstatistica on the depend
not does and aplitudes of squaresmean on the dependsIt
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* - from Monin & Yaglom, vol. II, ref. B6.
Math Meth-Financial
Price Analys, Lecture 2
Mathematics GR5360 17
Fourier Expansion and Fourier Transform*
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before. described of properties of because
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* - from Monin & Yaglom, vol. II, ref. B6.
Math Meth-Financial
Price Analys, Lecture 2
Mathematics GR5360 18
Fourier Expansion and Fourier Transform*
[ ]
.Kolmogorovby derivedfirst )( process stationary
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* - from Monin & Yaglom, vol. II, ref. B6.
Math Meth-Financial
Price Analys, Lecture 2
Mathematics GR5360 19
Fourier Expansion and Fourier Transform*
. intervalfrequency the toingcorrespond
)( process theofcomponent spectral thegives expression This
.)(Re2)()(),(
:filterschosen properly of meansby
ally experiment isolated becan ranges spectralshort individualthat
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Math Meth-Financial
Price Analys, Lecture 2
Mathematics GR5360 20
Fourier Expansion and Fourier Transform*
[ ]
).()(]),([)( where
,)(2)()(t),u(
:bygiven is ],[ range in the sfrequenciewith
nsoscillatio harmonic the toingcorrespondenergy mean theThen,
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* - from Monin & Yaglom, vol. II, ref. B6.
Math Meth-Financial
Price Analys, Lecture 2
Mathematics GR5360 21
Fourier Expansion and Fourier Transform*
function. delta Dirac a is where,)()(
2
1
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11
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* - from Monin & Yaglom, vol. II, ref. B6.
Math Meth-Financial
Price Analys, Lecture 2
Mathematics GR5360 22
Fourier Expansion and Fourier Transform*
.components spectrum
individual theof energies theof sum theis )( process theofenergy
total that theshowsit :nexplanatio physical simplely particular a has
)()()0(
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* - from Monin & Yaglom, vol. II, ref. B6.
Math Meth-Financial
Price Analys, Lecture 2
Mathematics GR5360 23
Fourier Expansion and Fourier Transform
( )
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Math Meth-Financial
Price Analys, Lecture 2
Mathematics GR5360 24
Algebraic Scaling Laws
scale. sticcharacteri no haveThey similar.-self
called are processesSuch . "microscope out the andin zooms"
only laws, lstatistica governing theofany changenot does
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Math Meth-Financial
Price Analys, Lecture 2
Mathematics GR5360 25
A Reminder on Gaussian Distribution Properties
Gaussian"." are on distributi theand function sticcharacteriboth ,0 if and,
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:function sticcharacteri Its
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Theorem,Limit Central torelated is importance Its science.in everywherealmost used
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