Encyclopedia of Quantitative Finance

2048 Pages·2010·20.44 MB·English

by  Rama Cont

Bachelier, Louis Let us say a few words about this extraordinary
thesis. The problem investigated by Bachelier is
(1870–1946)
described in less than a page. The stock market is
subject to innumerable random influences, and so it
is unreasonable to expect a mathematically precise
Formation Years forecast of stock prices. However, we can try to
establish the law of the changes in stock prices over
Louis Bachelier was born in Le Havre, France, on a fixed period of time. The determination of this law
March 11, 1870. His father, a native of Bordeaux, was the subject of Bachelier’s thesis. The thesis was
movedtoLeHavreafterhismarriagetothedaughter not particularly original. Since the early nineteenth
of a notable citizen of Le Havre. He started a wine century, people had applied probability theory to
andspiritsshop,andboughtandexportedwinesfrom studyexchangerates.InFrance,inparticular,wecan
Bordeaux and Champagne. At the time, Le Havre cite the work of Bicquilley (around 1800) or Jules
was an important port. The Protestant bourgeoisie in Regnault (around 1850). In his thesis, Bachelier [1]
thecity,whichdominatedthelocalcottonandcoffee intended to revisit this issue from several viewpoints
markets,occupiedtheupperechelonsofsociety.The taken from physics and probability theory, as these
young Louis was educated at a high school in Le subjects were taught in Europe, including Paris,
Havre. He seems to have been a fairly good student, around 1900. He adapted these viewpoints to aid his
but he interrupted his studies after earning his high investigation.Thefirstmethodheusedisthemethod
school diploma in 1889, when both of his parents adopted by Einstein, five years later, to determine
died in the span of a few weeks. To provide for his the law of Brownian motion in a physical context. It
youngest brother and his older sister, most likely, he consistsofstudyingtheintegralequationthatgoverns
took over his father’s business, but he sold it after a theprobabilitythatthechangeinpriceisy attime t,
fewyears.In1892,hecompletedhismilitaryservice under two natural assumptions: the change in price
asaninfantrymanandthenmovedtoParis,wherehis duringtwoseparatetimeintervalsisindependentand
activities are unclear. What is clear, however, is that the expectation of the change in price is zero. The
Bachelierfocusedonhisinterestsinthestockmarket resulting equation is a homogeneous version of the
and undertook university studies at the University diffusionequation,nowknownastheKolmogorov(or
of Paris, where in 1895 he obtained his bachelor’s Chapman–Kolmogorov)equation,inwhichBachelier
degree in the mathematicalsciences, withoutbeing a boldlyassertsthattheappropriatesolutionisgivenby
particularly distinguished student. After earning his a centered Gaussian law with variance proportional
degree, he continued to attend the lectures of the to time t. He proved a statement already proposed,
Faculty, including courses in mathematical physics without justification, by Regnault in 1860 that the
taught by Poincare´ and Boussinesq. expectation of the absolute change in price after
Although we cannot be absolutely certain, it is time t is proportional to the square root of t.
likely that in 1894, Bachelier attended lectures in But this first method, which would eventually be
probability theory given by Poincare´, which were used in the 1930s by physicists and probabilists, did
published in 1896 and were based on the remarkable not seem to satisfy Bachelier, since he proposed a
treatise that Joseph Bertrand published in 1888. second method, which was further developed in the
His attendance at these lectures, his reading of 1930s by the Moscow School: the approximation of
treatises by Bertrand and Poincare´, and his interest the law of Brownian motion by an infinite sequence
in the stock market probably inspired his thesis, of coin flips, properly normalized. Since the change
“theory of speculation”, which was defended by in price over a given period of time is the result of a
Bachelier [1] in Paris on March 29, 1900, before a very large number of independent random variables,
jury composed of Appell, Boussinesq, and Poincare´. it is not surprising that this change in price is
OnthereportbyHenriPoincare´,hewasconferredthe Gaussian. But the extension of this approximation
rank of Doctor of Mathematics with an “honorable” to a continuous-time version is not straightforward.
designation,thatis,adesignationinsufficientforhim Bachelier, who already know the result he wanted
toobtainemploymentinhighereducation,whichwas to obtain, states and prepares the way to the first
extremely limited at the time. known version of a theorem, which in the current
2 Bachelier, Louis (1870–1946)
language reads as follows: let {X ,X ,...,X ,...} whose solution is the law of a centered Gaussian
1 2 n
beasequenceofindependentrandomvariablestaking random variable with variance n.
values 1 or −1 with probability 1/2. If we let S =
n
X +···+X and let [x] denote the integer part of
1 n Theory of Speculation
a real number x, then
(cid:1) (cid:2)
1 (cid:3) (cid:4) At the stock market, probability radiates like heat.
√ S[nt],t ≥0 −−−→ Bt,t ≥0 (1) This “demonstrates” the role of Gaussian laws in
n
problems related to the stock market, as acknowl-
inlawasn−−−→∞,where(Bt,t ≥0)isastandard edged by Poincare´ himself in his report: “A little
Brownian motion. reflection shows that the analogy is real and the
This second method, which is somewhat difficult comparison legitimate. The arguments of Fourier are
to read and not very rigorous, naturally leads to applicable, with very little change, to this problem
the previous solution. But it is still not sufficient. that is so different from the problem to which these
Bachelier proposes a third method,the “radiation (or arguments were originally applied.” And Poincare´
diffusion)ofprobability”.Bachelier,havingattended regretted that Bachelier did not develop this point
thelecturesofPoincare´ andBoussinesqonthetheory further, though this point would be developed in a
of heat, was aware of the “method of Laplace”, masterly way by Kolmogorov in a famous article
which gives the fundamental solution of the heat published in 1931 in the MathematischeAnnalen. In
equation, a solution that has exactly the form given fact, the first and third methods used by Bachelier
by the first (and second) methods used by Bachelier. are intrinsically linked: the Chapman–Kolmogorov
Hence, there is a coincidence to be elucidated. We equation for any regular Markov process is equiva-
know that Laplace probably knew the reason for lenttoapartialdifferentialequationofparabolictype.
this coincidence. Lord Rayleigh had recently noticed InallregularMarkovianschemesthatarecontinuous,
this coincidence in his solution to the problem of probabilityradiateslikeheatfromafirefannedbythe
“random phases”. It is likely that neither Bachelier thousandwindsofchance.Andfurtherwork,exploit-
norPoincare´hadreadtheworkofRayleigh.Anyway, ing this real analogy, would transform not only the
Bachelier, in turn, explains this curious intersection theory of Markov processes but also the century-old
betweenthetheoryofheatandthepricesofannuities theory of Fourier equations and parabolic equations.
ontheParisstockexchange.Thisishisthirdmethod, Now,havingdeterminedthelawofpricechanges,
which can be summarized as follows. all calculations of financial products involving time
Considerthegameofflippingafaircoinaninfinite follow easily. But Bachelier did not stop there. He
numberoftimesandsetf(n,x)=(cid:1)(S =x).Ithas proposed a general theory of speculation integrat-
n
beenknownsinceatleasttheseventeenthcenturythat ing all stock market products that could be proposed
to clients, whose (expected) value at maturity—and
f(n+1,x)= 1f(n,x−1)+ 1f(n,x+1) therefore whose price—canbe calculatedusing gen-
2 2 eralformulasresultingfromtheory.Themostremark-
(2) able product that Bachelier priced was based on the
maximumvalueofastockduringtheperiodbetween
Subtracting f(n,x) from both the sides of the equa-
its purchase and a maturity date (usually one month
tion, we obtain
later).Inthiscase,onemustdeterminethelawofthe
(cid:5) maximumofastockpriceoversomeintervaloftime.
f(n+1,x)−f(n,x)= 1 f(n,x+1) ThisproblemwouldbeofconcerntoNorbertWiener,
2
(cid:6) the inventor of the mathematicaltheory of Brownian
−2f(n,x)+f(n,x−1) (3) motion, in 1923. It involves knowing a priori the
law of the price over an infinite time interval, but it
It then suffices to take the unit 1 in the preceding was notknown—eitherin 1923 orin 1900—howto
equation to be infinitely small to obtain the heat easily calculate the integrals of functions of an infi-
equation nitenumberofvariables.Letusexplainthereasoning
∂f 1∂2f used by Bachelier [1] as an example of his methods
= (4)
∂n 2 ∂x2 of analysis.
Bachelier, Louis (1870–1946) 3
Bachelier proceeded in two different ways. The asimpleformulabyusingaverysimpleprobabilistic
firstwaywasbasedonthesecondmethoddeveloped (or combinatorial) argument.
in Bachelier’s thesis. It consists of discretizing time Of course, Bachelier had to do his mathematics
in steps of (cid:2)t, and introducing a change in price without a safety net. What could his safety net have
at each step of ±(cid:2)x. Bachelier wanted to calculate been? The mathematical analysis available during
the probability that before time t =n(cid:2)t, the game his time could not deal with such strange objects
(or price) exceeds a given value c=m(cid:2)x. Let n= and calculations. It was not until the following
m+2p. Bachelier proposed to first calculate the year, 1901, that Lebesgue introduced the integral
probability that the price c is reached for the first based on the measure that Borel had just recently
time at exactly time t. To this end, he uses the constructed.TheDaniellintegral,whichWienerused,
gambler’s ruin argument: the probability is equal dates to 1920 and it was not until the 1930s that
to (m/n)Cp2−n, which Bachelier obtained from the European mathematicians realized that computing
n
ballot formula of Bertrand, which he learned from probabilities with respect to Brownian motion, or
Poincare´ or Bertrand’s work, or perhaps both. It with respect to sequences of independent random
suffices to√then pass properly to the limit so that variables, could be done using Lebesgue measure on
(cid:2)x =O( (cid:2)t).Onethenobtainstheprobabilitythat theunitinterval.SinceLebesgue’stheorycametobe
the price exceeds c before t. Bachelier then noted viewed as one of the strongest pillars of analysis in
that this probability is equal to twice the probability the twentieth century, this approach gave probability
that the price exceeds c at time t. theory a very strong analytic basis. We will have to
The result is Bachelier’s formula for the law of wait much longer to place the stochastic calculus
the maximum M of the price B over the interval of Brownian motion and sample path arguments
t t
[0,t]; that is, involving stopping times into a relatively uniform
analytical framework. Anyway, Bachelier had little
(cid:1)(M >c)=2(cid:1)(B >c) (5) concern for either this new theory in analysis or the
t t
workofhiscontemporaries,whomhenevercites.He
It would have been difficult to proceed in a simpler referstotheworkofLaplace,Bertrand,andPoincare´,
fashion. Having obtained this formula, Bachelier who never cared about the Lebesgue integral, and so
had to justify it in a simple way to understand Bachelier always ignored its existence.
why it holds. Bachelier therefore added to his first It seems that in 1900, Bachelier [1] saw very
calculation (which was somewhat confusing and clearly how to model the continuous movement of
difficult to follow) a “direct demonstration” without stock prices and he established new computational
passing to the limit. He used the argument that “the techniques, derived notably from the classical tech-
pricecannotpassthethresholdc overatimeinterval niques involving infinite sequences of fair coin flips.
of length t without having done so previously” and He provided an intermediate mathematical argument
hence that to explain a new class of functions that reflected the
vagaries of the market, just as in the eighteenth cen-
(cid:1)(B >c)=(cid:1)(M >c)α (6) tury,whenoneusedgeometricreasoningandphysical
t t
intuition to explain things.
where α is the probability that the price c, having
been attained before time t, is greater than c at
time t. The latter probability is obviously 1/2, due After the Thesis
to symmetry of the sample paths that go above
and that remain below c by time t. And Bachelier His Ph.D.thesis defended,Bachelier suddenlyseem-
concludes:“Itisremarkablethatthemultipleintegral ed to discover the immensity of a world in which
that expresses the probability (cid:1)(M >c) does not randomness exists. The theory of the stock market
t
seem amenable to ordinary methods of calculation, allowed him to view the classical results of proba-
but can be determined by very simple probabilistic bility with a new eye, and it opened new viewpoints
reasoning.” It was, without doubt, the first example for him. Starting in 1901, Bachelier showed that the
of the use of the reflection principle in probability known results about infinite sequences of fair coin
theory.Intwosteps,acomplicatedcalculationyields flips could all (or almost all) be obtained from stock
4 Bachelier, Louis (1870–1946)
market theory and that one can derive new results Bachelier essentially did not publish any original
thataremoreprecisethananyonehadpreviouslysus- work. He married in 1920, but his wife died a few
pected.In1906,Bachelierproposesanalmostgeneral months later. He was often ill and he seems to have
theory of “related probabilities”, that is to say, a been quite isolated.
theory about what would, 30 years later, be called In 1937, he moved with his sister to Saint-Malo
Markov processes. This article by Bachelier was the inBrittany.DuringWorldWarII,hemovedtoSaint-
starting point of a major study by Kolmogorov in Servan, where he died in 1946. He seemed to be
1931 that we already mentioned. All of Bachelier’s aware of the new theory of stochastic processes that
work was published with the distant but caring rec- was then developing in Paris and Moscow, and that
ommendationofPoincare´,sothatby1910,Bachelier, was progressively spreading all over the world. He
whose income remains unknown and was proba- attempted to claim credit for the things that he had
bly modest, is permitted to teach a “free course” in done, without any success. He regained his appetite
probability theory at the Sorbonne, without compen- for research, to the point that in 1941, at the age
sation. Shortly thereafter, he won a scholarship that of 70, he submitted a note for publication to the
allowed him to publish his Calculus of Probability, Academy of Sciences in Paris on the “probability of
Volume I, Paris, Gauthier-Villars, 1912 (Volume II maximum oscillations”, in which he demonstrated a
neverappeared),whichincludedallofhisworksince finemasteryofthetheoryofBrownianmotion,which
his thesis. This very surprising book was not widely was undertaken systematically by Paul Levy starting
circulated in France, and had no impact on the Paris in 1938. Paul Levy, the principal French researcher
stock market or on French mathematics, but it was of the theory of Brownian motion, recognized,albeit
one of the sources that motivated work in stochastic belatedly, the work of Bachelier, and his work
processesattheMoscowSchoolinthe1930s.Italso provided a more rigorous foundation for Bachelier’s
influenced work by the American School on sums “theory of speculation”.
of independent random variables in the 1950s, and
at the same time, influenced new theories in math-
Reference
ematical finance that were developing in the United
States. And, as things should rightly be, these theo-
[1] Bachelier, L. (1900). The´orie de la spe´culation, The`se
ries traced back to France, where Bachelier’s name
Sciences mathe´matiques Paris. Annales Scientifiques de
had become so well recognized that in 2000, the
l’Ecole Normale Supe´rieure 17, 21–86; The Random
centennialanniversaryofhisworkin“theoryofspec- Character of Stock Market Prices, P. Cootner, ed, MIT
ulation” was celebrated. Press,Cambridge,1964,pp.17–78.
The First World War interrupted the work of
Bachelier, who was summoned for military service Further Reading
in September 1914 as a simple soldier. When the
warendedinDecember1918,hewasasublieutenant
Courtault,J.M.&Kabanov,Y.(eds)(2002).LouisBachelier:
in the Army Service Corps. He served far from the
AuxoriginesdelaFinanceMathe´matique, Presses Univer-
front, but he carried out his service with honor. As a sitairesFranc-Comtoises,Besanc¸on.
result,in1919,theDirectorateofHigherEducationin Taqqu, M.S. (2001). Bachelier and his times: a conversation
Paris believed it was necessary to appoint Bachelier withBernardBru,FinanceandStochastics5(1),3–32.
to a university outside of Paris, since the war had
decimatedtheranksofyoungFrenchmathematicians Related Articles
and there were many positions to be filled. After
many difficulties, due to his marginalization in the
Black–Scholes Formula; Markov Processes;
Frenchmathematicalcommunityandtheincongruent
Martingales; Option Pricing: General Principles.
nature of his research, Bachelier finally received
tenure in 1927 (at the age of 57) as a professor at
BERNARD BRU
the University of Besanc¸on, where he remaineduntil
hisretirementin1937.Throughoutthepostwaryears,
Samuelson, Paul A. fat-tailed, infinite-variance return distributions [14],
and, over a span of nearly four decades, analyzing
the systematic dependence on age of optimal port-
folio strategies, in particular, optimal long-horizon
PaulAnthonySamuelson(1915–) is Institute Profes-
investment strategies, and the improper use of the
sor Emeritus at the Massachusetts Institute of Tech-
Law of Large Numbers to arrive at seemingly domi-
nology where he has taught since 1940. He earned
natingstrategiesforthelongrun [10,15,17,21–27].
a BA from the University of Chicago in 1935 and
Ininvestigatingtheoft-toldtalethatinvestorsbecome
his PhD in economics from Harvard University in
systematically more conservative as they get older,
1941. He received the John Bates Clark Medal in
Samuelson shows that perfectly rational risk-averse
1947 and the National Medal of Science in 1996.
investors with constant relative risk aversion will
In 1970, he became the first American to receive the
select the same fraction of risky stocks versus safe
Alfred Nobel Memorial Prize in Economic Sciences.
cashperiodbyperiod,independentlyofage,provided
Histextbook,Economics,firstpublishedin1948,and
that the investment opportunity set is unchanging.
inits18thedition,isthebest-sellingandarguablythe
Havingshownthatgreaterinvestmentconservatismis
most influential economics textbook of all time.
not an inevitable consequence of aging, he later [24]
Paul Samuelson is the last great general
demonstrates conditions under which such behavior
economist—never again will any one person make
such foundational contributions to so many distinct canbeoptimal:withmean-revertingchangingoppor-
areas of economics. His prolific and profound theo- tunity sets, older investors will indeed be more con-
reticalcontributionsoversevendecadesofpublished servative than in their younger days, provided that
research have been universal in scope, and his ram- they are more risk averse than a growth-optimum,
ified influence on the whole of economics has led log-utilitymaximizer.Tocompletetherichsetofage-
to foundational contributions in virtually every field dependent risk-taking behaviors, Samuelson shows
of economics, including financial economics. Repre- thatrationalinvestorsmayactuallybecomelesscon-
senting 27years of scientific writing from 1937 to servative with age, if either they are less risk averse
themiddleof1964,thefirsttwovolumesofhisCol- than log or if the opportunity set follows a trend-
lectedScientificPapers contain129articlesand1772 ing, momentum-like dynamic process. He recently
pages. These were followed by the publication of confided that in finance, this analysis is a favorite
the 897-page third volume in 1972, which registers brainchild of his.
the succeeding seven years’ product of 78 articles PublishedinthesameissueoftheIndustrialMan-
published when he was between the ages of 49 and agement Review, “Proof That Properly Anticipated
56 [18]. A mere five years later, at the age of 61, PricesFluctuateRandomly”and“RationalTheoryof
Samuelson had published another 86 papers, which Warrant Pricing” are perhaps the two most influen-
fill the 944 pages of the fourth volume. A decade tial Samuelson papers in quantitative finance. Dur-
later,thefifthvolumeappearedwith108articlesand ing the decade before their printed publication in
1064 pages.A glance at his list of publicationssince 1965, Samuelson had set down, in an unpublished
1986 assures us that a sixth and even seventh vol- manuscript, many of the results in these papers and
ume could be filled. That Samuelson paid no heed had communicated them in lectures at MIT, Yale,
to the myth of debilitating age in science is particu- Carnegie, the American Philosophical Society, and
larlywell-exemplifiedinhiscontributionstofinancial elsewhere. In the early 1950s, he supervised a PhD
economics,withall but6 ofhis morethan 60papers thesis on put and call pricing [5].
being published after he had reached the age of 50. The sociologist or historian of science would
Samuelson’s contribution to quantitative finance, undoubtedly be able to develop a rich case study
as with mathematical economics generally, has been of alternative paths for circulating scientific ideas
foundational and wide-ranging: these include recon- by exploring the impact of this oral publication of
ciling the axioms of expected utility theory first with research in rational expectations, efficient markets,
nonstochastictheoriesofchoice [9]andthenwiththe geometric Brownian motion, and warrant pricing in
ubiquitous and practical mean–variance criterion of the period between 1956 and 1965.
choice [16], exploring the foundations of diversifica- Samuelson (1965a) and Eugene Fama indepen-
tion [13]andoptimalportfolioselectionwhenfacing dentlyprovidethefoundationoftheEfficientMarket
2 Samuelson, Paul A.
theory that developed into one of the most impor- themostpartinthoseensuingyears,hisinterpretation
tant concepts in modern financial economics. As of the data is that organized markets where widely
indicated by its title, the principal conclusion of owned securities are traded are well approximated
the paper is that in well-informed and competitive as microefficient,meaningthat the relative pricingof
speculative markets, the intertemporal changes in individual securities within the same or very similar
prices will be essentially random. Samuelson has asset classes is such that active asset management
described the reaction (presumably his own as well applied to those similar securities (e.g., individual
as that of others) to this conclusion as one of “initial stock selection) does not earn greater risk-adjusted
shock—and then, upon reflection, that it is obvi- returns.
ous”. The argument is as follows: the time series of However, Samuelson is discriminating in his
changes in most economic variables gross national assessment of the efficient market hypothesis as it
product (GNP, inflation, unemployment, earnings, relates to real-world markets. He notes a list of
and even the weather) exhibit cyclical or serial the “few not-very-significantapparentexceptions”to
dependencies. Furthermore, in a rational and well- microefficient markets [23, p. 5]. He also expresses
informed capital market, it is reasonable to presume beliefthatthereareexceptionallytalentedpeoplewho
that the prices of common stocks, bonds, and com- can probably garner superior risk-corrected returns,
modityfuturesdependuponsucheconomicvariables. andevennamesafew.Hedoesnotseethemasoffer-
Thus, the shock comes from the seemingly inconsis- ing a practical broad alternative investment prescrip-
tentconclusionthatinsuchwell-functioningmarkets tionforactivemanagementsincesuchtalentsarefew
the changes in speculative prices should exhibit no andhardtoidentify.AsSamuelsonbelievesstrongly
serial dependencies. However, once the problem is inmicroefficiencyofthemarkets,heexpressesdoubt
viewedfromtheperspectiveofferedinthepaper,this about macromarket efficiency: namely that indeed
seeming inconsistency disappears and all becomes asset-value “bubbles” do occur.
obvious. There is no doubt that the mainstream of the pro-
Starting from the consideration that in a competi- fessional investment community has moved signifi-
tivemarket,ifeveryoneknew thataspeculativesecu- cantly in the direction of Paul Samuelson’s position
rity was expected to rise in price by more (less) than during the 35years since he issued his challenge to
the required or fair expected rate of return, it would that community to demonstrate widespread superior
already be bid up (down) to negate that possibility, performance [20]. Indexing as either a core invest-
Samuelson postulatesthatsecuritieswillbepricedat ment strategy or a significant component of insti-
each point in time so as to yield this fair expected tutional portfolios is ubiquitous, and even among
rate of return. Using a backward-in-time induction those institutional investors who believe they can
argument, he proves that the changes in speculative deliver superior performance, performance is typi-
prices around that fair return will form a martingale. cally measured incrementally relative to an index
And this follows no matter how much serial depen- benchmark and the expected performance increment
dency there is in the underlying economic variables to the benchmark is generally small compared to the
uponwhichsuchspeculativepricesareformed.Inan expected return on the benchmark itself. It is there-
informedmarket,therefore,currentspeculativeprices fore with no little irony that as investment practice
will already reflect anticipated or forecastable future has moved in this direction, for the last 15years,
changesintheunderlyingeconomicvariablesthatare academic research has moved in the opposite direc-
relevant to the formation of prices, and this leaves tion, strongly questioning even the microefficiency
only the unanticipated or unforecastable changes in case for the efficient market hypothesis. The con-
these variables as the sole source of fluctuations in ceptual basis of these challenges comes from the-
speculative prices. ories of asymmetric information and institutional
Samuelson is careful to warn the reader against rigidities that limit the arbitrage mechanisms that
interpreting his mathematically derived theoretical enforce microefficiency and of cognitive dissonance
conclusions about markets as empirical statements. and other systematic behavioral dysfunctions among
Nevertheless,for40years,hismodelhasbeenimpor- individual investors that purport to distort market
tant to the understanding and interpretation of the prices away from rationally determined asset prices
empiricalresultsobservedinreal-worldmarkets.For inidentifiedways.Asubstantialquantityofempirical
Samuelson, Paul A. 3
evidence has been assembled, but there is consider- his paper, Samuelson thus chose the term European
able controversy over whether it does indeed make for the relatively simple(-minded)-to-value option
a strong case to reject market microefficiency in the contract that can only be exercised at expiration and
Samuelsonian sense. What is not controversial at all American for the considerably more-(complex)-to-
is that Paul Samuelson’s efficient market hypothesis value option contract that could be exercised early,
has had a deep and profound influence on finance any time on or before its expiration date.
research and practice for more than 40years and all Although real-world options are almost always
indicationsarethatitwillcontinuetodosowellinto of the American type, published analyses of option
the future. pricing prior to his 1965 paper focused exclusively
If one were to describe the 1960s as “the decade on the evaluation of European options and therefore
of capital asset pricing and market efficiency” in didnotincludetheextravaluetotheoptionfromthe
view of the important research gains in quantitative right to exercise early.
finance during then, one need hardly say more than The most striking comparison to make between
“the Black-Scholes option pricing model” to justify
theBlack–ScholesoptionpricingtheoryandSamuel-
describing the 1970s as “the decade of option and
son’s rational theory [12] is the formula for the
derivativesecuritypricing.”Samuelsonwasaheadof
optionprice.TheSamuelsonpartialdifferentialequa-
the field in recognizing the arcane topic of option
tionfortheoptionpriceisthesameasthecorrespond-
pricingasarichareaforproblemchoiceandsolution.
ing equation for the Black–Scholes option price if
By at least the early 1950s, Samuelson had shown
one sets the Samuelson parameter for the expected
that the assumption of an absolute random walk or
return on the underlying stock equal to the riskless
arithmetic Brownian motion for stock price changes
interest rate minus the dividend yield and sets the
leads to absurd prices for long-lived options, and
Samuelson parameter for the expected return on the
this was done before his rediscovery of Bachelier’s
option equal to the riskless interest rate. It should,
pioneering work [1] in which this very assumption
however,beunderscoredthatthemathematicalequiv-
is made. He introduced the alternative process of a
alencebetweenthetwoformulaswiththeredefinition
“geometric” Brownian motion in which the log of
of parameters is purely a formal one. The Samuel-
price changes follows a Brownian motion, possibly
son model simply posits the expected returns for the
with a drift. His paper on the rational theory of
stock and option. By employing a dynamic hedging
warrant pricing [12] resolves a number of apparent
or replicating portfolio strategy, the Black–Scholes
paradoxesthathadplaguedtheexistingmathematical
analysis derives the option price without the need
theory of option pricing from the time of Bachelier.
to know either the expected return on the stock or
In the process (with the aid of a mathematical
the required expected return on the option. There-
appendixprovided byH.P. McKean,Jr),Samuelson
fore, the fact that the Black–Scholes option price
also derives much of what has become the basic
satisfies the Samuelson formula implies neither that
mathematicalstructureofoptionpricingtheorytoday.
the expected returns on the stock and option are
Bachelier [1] considered options that could only
equal nor that they are equal to the riskless rate of
be exercised on the expiration date. In modern
times, the standard terms for options and warrants interest. Furthermore, it should also be noted that
permit the option holder to exercise on or before Black–Scholes pricing of options does not require
the expiration date. Samuelson coined the terms knowledgeofinvestors’preferencesandendowments
European optiontorefertotheformerandAmerican as is required, for example, in the sequel Samuelson
option to refer to the latter. As he tells the story, and Merton [28] warrant pricing paper. The “ratio-
to get a practitioner’s perspective in preparation for nal theory” put forward in 1965 is thus clearly a
his research, he went to New York to meet with a “miss” with respect to the Black–Scholes develop-
well-knownputandcalldealer(therewerenotraded ment. However, as this analysis shows, it is just as
options exchanges until 1973) who happened to be clearly a “near miss”. See [6, 19] for a formal com-
Swiss. Upon his identifying himself and explaining parison of the two models.
what he had in mind, Samuelson was quickly told, Extensive reviews of Paul Samuelson’s remark-
“You are wasting your time—it takes a European able set of contributions to quantitative finance can
mind to understand options.” Later on, when writing be found in [2–4, 7, 8].
4 Samuelson, Paul A.
References [16] Samuelson, P.A. (1970). The fundamental approxima-
tion theorem of portfolio analysis in terms of means,
variances and higher moments, Review of Economic
[1] Bachelier, L. (1900, 1966). Theory de la Specula-
Studies 37, 537–542, Collected Scientific Papers, III,
tion, Gauthier-Villars, Paris, in TheRandomCharacter
Chap.203.
of Stock Market Prices, P. Cootner, ed, MIT Press,
[17] Samuelson, P.A.(1971b). The‘Fallacy’ of maximizing
Cambridge.
the geometric mean in long sequences of investing
[2] Bernstein, P.L. (2005). Capital Ideas: The Improbable
Origins of Modern Wall Street, John Wiley & Sons, or gambling, Proceedings of the National Academy of
Hoboken. Sciences of United States of America 68, 2493–2496,
[3] Carr, P. (2008). The father of financial engineering, CollectedScientificPapers,III,Chap.207.
BloombergMarkets17,172–176. [18] Samuelson,P.A.(1972).TheCollectedScientificPapers
[4] Fischer,S.(1987).Samuelson,PaulAnthony, TheNew of Paul A. Samuelson, R.C. Merton, ed, MIT Press,
Palgrave:ADictionaryofEconomics, MacMillan Pub- Cambridge,Vol.3.
lishing,Vol.4,pp.234–241. [19] Samuelson, P.A. (1972). Mathematics of speculative
[5] Kruizenga, R. (1956). Put and Call Options: A Theo- price, in MathematicalTopicsinEconomicTheoryand
reticalandMarketAnalysis,Doctoraldissertation,MIT, Computation, R.H.Day&S.M.Robinson,eds,Society
Cambridge,MA. for Industrial and Applied Mathematics, Philadelphia,
[6] Merton, R.C. (1972). Continuous-time speculative pro- pp.1–42,reprintedinSIAMReview 15,1973,Collected
cesses: appendix to P. A. Samuelson’s ‘mathematics ScientificPapers,IV,Chap.240.
of speculative price’, in Mathematical Topics in Eco-
[20] Samuelson,P.A.(1974).Challengetojudgment,Journal
nomic Theory and Computation, R.H., Day & S.M.
ofPortfolioManagement1,17–19,CollectedScientific
Robinson, eds, Philadelphia Society for Industrial and
Papers,IV,Chap.243.
Applied Mathematics, pp. 1–42, reprinted in SIAM
[21] Samuelson,P.A.(1979).Whyweshouldnotmakemean
Review 15,1973.
logofwealthbigthough yearstoactarelong,Journal
[7] Merton, R.C. (1983). Financial economics, in Paul
ofBankingandFinance3,305–307.
SamuelsonandModernEconomicTheory,E.C.Brown&
[22] Samuelson, P.A. (1989). A case at last for age-
R.M.Solow,eds,McGrawHill,NewYork.
phasedreductioninequity,ProceedingsoftheNational
[8] Merton, R.C. (2006). Paul Samuelson and financial
economics,inSamuelsonianEconomicsandtheTwenty- Academy of Science of United States of America 86,
FirstCentury,M.Szenberg,L.Ramrattan&A.Gottes- 9048–9051.
man, Oxford University Press, Oxford, Reprinted in [23] Samuelson, P.A. (1989). The judgment of economic
AmericanEconomist 50,no.2(Fall2006). science on rational portfolio management: indexing,
[9] Samuelson,P.A.(l952).Probability,utility,andtheinde- timing, and long-horizon effects, Journal of Portfolio
pendenceaxiom,Econometrica20,670–678,Collected ManagementFall,16,4–12.
ScientificPapers,I,Chap.14. [24] Samuelson, P.A. (1991). Long-run risk tolerance when
[10] Samuelson,P.A.(1963).Riskanduncertainty:afallacy equity returns are mean regressing pseudoparadoxes
oflargenumbers,Scientia57,1–6,CollectedScientific and vindication of ‘businessmen’s risk, in Money,
Papers,I,Chap.16. Macroeconomics,andEconomicPolicy:EssaysinHonor
[11] Samuelson, P.A. (l965). Proof that properly antici- of James Tobin, W.C. Brainard, W.D. Nordhaus &
pated prices fluctuate randomly, Industrial Manage- H.W. Watts, eds, The MIT Press, Cambridge, pp.
mentReview6, 41–49,Collected Scientific Papers,III, 181–200.
Chap.198.
[25] Samuelson,P.A.(1992).Atlastarationalcaseforlong
[12] Samuelson,P.A.(l965).Rationaltheoryofwarrantpric-
horizon risk tolerance and for asset-allocation timing?
ing,IndustrialManagementReview6,13–39,Collected
inActiveAssetAllocation,D.A.Robert&F.J.Fabozzi,
ScientificPapers,III,Chap.199.
eds,ProbusPublishing,Chicago.
[13] Samuelson, P.A. (1967). General proof that diversi-
[26] Samuelson, P.A. (1994). The long-term case of equi-
fication pays, Journal of Financial and Quantitative
ties and how it can be oversold, Journal of Portfolio
Analysis 2, 1–13, Collected Scientific Papers, III,
ManagementFall,21,15–24.
Chap.201.
[27] Samuelson, P.A. (1997). Proof by certainty equiva-
[14] Samuelson, P.A. (1967). Efficient portfolio selection
for Pareto-Levy investments, Journal of Financial and lents that diversification-across-time does worse, risk-
Quantitative Analysis 2, 107–122, Collected Scientific corrected, than diversification-throughout-time, Journal
Papers,III,Chap.202. ofRiskandUncertainty14,129–142.
[15] Samuelson, P.A.(l969). Lifetimeportfolio selectionby [28] Samuelson, P.A. & Merton, R.C. (1969). A complete
dynamicstochasticprogramming,ReviewofEconomics model ofwarrantpricing thatmaximizesutility, Indus-
andStatistics51,239–246,CollectedScientificPapers, trial Management Review 10, 17–46, Collected Scien-
III,Chap.204. tificPapers,III,Chap.2000.
Samuelson, Paul A. 5
Further Reading Samuelson, P.A. (1977). The Collected Scientific Papers of
Paul A. Samuelson, H. Nagatani & K. Crowley, eds, MIT
Samuelson, P.A. (1966). The Collected Scientific Papers of Press,Cambridge,Vol.4.
PaulA.Samuelson,J.E.Stiglitz,ed,MITPress,Cambridge, Samuelson, P.A. (1986). The Collected Scientific Papers of
Vols.1and2. PaulA.Samuelson,K.Crowley,ed,MITPress,Cambridge,
Samuelson,P.A.(l971).Stochasticspeculativeprice,Proceed- Vol.5.
ingsoftheNationalAcademyofSciencesoftheUnitedStates
of America 68, 335–337, Collected Scientific Papers, III, ROBERT C. MERTON
Chap.206.
Black, Fischer importantuseofthetoolswasinhisworkoninterest
rate derivatives, in the famous Black–Derman–Toy
term structure model [10].
Black got his start in finance after already earn-
The central focus of the career of Fischer Black
ing his PhD in applied mathematics (Harvard, 1964)
(1938–1995) was on teasing out the implications
whenhelearnedaboutCAPMfromTreynor[18],his
of the capital asset pricing model (CAPM) for the
colleague at the business consulting firm Arthur D.
changinginstitutionalframeworkoffinancialmarkets
Little,Inc.Fischerhadnevertakenasinglecoursein
ofhisday.HebecamefamousfortheBlack–Scholes
options formula [14], an achievement that is now economics or finance, nor did he ever do so subse-
widelyrecognizedashavingopenedthedoortomod- quently. Nevertheless, the field was underdeveloped
ern quantitative finance and financial engineering. at the time, and Fischer managed to set himself up
Fischerwasthefirstquant,butaveryspecialkindof as a financial consultant and to parlay his success
quant because of his taste for the big picture [16]. in that capacity into a career in academia (Univer-
Regarding that big picture, as early as 1970, he sity of Chicago 1971–1975, Massachusetts Institute
sketchedavisionofthefuturethathasbynowlargely of Technology 1975–1984), and then into a partner-
come true: ship at the Wall Street investment firm of Goldman
Sachs (1984–1995). There can be no doubt that his
Thus a long term corporate bond could actually be
earlysuccesswiththeoptionspricingformulaopened
soldtothreeseparatepersons.Onewouldsupplythe
thesedoors.Themoreimportantpointishow,ineach
moneyforthebond;onewouldbeartheinterestrate
risk;andonewouldbeartheriskofdefault.Thelast ofthesesettings,Fischerusedtheopportunityhehad
two would not have to put up any capital for the been given to help promote his vision of a CAPM
bonds, although they might have to post some sort future for the financial side of the economy.
ofcollateral. CAPM is only about a world of debt and equity,
Today we recognize the last two instruments as an andthedebtinthatworldisbothshorttermandrisk
interest rate swap and a credit default swap, the free.Insuchaworld,everyoneholdsthefullydiver-
two instruments that have been the central focus of sifiedmarketportfolioofequityandthenadjustsrisk
financial engineering ever since. exposure by borrowing or lending in the market for
All of the technology involved in this engineer- risk-freedebt.Asequityvaluesfluctuate,outstanding
ing can be traced back to roots in the original debt also fluctuates, as people adjust their portfolios
Black–Scholes option pricing formula [14]. Black tomaintaindesiredriskexposure.Oneimplicationof
himself came up with a formula through CAPM, by CAPM,therefore,isthatthereshouldbeamarketfor
thinkingabouttheexposuretosystemicriskthatwas passivelymanagedindexmutualfunds[15].Another
involvedinanoption,andhowthatexposurechanges implicationisthattheregulatoryapparatussurround-
asthepriceoftheunderlyingchanges.Todaythefor- ing banking, both lending and deposit taking, should
mulaismorecommonlyderivedusingtheItoformula bedrasticallyrelaxedtofacilitatedynamicadjustment
and theoption replicationideaintroduced byMerton of risk exposure [3]. And yet a third implication is
[17].Foralongtime,Blackhimselfwasunsureabout thattheremightbearoleforanautomaticriskrebal-
the social utility of equity options. If all they do is ancing instrument, essentially what is known today
to allow people to achieve the same risk exposure as portfolioinsurance [6, 13].
they could achieve by holding equity outright with Even while Black was working on remaking the
leverage, then what is the point? world in the image of CAPM, he was also expand-
The Black–Scholes formula and the hedging ing the image of the original CAPM to include a
methodologybehinditsubsequentlybecameacentral worldwithoutarisklessassetinhisfamouszero-beta
pillar in the pricing of contingent claims of all kinds model [1] and to include a world with multiple cur-
and in doing so gave rise to many innovations that rencies in his controversial universal hedging model
contributed to making the world more like his 1970 [2,7]thatsubsequentlyformedtheanalyticalcoreof
vision.BlackandCox[9]representsanearlyattempt theBlack–Littermanmodelofglobalassetallocation
to use the option pricing technology to price default [11, 12].
risk.Black[4]similarlyusestheoptionpricingtech- These and other contributions to quantitative
nology to price currency risk. Perhaps, Black’s most finance made Fischer Black famous, but according