Bacheliers Predecessors

HansJoachim Girlich

University of Leipzig Germany

Abstract

Recent pap ers by JeanMichel Cortault et al and Murad S Taqqu

have cast new light on Bacheliers work and his times see S S In

b oth articles Bernard Brus research seems to b e the most inuential In the case

of Louis Bachelier and his area of activity the dominantFrench point of view is

the most natural thing in the world and every b o dy is convinced by the results

The aim of the present note is to add a few tesseras also from other countries

to the picture which is known ab out the birth of mathematical nance and its

probabilistic environment

CF Gauss

Under Bacheliers predecessors who have prepared to ols and rst steps to sto chastic

nance should be mentioned rst Bachelier cited la loi

de Gauss which was intro duced as an error distribution by Gauss in see

We enlighten a fewer known side of the mathematicorum princeps

his work as a successful investor Since the s in deance or b ecause of his

mo dest salary Gauss invested his savings in bank sto cks and obligations and leaved

an estate of more than thalers see His great exp erience with

nancial op erations on a b ourse to ok eect also in public aairs He reorganized

the fund for professors widows at the universityofGottingen during

For this purp ose he calculated sp ecial lifeinsurance tables see

GT Fechner

Gustav Theo dor Fechner learned as a Physicist from Jean B Biot and Georg

Simon Ohm mathematical mo delling His own work in psychology and statistics

included the foundation of psychophysics and the theory on the measurementof

collectives The Web erFechner law based on the assumption that the sensation

as a function of a stimulus is Lipschitz continuous Bachelier and Einstein worked

later with smo oth functions and the Taylor formula It found applications also in

nance by MFM Osb orne see S S S

Revised version presented ad the nd World Congress of the Bachelier Finance So ciety Crete June

HD Macleo d Ch Castelli FY Edgeworth

In the s Henry Dunning Macleo d showed in his main work the principles of

Currency and Banking their progressive development in practice and the laws at

present aecting them see He gave juridical denition of the terms used in

monetary science

Bachelier refers in his Thesis to sp ecialized works on options Probably his trans

lator A J Boness added in S the title of a b o ok written by the sto ck and share

broker Charles Castelli in In this little book Castelli explains very clearly

the mo des practised in the Sto ck Exchange in relation to the call and put op

erations for brokers and clients A French translation was published in see

Francis Ysidro Edgeworth found in the s a connection between nance and

probability He investigated the problem of bankruptcy and p ostulated the law

of error as the foundation of banking Edgeworth tested his hyp othesis with real

data the amount of returns of Bank of England notes in the hand of the public

in the p erio d of to Up on this foundation he computed the probability

that the demand will not exceed any prop osed limit Furthermore he presented

a complex mo del of banking as a new game of chance formulated as a certain

p ortfolio problem with three p ossibilities to handle a disp osable fund see

J Bresson J Regnault H Lefevre

One of the rst imp ortant b o oks ab out op erations on the Sto ck Exchanges in

Paris is written by Jacques Bresson It followed later the handb o ok byAmbroise

Buchere

Probably the most imp ortant predecessors of Bachelier with resp ect to nance

were Henri Lefevre private secretary to Baron James de Rothschild and Jules

Regnault FranckJovanovic and Philipp e Le Gall investigated their work in a series

of pap ers S S

We refer esp ecially to Lefevres b o oklet comparable with Castellis see

which is only the third part Livre I I I of the b o ok A graphical depiction of a

long p osition in a Europ ean call long b efore Bachelier should b e emphasized

see S S S

Regnaults book contain an abundance of ideas for mo delling the price be

havior of a nancial market He empirically tested the law the deviation of the

prices increases with the square ro ot of time evaluating the mean value of the

French b ond This meanvalue approach analyzing time series was extended

signicantly by Thorwald Nicolai Thiele see S S S The use of the

Gaussian normal law in Regnaults research based on a visual approach see S

L Bacheliers Thesis

There were a few dissertations ab out the markets of the exchange but all presented

to faculties of law see The rst thesis on sp eculation from a sto chastic p oint

of view was written by Louis Bachelier It has b een seen and allowed for publication

in January by Jean Darb oux the dean of the Paris Faculty of Science and

was published inmediately in February see S

It is interesting how Bachelier derived the law of probability of relative prices

in comparison with Rayleighs approach for vibrations and Einsteins mo del of

Brownian motion see

The time after

Henry Poincare suggested in his rep ort on Bacheliers thesis to study further into

the details of Fouriers analysis the relationship of sto chastic pro cesses with partial

dierential equations Bachelier realized this hint

In the year three attemps were published to handle dep endent random phe

nomena They were given in the case of discrete time by Andrei Markovin St

Petersburg and following GT Fechners ideas by Heinrich Bruns in Leipzig

but in continuous time by Louis Bachelier in Paris see Bachelier

presented a framework which covered not only the Wiener pro cess as in but

also the OrnsteinUhlenbeck pro cess Furthermore he disclosed a connection with

the work of Pierre Simon Laplace see p I I x Unfortunately

Bacheliers approach failed the necessary rigour for a general acceptance In the

sp ecial case of the OrnsteinUhlenbeck pro cess Markovs metho d of moments was

successful yet The general classical case in Poincares sense has been solved by

Andrei Kolmogorov see

References

L Bachelier Theorie de la Speculation Annales Scientiques de l Ecole

e

Normale Superieure Series

L Bachelier Theorie des probabilites continues Journal de Mathematiques

Pures et Appliquees Sixieme Serie

J Bertrand Calcul des Probabilites GauthierVillars Paris

AJ Boness Elements of a theory of stockoption value Journal of Political

Economy

J Bresson Des Fonds Publics Francais et Etrangers et des Operations de

la Bourse de Parised BachelierCourcier Paris ed Bureau du

Cours general des Actions Paris

H Bruns Wahrscheinlichkeitsrechnung und Kol lektivmasslehreTeubner

Leipzig

H Bruns Das Gruppenschema fur zufal lige Ereignisse Abhandlungen d

Konigl Sachs Ges d Wiss Leipzig

A Buchere Traitetheorique et pratique des Operations de la Bourse

A Marescq Aine Paris

C Castelli The Theory of Options in Stocks and Shares FC Mathieson

Sons London

C Castelli Traite des Operations de Bourse a Primes Lecuir Paris

EB Dynkin Markov Processes Grundlehren Vol Springer

Berlin Moscow

FY Edgeworth Problems in Probability London Philosophical

Magazine Series

FY Edgeworth The Mathematical Theory of Banking Journal of the

Royal Statistical So ciety of London

A Einstein Uber die von der molekularkinetischen Theorie der Warme

geforderte Bewegung von in ruhenden Flussigkeiten suspendierten Teilchen

Annalen der Physik Folge

GT Fechner Drei Versuchsreihen welche die elektromotorische Kraft und

die einzelnen Elemente des Leitungswiderstandes betreen BIOT Lehrbuch

des Galvanismus und der Elektro chemie Leipzig Nachtrage S

GT Fechner Elemente der Psychophysik Breitkopf und Hartel Leipzig

GT Fechner Kol lektivmasslehre edited by GF Lipps Engelmann Leipzig

JB J de Fourier Theorie analytique de la Chaleur Didot Paris

CF Gauss Theoria Motus Corporum Coelestium in Carl Friedrich Gauss

Werke Vol Teubner Leipzig

Gauss an Olbers Maerz InCSchilling ed Wilhelm Olb ers sein

Leb en und seine Werke Vol Berlin

WertpapierVerzeichnisse Nachlass von Carl Friedrich Gauss G IX

Stadtarchiv Braunschweig

CF Gauss Nachlass Anwendung der Wahrscheinlichkeitsrechnung auf die

Bestimmung der Bilanz fur Witwenkassen in Carl Friedrich Gauss Werke

Vol Gottingen

JF Jeanneret Le Jeu le Pari et les Marches de BourseThese de do ctorat

Faculte de Droit de lUniversite de Berne Neuchatel

A Kolmogoro Uber die analytischen Methoden in der Wahrscheinlichkeits

rechnung Mathematische Annalen

PS Laplace Theorie analytique des Probabilites Courcier Paris

H Lefevre Traite des Valeurs mobiliers et des Operations de Bourse La

chaud Paris

H Lefevre Theorie elementaire des Operations de Bourse Bureau du

Journal des Placements Financiers Paris

H D Macleo d The Theory and Practice of Banking Longmans Green

ReaderDyer London

A Markov Extension of the Law of Large NumberstotheDependent Case

Notices of the Physico Mathematical So ciety on the University of Kazan

nd Series in Russian

A Markov On a Problem of Laplace Bulletin de lAcademie Imperiale des

Sciences Petrograd VI Serie in Russian

GS Ohm Vorlauge Anzeige des Gesetzes nach welchem Metal le die Kon

taktElektrizitat leiten Schweiggers Journal fur Chemie und Physik

GS Ohm Bestimmung des Gesetzes nach welchem Metal le die Kontakt

Elektrizitat leiten nebst einem Entwurf zu einer Theorie des Voltaschen

Apparates und des Schweiggerschen MultiplikatorsSchweiggers Journal fur

Chemie und Physik

GS Ohm Die galvanische Kette mathematisch bearbeitet Riemann Berlin

MFM Osb orne in the Stock Market Op erations Re

search

H Poincare Theorie analytique de la Propagation de la Chaleur Gauthier

Villars Paris

H Poincare Calcul des Probabilites GauthierVillars Paris

A Quetelet Uber den Menschen und die Entwicklung seiner Fahigkeiten oder

Versuch einer Physik der Gesel lschaft deutsche Ausgab e von VA Riecke

Schweizerbart Stuttgart

J Regnault Calcul des Chances et Philosophie de La Bourse MalletBachelier

Paris

JW Strutt Baron Rayleigh The Theory of Sound Macmillan and Co

London

TN Thiele Om Anvendelse af mindste Kvadraters Methode i nogle Til

faelde hvor en Komplikation af visse Slags uensartede tilfaeldige Fejlkilder

giver Fejlene en systematisk Karakter Lunos Kjb enhavn

Secondary literature

S GA Barnard Pivotal Models and the Fiducial ArgumentInternational

Statistical Review

S JM Courtault Ed LHeritage de Louis Bachelier Presses Universetaires

de Franche Comte Besancon

S JM Courtault Y Kabanov B Bru PCrep el I Leb on and A Le Mar

chand Louis Bachelier On the Centenary of Theorie de la Speculation

Mathematical Finance

S P Co otner ed The Random Character of Stock Marked Prices The MIT

Press Cambridge

S HJ Girlich Hausdors Beitrage zur Wahrscheinlichkeitstheorie in

E Brieskorn Hrsg Felix Hausdor zum Gedachtnis Band I Asp ekte seines

Werkes Vieweg BraunschweigWiesbaden

S HJ Girlich First Steps to Stochastic Finance in M Kohlmann S Tang

Eds Workshop of the Mathematical Finance Re

search Pro ject Konstanz Germany Birkhauser Basel

S A Hald A History of Mathematical Statistics from to J Wiley

New York

S FR Helmert Die Ausgleichsrechnung nach der Methode der kleinsten Qua

drateTeubner Leipzig

S CC Heyde E Seneta Eds Statisticians of the Centuries Springer New

York

S M Jacobsen Laplace and the origin of the OrnsteinUhlenbeck process

Bernolli

S F Jovanovic Lorigine de la theorie nanciere une reevaluation de lapport

de Louis Bachelier Revue dEconomie Politique

S F Jovanovic Pourquoi lhypothese de marche aleatoireentheorie nanciere

Les raisons historiques duni choise ethique Revue dEconomie Financiere

n mai pages

S F Jovanovic Instruments et theorie economiques dans la construction de

la Science de la Bourse dHenri Lefevre will be published in Revue

dHistoire des Sciences Humaines pages

S F Jovanovic Ph Le Gall Does God practice a The nan

cial physics of a th century forerunner Jules REGNAULT Europ ean

Journal of the History of Economic Thought pages

S F Jovanovic Ph Le Gall Les Jeux de lEnvers Une Autre Histoire de la

Contribution de Louis Bachelier alaTheorie Financiereeta lEconometrie

in S pages

S SL Lauritzen Time Series Analysis in A Discussion of Contributions

Made by TN ThieleInternational Statistical Review

S P Samuelson Mathematics of Speculative Price SIAM Review

S PSamuelson Modern Finance Theory Within One Lifetime in H Geman

D Madan SR Pliska T Vorst Eds Mathematical Finance Bachelier

Congress Springer Berlin

S SM Stigler The History of Statistics the Measurement of Uncertainty be

fore Harvard University Press Cambridge

S EJ Sulivan TM Weithers Louis Bachelier The Father of Modern Option

Pricing Theory Journal of Economic Education

S MS Taqqu Bachelier and his Times A Conversation with Bernard Bru

in H Geman D Madan SR Pliska T Vorst Eds Mathematical Fi

nance Bachelier Congress Springer Berlin