Thales of Miletus used a money-spinning device which,
though it was ascribed to his prowess as a philosopher, is in
principle open to anybody. The story is as follows: people had
been saying reproachfully to him that philosophy was useless, as
it had left him a poor man. But he, deducing from his knowledge of
the stars that there would be a good crop of olives, while it was
still winter and he had little a money to spare, used it to pay
deposits on all the oil-presses in Miletus and Chios, thus
securing their hire. This cost him only a small sum, as there were
no other bidders. Then the time of the olive-harvest came, and as
there was a sudden and simultaneous demand for oil-presses he
hired them out at any price he liked to ask. He made a lot of
money, and so demonstrated that it is easy for philosophers to
become rich, if they want to; but that is not their objective in
life.1.1
Derivative contracts in general and options in particular are not
novel securities. It has been nearly 25 centuries since the above
abstract appeared in Aristotle's Politics, describing the purchase
of a call option on oil-presses. More recently, De La Vega (1688),
in his account of the operation of the Amsterdam Exchange,
describes traded contracts that exhibit striking similarities to
the modern traded options.1.2
Nevertheless, the modern treatment of derivative contracts has its
roots in the inspired work of the Frenchman Louis Bachelier in
1900. This was the first attempt of a rigorous mathematical
representation of an asset price evolution through time. Bachelier
used the concepts of random walk in order to model the
fluctuations of the stock prices, and developed a mathematical
model in order to evaluate the price of options on bond futures.
Although the above model was incomplete and based on assumptions
that are virtually unacceptable in recent studies, its importance
lies on the novelty of its ideas, both from an economist's and a
mathematician's point of view.1.3 Unfortunately, this work was not developed
further, despite the publication of the Einstein paper on Brownian
motion in 1905, which would shed light on the properties of the
model and perhaps highlight its misspecifications.
The above treatment of security prices was long forgotten until
the 70s, when Professor Samuelson and his co-workers at MIT
rediscovered Bachelier's work and questioned its underlying
assumptions. By construction, the payoff of a call option on
the expiration day will depend on the price of the underlying
asset on that day, relative to the option's exercise price. Common
reasoning declares that therefore, the price of the call option
today has to depend on the probability of the stock price
exceeding the exercise price. One could then argue that a
mathematical model that can satisfactory explain the underlying
asset's price is sufficient in order to price the call option
today, just by constructing the probabilistic model of the price
on the expiration day. Professors Black, Merton and Scholes
recognized that the above reasoning is incorrect: Since
today's price incorporates the probabilistic model of the
future behavior of the asset price, the option can (and has to) be
priced relative to today's price alone. They realized that
a levered position, using the stock and the riskless bond, that
replicates the payoff of the option is feasible, and therefore the
option can be priced using no-arbitrage restrictions.
Equivalently, they observed that the true probability
distribution for the stock price return can be transformed into
one which has an expected value equal to the risk free rate, the
so called risk adjusted or risk neutral
distribution; the pricing of the derivative can be carried out
using the risk neutral distribution when expectations are taken.
The classic papers produced by this work, namely Black and Scholes
(1973) and Merton (1976), triggered an avalanche of papers on
option pricing, and resulted in the 1997 Nobel Prize in economics
for the pioneers of contingent claims pricing. Even today, nearly
thirty years after its publication, the original Black and Scholes
paper is one of the most heavily cited in finance.
Kyriakos
2003-03-17
