Claiming that the Black-Scholes equation had anything to do with the
Credit Crisis of 2007-2009 is a bit like claiming Daimler-Maybach were
responsible for bombing Hiroshima. Sure the planes used the internal
combustion engine, but the causal relationship is being stretched beyond
reason.

The claim that Black-Scholes was involved in the Credit Crisis has been made
by Ian Stewart in a piece, apparently promoting his new book,

*Seventeen Equations that Changed the World*, in*The Observer*.
Prof Stewart is the most important promoter of mathematics in the United
Kingdom, writing his ﬁrst book in the early 1970s,

*Concepts of Modern Mathematics*, as an exposition of Bourbaki mathematics. In the 1990s, Bourbaki and ‘New Math’ had faded/failed and Prof Stewart turned his attention tho the more pressing question of introducing mathematics into industry, as explained in the Preface to the Dover Publications edition.
Given this track-record I am at something of a loss to understand what Prof
Stewart was trying to achieve in his

*Observer*article, beyond generating publicity for his book. It is unfortunate that he has chosen to approach a critically important issue, the use of mathematics in ﬁnance, in such an unthoughtful way as he has.
His article is little more than a string of factually incorrect statements,
my guess is they are culled from the BBC’s

*Midas Formula*which in turn is based on Lowenstein’s*When Genius Failed*. For example, the piece begins with “It was the holy grail of investors.”. At its heart is the rather depressing statement, from an investors perspective, that proﬁts should equal the riskless rate, that's more a poison chalice than a holy grail - as one student asked in lectures last week “what's the point of that then”.
What is more, if it was an esoteric secret, why was the paper initially rejected
by the Journal of Political Economy on the basis that there was not enough
economics in it. Peter Bernstein, in

*Against the Gods*reports that Black felt it was because he was a mathematician and had no qualiﬁcations in economics. The paper then went to the Review of Economics and Statistics, again without success. Bernstein reports that the paper was only published by the Journal of Political Economy after the intervention of inﬂuential Chicago academics.
A more thoughtful assessment is that Black-Scholes enabled the CME
and CBOT to justify the introduction of ﬁnancial options trading, and
this is touched upon. But there is no accompanying explanation of the
collapse of Bretton-Woods, destroying ﬁxed exchange rate and broadly ﬂat
bond yields, meaning that volatility suddenly became an issue for the
markets.

The article, then links Black-Scholes with the sub-prime crisis, a gross
mis-representation since Black-Scholes played pretty much no (pricing) role in the
collapse of LTCM, let alone in the credit crisis. What Stewart fails miserably to
understand is that, in typical practice since the 1987 crash, Black-Scholes has not
taken volatility as an input and produced prices, but taken prices to imply
volatility. The failure is miserable, because this is the genius of the equation, a
consistent measurement tool of the markets’ assessment of risk in the
future.

The piece goes on to claim “The idea behind many ﬁnancial models goes
back to Louis Bachelier in 1900”. Well, Prof Stewart should know better. It would
be better to write that “The idea behind many mathematical models goes back to
the ﬁnancial mathematician, Fibonacci in 1202” or “The origins of the
Black-Scholes formula lie in the Pascal–Fermat solution to the Problem of Points
in 1654. The generally accepted origin of mathematical probability”. These are far
more interesting points.

The article continues

The Black-Scholes equation was based on arbitrage pricing theory, in which both drift and volatility are constant. This assumption is common in ﬁnancial theory, but it is often false for real markets.

This is nonsense, arbitrage pricing is a concept, constant drift and volatility are
implementations. Stewart is happy to talk about the economic-physics
Black-Scholes equation but seems ignorant of the mathematical

**Fundamental Theorem of Asset Pricing**A market is free from arbitrage if and only if a martingale measure exists.

A market is complete and free from arbitrage if and only if a unique martingale measure exists.

Why oh why didn’t Prof Stewart talk about this, admittedly not an equation, but
profound mathematics that tells us something signiﬁcant about markets.

Mathematics, in my humble opinion, is not equations, it is concerned with ideas and understanding. Understanding
the Fundamental Theorem requires an appreciation that probability is not relative
frequency, of epistemology (Black Swans), through the idea of completeness,
and ethics, being arbitrage free is about fairness, a martingale measure
is about equality. This is all in Aristotle and Aquinas, as described by
the mathematician James Franklin, and all in ﬁnancial mathematics,
where contemporary papers include words like ‘belief’ and ‘greed’ in their
title.

Stewart ﬁnishes his article by suggesting the solution lies in the mathematics
of dynamical systems and complexity and adopting analogues from ecology. The
Bank of England has followed this line of thinking building on work of Lord May,
the biologist and past President of the Royal Society. The problem is, Lord May’s
analysis is based on the model of ‘contagion’, the banking system is an ecology
through which default spreads, in the same way that mad-cow disease spreads
through farms. The response is to build ﬁrewalls, quarantine zones, around banks.
The Bank of England seems reluctant to develop this line of thinking,
possibly because they now realise that the model is just plain wrong. A more
appropriate model for banking is the internet, the Credit Crisis was a result of
linkages in a network collapsing not of some invisible infectious agent
spreading throughout the network. The preferred model now seems to be
electricity grids (the article was apparently prompted by a discussion with
Andy Haldane). So the Bank is still using a physical analogue, but a better analogue.

Stewart would see the solution in dynamical systems and complexity, because
that is an area of mathematics that the British are strong in. But, dynamical
systems are typically ergodic (in the 2003 Review, “dynamical systems and
complexity” was labelled “dynamical systems and ergodic theory”) , and,
unfortunately, the economy is not ergodic.

Mathematicians must start taking the economy seriously, and not try and
shoe-horn economic problems into a framework of the mathematics they
understand. Ian Stewart is extraordinarily inﬂuential in deﬁning what
mathematics is and how it can be applied. Academics should not publicly discuss
topics they are not expert on, that is the realm of journalism (or blogs). In this
article, Prof Stewart has misrepresented mathematics, damaging its reputation.
For this he should be ashamed.