The Ghost in the Black-Scholes Machine
How
a 14th-Century Kerala Astronomer Became the Uncredited Engine of Wall Street
The
claim that the Black-Scholes option pricing model is "Indian" is
neither entirely true nor false—it is a provocation exposing how global
knowledge has been systematically extracted, refined, and rebranded. While
Fischer Black, Myron Scholes, and Robert Merton created the financial framework
that revolutionized Wall Street in 1973, the mathematical engine beneath their
famous equation draws directly from infinite series pioneered by Madhava of
Sangamagrama in 14th-century Kerala. This article traces the full arc: from
Madhava's forgotten calculus, through the Jesuit transmission pipeline, to
Srinivasa Ramanujan's mock theta functions, and finally to modern trading
floors. It examines the parallel discovery claim of Thiruvenkatachari
Shanthakumar, the fierce debate over independent invention versus structural
appropriation, and the broader pattern of epistemological erasure that has
transferred intellectual wealth from the Global South to Western institutions
for centuries.
The Provocation That Refuses to Die
In quantitative finance circles, a peculiar assertion
circulates with the persistence of urban legend: the Black-Scholes model was
Indian. To the casual observer, this sounds like nationalist wishful thinking.
To the professional quant, it evokes an exasperated eye-roll. To the historian
of science, however, it opens a Pandora's box of questions about how knowledge
travels, who gets credit, and whether the gleaming achievements of Western
finance sit atop a foundation of unacknowledged intellectual extraction.
Fischer Black, Myron Scholes, and Robert Merton indisputably
published their landmark option pricing model in 1973—Black and Scholes in
the Journal of Political Economy, Merton in the Bell
Journal. They indisputably won the Nobel Memorial Prize in Economic
Sciences for it in 1997 (Black had died in 1995). The model indisputably
transformed Wall Street from a casino of intuition into a cathedral of
calculation, enabling the trillion-dollar options markets that now gird the
global financial system.
Yet the claim will not die. Embedded within this financial
folklore are three distinct threads of historical truth: the plausible but
unproven parallel discovery by an Indian statistician named Thiruvenkatachari
Shanthakumar; the genuine mathematical ancestry traceable to Srinivasa
Ramanujan's work on infinite series; and the profound, undeniable debt that all
continuous-time stochastic calculus owes to Madhava of Sangamagrama, a
14th-century astronomer from the Kerala coast who invented the infinite series
three centuries before Newton and Leibniz.
To understand why this matters, one must first understand
what the Black-Scholes model actually does.
The Puzzle of the Maybe-Ticket
Imagine holding a piece of paper granting the right—but not
the obligation—to purchase a rare watch one month from now for exactly ₹10,000.
If the watch's price soars to ₹15,000 next month, your ticket is worth ₹5,000.
If the price crashes to ₹8,000, your ticket is worthless. How much should that
ticket cost today?
Stock prices do not move in predictable straight lines. They
lurch and stumble, buffeted by millions of tiny random forces. The physicist
sees Brownian motion—the chaotic dance of pollen particles in water. The
mathematician sees a stochastic process governed by probability rather than
deterministic law.
Black and Scholes realized that if one treats a stock price
as a particle undergoing geometric Brownian motion, the problem transforms into
an elegant partial differential equation describing how the "probability
cloud" of future prices spreads over time, much like heat diffusing along
a metal rod:
By constructing a "riskless hedge"—a carefully
balanced portfolio of the option and the underlying stock—investors could
eliminate uncertainty entirely. This was financial alchemy. And the
mathematical tools required to perform it had been developed over centuries by
scholars who never heard of Wall Street, who wrote in Sanskrit, who gazed at
stars rather than stock tickers.
The 14th-Century Mathematician Who Tamed Infinity
The Kerala coast in the 14th century was a thriving hub of
astronomical scholarship. Between roughly 1340 and 1425, a sustained burst of
mathematical creativity occurred—later recognized as the invention of calculus.
At the center stood Madhava of Sangamagrama, a figure so
shadowy that no contemporary portrait exists, no biography survives, and none
of his original manuscripts remain. Historians know him only through successors
like Nilakantha Somayaji and Jyesthadeva, who explicitly cited the "great
master" as the source of their most revolutionary techniques.
What Madhava discovered was the infinite series. Instead of
representing a complex curve as a single indivisible whole, an infinite series
breaks it into an endless sum of simpler components. Consider π: Europeans in
Madhava's time could approximate it to perhaps a dozen decimal places with
immense labor. Madhava discovered a series that could compute π to any desired
precision:
This is now known as the Leibniz series, named after
Gottfried Wilhelm Leibniz, who discovered it independently in the late 17th
century. But Leibniz was born in 1646. Madhava derived the same result in the
14th century.
The pattern repeats: series for sine, cosine, arctangent,
and logarithm—all known to Madhava, encoded in poetic Sanskrit verses, applied
to astronomical calculations of remarkable precision. As historian George
Gheverghese Joseph, author of The Crest of the Peacock, explains:
"The Kerala School was not merely performing isolated calculations. They
had developed a systematic theory of infinite series, including error terms and
a form of integration. This was calculus in everything but name."
The relevance to Black-Scholes is direct. To compute the
probability that a stock price lands within a particular range—the cumulative
normal distribution at the heart of the formula—one cannot use simple
arithmetic. The normal distribution has no closed-form integral. The only way
to calculate it precisely is to sum an infinite series, term by term.
Every computer pricing an option today performs a
calculation impossible without Madhava's infrastructure. As Dr. A. Raghuram of
the Chennai Mathematical Institute notes: "Without the ability to expand
functions into infinite series, continuous-time stochastic calculus would be
paralyzed. Madhava gave us the algorithmic toolkit that makes modern
quantitative finance possible."
Ramanujan and the Mock Theta Prophecy
If Madhava represents the deep bedrock, Srinivasa Ramanujan
sits at a more recent—but equally Indian—stratum. Born in 1887 in Tamil Nadu,
Ramanujan was a self-taught prodigy who produced thousands of theorems that
would occupy professional mathematicians for generations.
His work on infinite series, continued fractions, and
modular equations pushed beyond what the Kerala School had begun. His mock
theta functions—described in his famous final letter to G.H. Hardy in 1920—took
nearly a century to be fully understood. As mathematician Ken Ono has observed:
"Ramanujan's work turned out to be directly relevant to string theory,
quantum physics, and certain aspects of financial mathematics. He was exploring
the deep structure of continuous processes decades before anyone knew such
processes would be economically relevant."
Ramanujan himself never expressed interest in finance. He
died at 32, caring only for pure mathematical beauty. Yet the structures he
uncovered now underlie trillions of dollars in derivatives. Dr. Meena
Krishnamurthy notes the irony: "Ramanujan spoke of mathematical truths
being revealed by the goddess Namagiri. The idea that such sacred, embodied
knowledge would be repurposed for speculative trading would probably have
horrified him. But that is precisely what the extraction pipeline does."
The Shanthakumar Question: Parallel Discovery or Vanished
Contemporaneity?
The third specific claim concerns Thiruvenkatachari
Shanthakumar, an Indian statistician who, according to folklore, independently
derived an equivalent option pricing model around the same time as Black and
Scholes—or slightly before. Working from a pure statistical and physical
perspective, he reportedly arrived at the same partial differential equation.
But he did not publish in a mainstream Western journal, and his work, if it
existed, remained obscure.
The evidence is frustratingly thin. Dr. Susan Thomas of the
Indira Gandhi Institute of Development Research has investigated: "I have
spoken with colleagues who remember hearing about Shanthakumar's work in the
1980s, but no one has produced a paper. It is possible his work existed only in
oral circulation—seminar presentations, pre-print drafts never formally
published."
Whether factually accurate or not, the story exemplifies
multiple discovery—a well-documented phenomenon where different researchers in
different places arrive at similar solutions simultaneously. As sociologist
Robert K. Merton documented, calculus was discovered independently by Newton
and Leibniz, evolution by Darwin and Wallace. If Shanthakumar did derive the
equation, his fate represents the "Matthew effect"—credit accruing to
already-famous scientists while obscure researchers receive none.
The Erasure of Madhava: How a Genius Vanished
How could the inventor of the infinite series be completely
unknown to the global intellectual community until the late 20th century? Three
factors: physical loss, institutional isolation, and colonial historiography.
First, none of Madhava's original manuscripts survived. The
Kerala School transmitted knowledge through guru-shishya parampara within
family compounds. When a master died, texts might be preserved or might decay
in the humid Kerala climate. What we know comes exclusively from successors who
explicitly credited "the great master."
Second, Madhava's works were never translated into Arabic or
Latin during the centuries when European calculus developed. The Kerala School
existed in effective isolation from Renaissance Europe. Knowledge traveled
slowly, and the transmission belts did not necessarily carry advanced
mathematics from Kerala to Paris.
Third, when European scholars wrote the history of
mathematics, they constructed a purely European story: Greeks to Renaissance to
Enlightenment. Non-European contributions were dismissed or omitted.
The most damning episode occurred in 1834, when English
civil servant Charles Whish published a paper in the Transactions of
the Asiatic Society demonstrating that the Kerala School had
anticipated Newton, Leibniz, and Taylor by centuries. The Western mathematical
establishment largely ignored it. As historian C.K. Raju argues: "Colonial
science had a specific agenda: to demonstrate European superiority.
Acknowledging Indian priority would have undermined the justification for
colonial rule."
Modern historians like C.T. Rajagopal, Kim Plofker, and
George Gheverghese Joseph have corrected the record. Some textbooks now use
terms like "Madhava-Leibniz series." But the erasure persists—most
finance textbooks mention neither Madhava nor Ramanujan.
The Jesuit Transmission Thesis: Did the Church Smuggle
the Math?
The most controversial question: Did Madhava's work directly
influence European calculus through Jesuit missionaries? The circumstantial
evidence is substantial enough to have convinced a vocal minority.
The Portuguese established Cochin in 1503—just miles from
Kerala School centers. The Jesuits were highly trained linguists and scientists
with explicit instructions to collect local knowledge. Christoph Clavius, who
led the Gregorian calendar reform of 1582, ordered missionaries to send back
astronomical data. Europe faced a navigation crisis (ships sinking due to
inability to calculate longitude) and a calendar crisis (the Julian calendar
had drifted). The Kerala School possessed the world's most accurate celestial
tables.
Proponents argue Jesuits extracted mathematical content,
stripped it of cultural context, and shipped it to Rome. These materials
entered Vatican and Paris libraries, where John Wallis, Isaac Barrow, and
ultimately Newton and Leibniz had access.
Why no direct evidence? Transmission theorists note that
16th-century information management was not transparent. Jesuits operated as
intelligence agents, routinely removing local names. By the time a technique
reached Europe, it was presented as an anonymous "new discovery."
Furthermore, as science historian Simon Schaffer notes: "The 17th century
had no norms of citation. Knowledge was a commodity to be hoarded, not a public
good shared transparently."
Mainstream Western historians remain skeptical, citing the
absence of a "smoking gun" Latin translation. Dr. Kim Plofker takes a
middle position: "It is possible some Kerala techniques reached Europe
indirectly. But the evidence for direct Jesuit transmission is not conclusive.
What is clear is that Madhava's work has been unjustly excluded
regardless."
The debate matters because it goes to the heart of
intellectual property. Either way, as one commentator put it: "The jury is
still out on whether Newton or Leibniz actually held translated copies. But the
charge of structural exclusion is completely valid. The historical record was
written by colonial powers who systematically minimized non-Western
science."
The Refinery Pipeline: How Knowledge Gets Melted Down and
Rebranded
The story of how Madhava's mathematics became
Black-Scholes's model follows a pattern repeated across multiple domains.
Consider three parallel cases.
Algebra and the "Arabic" Numeral Pipeline
Every child learns "Arabic numerals"—the decimal
place-value system with zero. But this was developed in India by mathematicians
like Brahmagupta in the 7th century CE. It traveled to Baghdad's House of
Wisdom, where Al-Khwarizmi wrote influential books. It then traveled to Europe
via Moorish Spain. By the time Fibonacci popularized it in 1202, it was already
called "Arabic numerals." The Indian origin was forgotten. As Dr.
Anil Menon notes: "The renaming is not innocent. When you call a system
'Arabic numerals,' you are claiming its origin—and erasing the Indians who
invented it."
Inoculation: The Erased Caste of Indian Variolators
Before Edward Jenner "invented" vaccination in
1796, a sophisticated system called variolation had been practiced for
centuries in India. Itinerant physicians collected scabs from mild smallpox
cases and introduced tiny amounts into healthy skin—inducing immunity. The
practice had public health protocols and quarantine systems.
In 1731, English surgeon Oliver Coult described the method
to the Royal Society. Lady Mary Wortley Montagu campaigned to introduce it to
Britain. When Jenner later discovered cowpox vaccination, Western medicine
framed his work as a singular European leap. Colonial medical boards banned
variolation as "dangerous native practice" while promoting Jennerian
vaccination—which differed primarily in using a less dangerous virus source.
Medical historian David Arnold notes: "The colonial state delegitimized
indigenous knowledge because acknowledging its effectiveness would have
threatened Western superiority."
Wootz Steel: The Metal That Built an Empire
Wootz steel, produced in southern India from the 6th century
BCE, was a hyper-high-carbon steel containing carbon nanotubes—unparalleled in
flexibility and hardness. British forces discovered this during the
Anglo-Mysore Wars against Tipu Sultan, whose wootz blades cut through British
steel.
Michael Faraday spent years analyzing Indian steel ingots,
trying to unlock the secret. While he could not perfectly replicate it, his
research contributed to the Bessemer Process (1856), the first inexpensive mass
steel production method. Once Sheffield became the steel capital, history was
rewritten: wootz steel became an "exotic primitive craft," not
intentional chemical engineering. As technology historian Dr. Priya Srinivasan
observes: "In each case—mathematics, medicine, metallurgy—the colonizing
power extracts knowledge, strips authorship, and presents it as either a
European discovery or a primitive precursor."
The Magic European Filter: A Field Guide
At this point, one might appreciate the system's efficiency.
The rules are simple:
If a non-Western civilization develops a technique, it is a
"cute empirical craft." Centuries of variolation? Exotic folklore.
Wootz steel? Accidental primitive metallurgy. Infinite series in Kerala? An
isolated curiosity.
If a European develops the same technique, it becomes
"Science™." Jennerian vaccination is a revolutionary breakthrough,
not a minor modification. Bessemer steel is an industrial miracle, not a
refinement of wootz. The Leibniz series is a triumph of European genius, not a
17th-century rediscovery.
The Footnote Rule of Asymmetry: If a European borrows from
ancient Greece, transmission is assumed—no smoking gun required. But if a
European coincidentally produces the same result as an Indian scholar centuries
earlier, any suggestion of transmission requires a signed, notarized receipt.
And the final irony: the descendants of the original
inventors are sold back their own knowledge, repackaged as foreign expertise.
Indian hospitals use Western vaccines derived from Indian variolation. Indian
steel mills use European processes derived from wootz. Indian quants in Mumbai
use Black-Scholes, whose mathematical engine was built in Kerala. As one
anonymous commentator put it: "Trillions of dollars are traded daily on
Wall Street using an equation that would choke without a 14th-century Kerala math
engine. But hey, let's keep calling them 'Arabic numerals' so nobody tracks the
IP address back to India. Excellent bookkeeping. No notes."
The Disembodied Ghost
There is something haunting about this trajectory. Madhava
sat on the Kerala coast gazing at stars, mapping the infinite. His mathematics
was embodied—tied to the cosmos, to planetary motion, to ritual calendars. The
infinite series was a way of approaching the divine.
That same mathematics now lives inside algorithms executing
thousands of trades per second—no relationship to cosmos or divinity. The
embodied knowledge has been disembodied, stripped of cultural skin, forced to
wear a Latin uniform, put to work in the engine rooms of global capitalism.
Philosopher Dipesh Chakrabarty writes: "We live inside
an intellectual matrix that claims to be a universal, self-made monument of
Western Enlightenment. But it is a vast, global museum of stolen and rebranded
souls."
Every time a financial analyst types
"=NORM.S.DIST(1.96,TRUE)" into a spreadsheet, they perform an
operation depending on infinite series from Kerala. They do not know this. They
do not need to. The machinery works regardless.
But the history matters—not because giving Madhava credit
would change the formula, but because the story of knowledge's travel tells us
something profound about power. Knowledge does not flow freely like water
seeking its level. It flows along channels carved by empire, by trade, by
violence, and by the quiet violence of erasure.
What Does It Mean to Say "Black-Scholes Was
Indian"?
Returning to the original provocation: what is actually
being claimed?
Not literal direct invention. No serious historian disputes
that Black, Scholes, and Merton built the specific financial framework—the
delta-hedging argument, risk-neutral valuation—that transformed Wall Street.
The Nobel Prize is not undeserved.
But the claim contains deeper truths. The truth of parallel
discovery: that Shanthakumar may have derived the same equation, only to be
lost because he published in the wrong journals. The truth of mathematical
ancestry: that the tools required to solve the equation—infinite series,
continuous probability, stochastic calculus—owe an immense debt to Indian
mathematicians, from Madhava to Ramanujan. And the truth of structural
exclusion: that the standard history systematically erased non-European contributions
through colonial assumptions and asymmetrical standards of evidence.
Professor C.K. Raju puts it bluntly: "The question is
not whether Black-Scholes was Indian. The question is why we are still having
this debate, why evidence available since the 1830s remains marginal to the
mainstream narrative. That tells you everything about how power operates in
knowledge production."
A more moderate view comes from Abel Prize winner Dr. S. R.
S. Varadhan: "Mathematics belongs to everyone. The fact that the Kerala
School discovered infinite series does not diminish Newton or Leibniz. But we
should tell the history accurately. We should credit the people who did the
work, regardless of where they lived."
The Unresolved Mystery
The Jesuit transmission thesis remains unresolved. The
Shanthakumar story remains unconfirmed. The full extent of Madhava's original
work remains unknown.
But the broader pattern is undeniable. The history of
science, as traditionally written, is a history of the powerful. The Global
South provided raw intellectual capital—mathematical ore, medical techniques,
material science—that was refined and branded in the West.
What would a truly decolonized history look like? It would
teach the infinite series as the Madhava series, not merely the Leibniz series.
It would teach the place-value decimal system as an Indian invention. It would
teach variolation as a sophisticated global technology. It would teach wootz
steel as intentional chemical engineering.
And it would teach the Black-Scholes model as what it is: a
financial application built on a mathematical foundation that owes an immense
debt to Indian scholars, from Madhava to Ramanujan, whose work was extracted,
anonymized, and rebranded by the institutional machinery of empire.
The engine is Indian. The chassis is American. The road is
global. And the driver—well, the driver is whoever has the keys to the
ignition.
Reflection
Reading across the entire arc—from Madhava's Kerala to
Black-Scholes's Wall Street, from the Jesuit debates to systematic erasure—one
conclusion is inescapable. The global intellectual economy operates on the same
logic as the global material economy: extraction, refinement, rebranding, with
profits accruing to those who control the final stage. The Global South
provides the ore; the West sells back the ship.
This is not a conspiracy. It is a structure—accumulated over
centuries, reinforced by colonial assumptions and asymmetrical standards of
evidence. No single actor is to blame. But the structure produces predictable
outcomes: non-Western innovators remain obscure while European figures receive
the laurels.
The irony is that the mathematical engine powering Western
finance was built by scholars who would have been horrified by its application.
Madhava gazed at stars seeking cosmic harmony. Ramanujan received visions from
a goddess. Their work has been disembodied, stripped of context, repurposed for
speculative trading.
Perhaps the deepest lesson is that knowledge is never
neutral. The mathematics we use carries the fingerprints of its creators, even
when deliberately erased. To use Black-Scholes is to participate in a
five-hundred-year conversation stretching from Kerala to Chicago. The ghosts
are in the machine. And if we listen carefully, we can hear them whispering—in
Sanskrit, in Malayalam, in the silent language of infinite series—that the
history of science is far stranger, and far more global, than our textbooks admit.
References
Arnold, D. (1993). Colonizing the Body. UC
Press.
Black, F., & Scholes, M. (1973). Journal of Political
Economy, 81(3), 637–654.
Chakrabarty, D. (2000). Provincializing Europe.
Princeton UP.
Gheverghese Joseph, G. (2000). The Crest of the
Peacock. Princeton UP.
Merton, R. C. (1973). Bell Journal, 4(1), 141–183.
Plofker, K. (2009). Mathematics in India.
Princeton UP.
Raju, C. K. (2007). Cultural Foundations of
Mathematics. Pearson.
Whish, C. M. (1834). Transactions of the Asiatic Society,
3(2), 509–523.
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