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Louis Bachelier on the Centenary of Théorie De La Spéculation Jean-Michel Courtault, Youri Kabanov, Bernard Bru, Pierre Crepel, Isabelle Lebon, Arnaud Le Marchand

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Louis Bachelier on the Centenary of Théorie De La Spéculation Jean-Michel Courtault, Youri Kabanov, Bernard Bru, Pierre Crepel, Isabelle Lebon, Arnaud Le Marchand Louis Bachelier On the centenary of Théorie de la Spéculation Jean-Michel Courtault, Youri Kabanov, Bernard Bru, Pierre Crepel, Isabelle Lebon, Arnaud Le Marchand To cite this version: Jean-Michel Courtault, Youri Kabanov, Bernard Bru, Pierre Crepel, Isabelle Lebon, et al.. Louis Bachelier On the centenary of Théorie de la Spéculation. Mathematical Finance, Wiley, 2000, 10 (3), pp.339-353. halshs-00447592 HAL Id: halshs-00447592 https://halshs.archives-ouvertes.fr/halshs-00447592 Submitted on 18 Jan 2010 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Louis Bachelier On the centenary of Theorie de la Speculation JM Courtault Yu Kabanov BBru P Crep el I Leb on A Le Marchand Centenary of mathematical nance The date March should be considered as the birthdate of mathematical nance on this day French p ostgraduate student Louis Bachelier successfully de fended at Sorb onne his thesis Theorie de la Speculation As a work of exceptional merit strongly supp orted by Henri Poincare Bacheliers sup ervisor it has been published in Annales scientiques de lEcole Normale Superieure one of the most inuential French scientic journals This pioneering study on the analysis of the sto ck and option markets contains several ideas of enormous value in b oth nance and probability In particular the theory of Brownian motion one of the most im portant mathematical discoveries of the twentieth centurywas initiated and used for the mathematical mo deling of price movements and evaluation of contingent claims in nancial markets The thesis of Louis Bachelier together with his subsequent w orks inuenced deeply the whole development of sto chastic calculus and mathematical nance As a testimony of his great contribution the newly created international Bachelier Finance So ciety is named after him The centenary of the famous thesis is widely celebrated as a landmark event in the history of mo dern science On this o ccasion we present to the readers attention a brief account of our research on Louis Bachelier In spite of his remarkable contributions he remained until recently in obscure darkness as one of the most mysterious gures in mathematics of the twentieth few facts ab out him could be found in the literature There is century Only a enormous public interest in his scientic biography which can be explained by the amazingly fast development of mathematical nance in the last two decades together with a deep er understanding of the fundamental role of Brownian motion We b elieve that our short note brings some new light on Louis Bachelier a human person a mathematician and a philosopher Dates of biography March Louis JeanBaptiste Alphonse Bachelier was b orn in Le Havre Octob er Graduated from secondary scho ol at Caen January Fathers death May Mothers death He is the head of Bachelier ls Military service Student at Sorb onne Octob er Bachelor in sciences at Sorb onne July Certicate in mathematical physics March Bachelier defended his thesis Free lecturer at Sorb onne The book Calcul des Probabilites The book Le Jeu la Chance et le Hasard September Drafted as a private in the F rench army December Back from the army December A member of the French Mathematical So ciety Assistant professor in Besancon September Married Augustine Jeanne Maillot she died so on Assistant professor in Dijon Asso ciated professor in Rennes January Blackballed in Dijon Octob er Professor in Besancon Professor Emeritus Octob er Retired The last publication April Died in SaintServansurMer buried in Sanvic near Le Havre The Bachelier Finance So cietyis founded Louis Bachelier was b orn in a resp ectful b ourgeois family known in Le Havre for its cultural and so cial traditions His father Alphonse Bachelier was not just a wine merchant but also the viceconsul of Venezuela at Le Havre and an amateur scientist Louis mother Cecile FortMeu was a bankers daughter her grandfather an imp ortant person in the nancial business was known also as the author of p o etry b o oks Unfortunately at the age of eighteen just after graduation from the secondary scho ol at Caen and getting the French bachelor degree baccalaureat es sciences Louis Bachelier lost his parents and was forced to in terrupt his studies to continue his fathers business and takecareofhissisterandthree year old brother These dramatic events had far reaching consequences for his academic career in particular they explain why Bachelier did not follow any grande ecole with the French scientic elite a weak point in his curriculum However quite probably as the head of the family enterprise ocially registered as Bachelier ls he got acquainted with the world of nancial markets one can nd some hints of p ersonal exp eriences in his works After the military service having in total a fouryear interruption of his stud ies an enormous handicap in a forming mathematician Louis arrived in Paris to continue his university education at Sorb onne Scarce information is available ab out these years He followed lectures of Paul App ell Joseph Boussinesq and not among the best students For example Henri Poincare Apparently he was Bacheliers marks in mathematics in the register were largely below those of his classmates Langevin and Lienard But in spite of a huge delay in his career Bacheliers development as a scientist was fast enough and he wrote his celebrated thesis Theorie de la Speculation on the application of probability to sto ck markets This was historically the rst attempt to use advanced mathematics in nance and witnessed the intro duction of Brownian motion was In accordance with the tradi tion of the ep o ch he also defended a second thesis on a sub ject chosen by the faculty namely on mechanics of uids Its title as well as the names of his professors may reect Bacheliers background Resistance dune masse liquide indenie pourvue de frottements interieurs regis par les formules de Navier aux p etits mouvements varies de translation dune sphere solide immergee dans cette masse et adherente a la couche uide qui la touche The thesis and Poincares rep ort The rst part of Bacheliers thesis contains a detailed description of pro ducts avail able in the French sto ck market of the epoch like forward contracts and options Their sp ecications were quite dieren t from the corresp onding pro ducts in the American market see comments in For example all payments were related to a single date and one had no need to think ab out the discounting or change of numeraire After nancial preliminaries Bachelier starts the mathematical mo d eling of the sto ck price movements and formulates the principle that the exp ec tation of the sp eculator is zero Obviously he understands here by exp ectation the conditional exp ectation given the past information In other words he im plicitly accepts as an axiom that the market evaluates assets using a martingale measure The further hyp othesis is that the price evolves as a continuous Markov pro cess homogeneous in time and space Bachelier shows that the density of one dimensional distributions of this pro cess satises the relation known now as the ChapmanKolmogorov equation and notes that the Gaussian density with the lin early increasing variance solves this equation The question of the uniqueness is not discussed but Bachelier provides some further arguments to conrm his conclusion considering the price pro cess as a limit of random He arrives at the same law by walks Bachelier also observes that the family of distribution functions of the pro cess satises the heat equation the probability diuses or radiates The results in these dozen pages are sp ectacular but this is not the end The mo del is applied to calculate various option prices Having in mind American and pathdep endent options Bachelier calculates the probability that the Brownian mo tion do es not exceed a xed level and nds the distribution of the supremum of the Brownian motion One hundred years after publishing the thesis it is quite easy to appreciate the imp ortance of his ideas It can be viewed as the origin of mathematical nance as well as several imp ortant branches of sto chastic calculus such as the theory of Brownian motion Markov pro cesses diusion pro cesses and even weak convergence in functional spaces Of course the reasoning of Bachelier was not rigorous but basically on the intuitive level correct This is really astonishing b ecause at the b eginning of the century the mathematical foundations of probability did not exist AMarkov started his studies on what are now called Markov chains only in and the concept of conditional exp ectations with resp ect to an arbitrary random variable or algebra was develop ed only in the s Poincares rep ort signed also by P App ell and J Boussinesq is a remarkable do cument showing that Bacheliers thesis was highly appreciated b y the outstanding mathematician It contains a deep analysis not only of the mathematical results but also an insightinto market laws In contrast to the legend that the evaluation note honorable means somehow that the examinators
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