Options have existed—at least
in concept—since antiquity. It wasn't until publication of the
Black-Scholes (1973) option pricing formula
that a theoretically consistent framework for pricing options became available.
That framework was a direct result of work by Robert Merton as well as Black and
Scholes. In 1997, Scholes and Merton won the Nobel Prize in economics for this
work. Black had died in 1995, but otherwise would have shared the prize.
Option pricing theory—also
called Black-Scholes theory or derivatives pricing
theory—traces its roots to
Bachelier (1900) who
invented Brownian motion to model options on French government bonds. This work
anticipated by five years Einstein's independent use of Brownian motion
Research picked up in the 1960's. Typical of efforts during this
period is Samuelson (1965). He considered long-term
equity options, and
value of the option at
exercise. The model required two assumptions. The first was the
return α for the stock price. The second was
the rate β at which the option's value at exercise
should be discounted back to the pricing date.
These two factors depended upon the unique risk characteristics of,
respectively, the underlying stock and the option. Neither factor was
observable in the market place. Depending upon their degree of
different observers might propose different values for the factors. Accordingly,
Samuelson's formula was largely arbitrary. It offered no means for a buyer and
seller with different risk aversions to agree on a price for an option.
Black and Scholes got around the problem with a completely new
Consider an options trader who is about to sell an
option. She intends to dynamically hedge the exposure until the option expires.
What price should she charge for the option? Black and Scholes propose that she
charge the cost of dynamically hedging the short option. Significantly, given
certain simplifying assumptions, they proposed that this cost could be known in advance.
the real world, dynamic hedging is an uncertain undertaking. Depending upon the
underlier and your hedging strategy, you might adjust your hedge a few times a
week, or a few times a day. Between those adjustments, a lot can happen. If the
underlier moves, you might rehedge immediately, or you might wait. If you wait,
the underlier might move back, saving you the need to rehedge, or the underlier
might keep moving in the same direction, causing a large loss. Real world dynamic
hedging entails risk.
a practical matter, there is a limit to how frequently a trader can rehedge,
however, as the frequency of rehedging increases, dynamic hedging becomes more
predictable. Using stochastic calculus and certain simplifying assumptions,
Black and Scholes took the limiting case as the frequency of rehedging
approaches infinity. In that limiting case, the cost of dynamic hedging is
independent of the actual path taken by the underlier's price. It depends only
upon the price's volatility. If that volatility is constant and known in
advance, the cost of dynamic hedging a
short option is certain. Being certain,
it entails no risk, so it can be discounted at a
risk free rate to obtain the
price of the option.
Based upon this approach, Black and Scholes derived a partial
differential equation for valuing claims contingent on a traded underlier. The
equation is general. By applying different boundary conditions, it can be solved
to price any such contingent claim. Black and Scholes applied the boundary
conditions for a European call option on a
non-dividend-paying stock and
obtained their famous (1973) option pricing formula.
C. Cox and Stephen A. Ross made an important contribution with their
method of risk neutral valuation.
Consider again Samuelson's (1965) approach to pricing options, where he modeled
an underlying stock price with some expected return
α and discounted option values at exercise back to the pricing date with some
rate β. There was nothing theoretically wrong with such an approach. Indeed, if
it were based upon the same assumptions as the Black-Scholes approach,
it would produce consistent option prices. This lead Cox and Rubinstein to an
interesting conclusion. The two approaches were equivalent, yet one required α
and β as inputs whereas the other did not. They concluded that the effects
of α and β must somehow cancel. As long as α and
β reflect the same degree of risk aversion, they do not affect the option
price—and this must be true
no matter what degree of risk aversion they reflect. If they can reflect any degree of risk aversion and still yield correct option prices,
then they can be based upon an assumption of no risk aversion whatsoever. If
an investor were risk neutral, he would require no excess return for taking
risk. He would discount all cash flows—irrespective of their risk—at the risk
free rate. The factors α and β would both equal the risk free rate. This brilliant
insight was the Cox and Rubinstein risk neutral approach to option pricing.
The risk neutral approach opened the door to a host of
option valuation techniques that used binomial trees or the Monte Carlo
method to model future asset values. Rather than attempt to ascribe
"realistic" expected returns and "realistic" discount rates in the
analyses, users could treat all financial assets as having expected
returns equal to the risk free rate. They could discount all cash flows at
the risk free rate. The risk neutral assumption is not reflective of the
real world. Real investors are not risk neutral, but this doesn't matter.
Correctly implemented, the risk neutral assumption produces correct
Cox and Ross didn't immediately perceive how profound risk neutral
valuation would be. They buried it in the middle of a (1976)
paper on pricing options with jump processes. But three years later, they
teamed up with
Mark Rubinstein to publish a (1979)
paper that used risk neutral valuation to develop the method of binomial
trees. The mathematics of risk neutral valuation was formalized in
continuous time by other authors to become the method of equivalent
martingale measures. Today, this is the predominant
methodology for derivatives
pricing in complete markets.
Work on financial engineering spawned the field of
financial engineering. Practitioners, called financial engineers,
design and implement derivatives pricing models. Top financial engineers
are highly paid professionals who typically hold advanced degrees in
mathematics or physics. Financial engineers are informally called
quants or rocket
The Black Scholes approach and generalizations employ
partial differential equations, so they are sometimes called the
differential equations approach.
Those differential equations often have
closed-form solutions, leading to
simple pricing formulas such as the original Black-Scholes (1973) formula.
Other times, the differential equations need to be solved numerically
using techniques such as the Monte Carlo
The risk neutral approach tends to entail extensive use of
stochastic calculus with changes of measure between a "real world" and a
"risk neutral" world. For this reason, it (and analogous approaches) tend
to be called the stochastic
calculus approach. It can lead to closed form solutions, but
numerical solutions tend to be the norm. It is more flexible than the
Black-Scholes approach. Sometimes, it can be used to price derivatives
that the Black-Scholes approach cannot.
Techniques of financial engineering have been extended to
fixed income derivatives, which generally require the modeling of entire
term structures. They have also been extended to commodities markets,
where risk neutral valuation becomes problematic.
To learn about the historical origins
of option pricing theory, Bernstein (1993)
offers a nice overview of 20th century finance. Mehrling (2005)
and Lehman (2004)
offer more detailed accounts of Fischer Black and his associates'
contributions. Haug (1997)
is a handy encyclopedia of published option pricing formulas.
is an excellent introduction to option pricing
theory and financial engineering. It is a bit dated, presenting
the subject more as it was first developed by Black, Scholes,
Merton and contemporaries. Once you have read Chriss, proceed to
more modern treatments by Back (2005)
and Bingham and Kiesel (2004).
Bachelier, Louis (1964). Theory of
Speculation, The Random Character of Stock Prices, Paul H.
Cootner (editor), Cambridge: MIT. Translated from the 1900
Samuelson, Paul A. (1965). Rational
theory of warrant pricing, Industrial Management Review, 6,
Black, Fischer and
Myron S. Scholes (1973). The pricing of options and corporate
liabilities, Journal of Political Economy, 81, 637-654.
Merton, Robert C. (1973). Theory of
rational option pricing, Bell Journal of Economics and
Management Science, 4 (1), 141-183. Available in Merton
Cox, John C. and Stephen A. Ross
(1976). The valuation of options for alternative stochastic
processes, Journal of Financial Economics, 3, 145-166.
John C., Stephen A. Ross and Mark Rubinstein (1979). Option
Pricing: A Simplified Approach, Journal of Financial Economics
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