**Black-Scholes: The maths formula linked to the financial crash**

It's not every day that someone
writes down an equation that ends up changing the world. But it does happen
sometimes, and the world doesn't always change for the better. It has been
argued that one formula known as Black-Scholes, along
with its descendants, helped to blow up the financial world.

Black-Scholes
was first written down in the early 1970s but its story starts earlier than
that, in the Dojima Rice Exchange in 17th Century

You can imagine why this kind of
contract might be useful. If I am running a big chain of hamburger restaurants,
but I don't know how much beef I'll need to buy next year, and I am nervous
that the price of beef might rise, well - all I need is to buy some options on
beef. But then that leads to a very ticklish problem. How much should I be
paying for those beef options? What are they worth? And that's where this
world-changing equation, the Black-Scholes formula,
can help. "The problem it's trying to solve is to define the value of the
right, but not the obligation, to buy a particular asset at a specified price,
within or at the end of a specified time period," says Professor Myron Scholes, professor of finance at the Stanford University
Graduate School of Business and - of course - co-inventor of the Black-Scholes formula.

The young Scholes
was fascinated by finance. As a teenager, he persuaded his mother to set up an
account so that he could trade on the stock market. One of the amazing things
about Scholes is that throughout his time as an
undergraduate and then a doctoral student, he was half-blind. And so, he says,
he got very good at listening and at thinking. When he was 26, an operation
largely restored his sight. The next year, he became an assistant professor at
MIT, and it was there that he stumbled upon the option-pricing puzzle. One part
of the puzzle was this question of risk: the value of an option to buy beef at
a price of - say - $2 (£1.23) a kilogram presumably depends on what the price
of beef is, and how the price of beef is moving around. But the connection
between the price of beef and the value of the beef option doesn't vary in a
straightforward way - it depends how likely the option is to actually be used. That
in turn depends on the option price and the beef price. All the variables seem
to be tangled up in an impenetrable way.

Scholes worked on the problem with his
colleague, Fischer Black, and figured out that if I own just the right
portfolio of beef, plus options to buy and sell beef, I have a delicious and
totally risk-free portfolio. Since I already know the price of beef and the
price of risk-free assets, by looking at the difference between them I can work
out the price of these beef options. That's the basic idea. The details are
hugely complicated. "It might have taken us a year, a year and a half to
be able to solve and get the simple Black-Scholes formula,"
says Scholes. "But we had the actual underlying
dynamics way before."

The Black-Scholes
method turned out to be a way not only to calculate value of options but all
kinds of other financial assets. "We were like kids in a candy story in
the sense that we described options everywhere, options were embedded in
everything that we did in life," says Scholes. But
Black and Scholes weren't the only kids in the candy
store, says Ian Stewart, whose book argues that Black-Scholes
was a dangerous invention. "What the equation did was give everyone the
confidence to trade options and very quickly, much more complicated financial
options known as derivatives," he says.

Scholes thought his equation would be
useful. He didn't expect it to transform the face of finance. But it quickly
became obvious that it would. "About the time we had published this
article, that's 1973, simultaneously or approximately a month thereafter, the
Chicago Board Options Exchange started to trade call options on 16 stocks,"
he recalls. Scholes had just moved to the

The Black-Scholes
formula had passed the market test. But as banks and hedge funds relied more
and more on their equations, they became more and more vulnerable to mistakes
or over-simplifications in the mathematics. After Black-Scholes
it was the computer that said yes, or no. "The equation is based on the
idea that big movements are actually very, very rare. The problem is that real
markets have these big changes much more often that this model predicts,"
says Stewart. "And the other problem is that everyone's following the same
mathematical principles, so they're all going to get the same answer." Now
these were known problems. What was not clear was whether the problems were
small enough to ignore, or well enough understood to fix. And then in the late
1990s, two remarkable things happened.

Fischer Black died young, in 1995.
When in 1997 Scholes won the Nobel memorial prize, he
shared it not with Black but with Robert Merton, another option-pricing expert.
Scholes' work had inspired a
generation of mathematical wizards on Wall Street, and by this stage
both he and Merton were players in the world of finance, as partners of a hedge
fund called Long-Term Capital Management. "The whole idea of this company
was that it was going to base its trading on mathematical principles such as
the Black-Scholes equation. And it actually was
amazingly successful to begin with," says Stewart. "It was
outperforming the traditional companies quite noticeably and everything looked
great."

But it didn't end well. Long-Term
Capital Management ran into, among other things, the Russian financial crisis.
The firm lost $4bn (£2.5bn) in the course of six weeks. It was bailed out by a
consortium of banks which had been assembled by the Federal Reserve. And - at
the time - it was a very big story indeed. This was all happening in August and
September of 1998, less than a year after Scholes had
been awarded his Nobel prize. Stewart says the lessons
from Long-Term Capital Management were obvious. "It showed the danger of
this kind of algorithmically-based trading if you don't keep an eye on some of
the indicators that the more conventional people would use," he says.
"They [Long-Term Capital Management] were committed, pretty much, to just
ploughing ahead with the system they had. And it went wrong."

Scholes says that's not what happened at
all. "It had nothing to do with equations and nothing to do with
models," he says. "I was not running the firm,
let me be very clear about that. There was not an ability to withstand the
shock that occurred in the market in the summer and fall of late 1998. So it
was just a matter of risk-taking. It wasn't a matter of modelling." This
is something people were still arguing about a decade later. Was the collapse
of Long-Term Capital Management an indictment of mathematical approaches to
finance or, as Scholes says, was it simply a case of
traders taking too much risk against the better judgement of the mathematical
experts?

Ten years after the Long-Term
Capital Management bail-out, Lehman Brothers collapsed. And the debate over
Black-Scholes and LTCM is now a broader debate over
the role of mathematical equations in finance. Ian Stewart claims that the
Black-Scholes equation changed the world. Does he
really believe that mathematics caused the financial crisis? "It was abuse
of their equation that caused trouble, and I don't think you can blame the
inventors of an equation if somebody else comes along and uses it badly,"
he says. "And it wasn't just that equation. It was a whole generation of
other mathematical models and all sorts of other techniques that followed on
its heels. But it was one of the major discoveries that opened the door to all
this."

Black-Scholes
changed the culture of Wall Street, from a place where people traded based on
common sense, experience and intuition, to a place where the computer said yes or
no. But is it really fair to blame Black-Scholes for
what followed it? "The Black-Scholes technology
has very specific rules and requirements," says Scholes.
"That technology attracted or caused investment banks to hire people who
had quantitative or mathematical skills. I accept that. They then developed
products or technologies of their own." Not all of those subsequent
technologies, says Scholes, were good enough.
"[Some] had assumptions that were wrong, or they used data incorrectly to
calibrate their models, or people who used [the] models didn't know how to use
them." Scholes argues there is no going back.
"The fundamental issue is that quantitative technologies in finance will
survive, and will grow, and will continue to evolve over time," he says.

But for Ian Stewart, the story of
Black-Scholes - and of Long-Term Capital Management -
is a kind of morality tale. "It's very tempting to see the financial
crisis and various things which led up to it as sort of the classic Greek
tragedy of hubris begets nemesis," he says. "You try to fly, you fly
too close to the sun, the wax holding your wings on melts and you fall down to
the ground. My personal view is that it's not just tempting to do that but
there is actually a certain amount of truth in that way of thinking. I think
the bankers' hubris did indeed beget nemesis. But the big problem is that it
wasn't the bankers on whom the nemesis descended - it was the rest of us."

*27 April 2012 Last updated at 23:06 GMT *

*By Tim Harford BBC Radio ,
additional reporting by Richard Knight*