# Arbitrage, Hedging, and the Law of One Price

- Nov 4, 2005

## Mispricing, Convergence, and Arbitrage

Arbitrage
exploits
violations
of the
Law
of One
Price
by buying
and
selling
assets,
separately
or in
combination,
that
should
be priced
the
same
but
are
not.
Implicit
in an
arbitrage
strategy
is the
expectation
that
the
prices
of the
misvalued
assets
will
ultimately
move
to their
appropriate
values.
Indeed,
arbitrage
should
push
prices
to their
appropriate
levels.
Thus,
an arbitrage
strategy
has
two
key
aspects:
execution
and
convergence. *Execution* includes
how
the
arbitrage
opportunity
is identified
in the
first
place,
how
the
strategy
is put
together,
how
it is
maintained
over
its
life,
and
how
it is
ultimately
closed
out. *Convergence* is
the
movement
of misvalued
asset
prices
to their
appropriate
values.^{15} Of
particular
importance
are
the *time
frame* over
which
convergence
is expected
to occur
and
the *process* driving
the
convergence.
These
two
are
the
primary
factors
that
determine
the
design
of the
appropriate
arbitrage
strategy
in a
given
situation.

The processes driving convergence fall into two categories: mechanical or
absolute, and behavioral or correlation. A *mechanical or absolute
convergence process* has an explicit link that forces prices to converge
over a well-defined time period. An example is index arbitrage, in which the
futures price of an index is mechanically linked to the spot (cash) value of the
index through the cost-of-carry pricing relation. This is examined in Chapter 3,
"Cost of Carry Pricing." In index arbitrage, the convergence time period
is deterministically dictated by the delivery/expiration date of the index
futures contract.

A *behavioral *or* correlation convergence process* exists when
there is historical evidence of a systematic relationship or a correlation in
the behavior of the assets’ prices. However, the mispriced assets fall
short of being linked mechanically. An example of a behavioral or correlation
convergence process is pairs trading. Pairs trading identifies two stocks that
have historically tended to move closely, as measured by the average spread
between their prices. It is common to identify pairs of stocks that are highly
correlated in large part due to being in the same industry. The essence of this
strategy is to identify pairs whose spreads are significantly higher or lower
than usual and then sell the higher-priced stock and buy the lower-priced stock
under the expectation that the spread will eventually revert to its historical
average. Thus, pairs trading relies on an estimated correlation and projected
convergence toward the historical mean spread. Importantly, no mechanical link
guarantees this convergence, and no deterministic model indicates how long such
convergence should take. Although they are commonly referred to as arbitrage,
behavioral/correlation convergence process-based strategies are not true
arbitrage, because they can be quite risky. This book is concerned primarily
with mechanical/absolute convergence process-based arbitrage because that is the
fertile soil from which modern finance has grown.