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Bachelier's Predecessors* 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 Carl Friedrich Gauss 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 option 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 Brownian
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