Behavior of asset prices

Efficient markets hypothesis
The efficient market approach to explaining asset prices views them as the present
values of the income streams they generate. Efficient market theory implies that all
available information regarding future asset prices is impounded in current asset
prices. It provides a useful starting point for analyzing derivatives.
One implication of market efficiency is that asset returns follow a random walk.
The motion of the asset price has two parts, a drift rate, that is, a deterministic rate
at which the asset price is expected to change over time, and a variance rate, that
is, a random change in the asset price, also proportional to the time elapsed, and
also unobservable. The variance rate has a mean of zero and a per-period variance
equal to a parameter p, called the volatility. This assumption implies that the
percent changes in the asset price are normally distributed with a mean equal to the
drift rate and a variance equal to p2.
The random walk hypothesis is widely used in financial modeling and has several
implications:
Ω The percent change in the asset price over the next time interval is independent
of both the percent change over the last time interval and the level of the asset
price. The random walk is sometimes described as ‘memoryless’ for this reason.
There is no tendency for an up move to be followed by another up move, or by a
down move. That means that the asset price can only have a non-stochastic trend
equal to the drift rate, and does not revert to the historical mean or other ‘correct’
level. If the assumption were true, technical analysis would be irrelevant.
Ω Precisely because of this lack of memory, the asset price tends over time to wander
further and further from any starting point. The proportional distance the asset
price can be expected to wander randomly over a discrete time interval q is the
volatility times the square root of the time interval, p q.
Ω Asset prices are continuous; they move in small steps, but do not jump. Over a
given time interval, they may wander quite a distance from where they started,
but they do it by moving a little each day.
Ω Asset returns are normally distributed with a mean equal to the drift rate and a
standard deviation equal to the volatility. The return distribution is the same each
period.
The Black–Scholes model assumes that volatility can be different for different asset
prices, but is a constant for a particular asset. That implies that asset prices are
homoskedastic, showing no tendency towards ‘volatility bunching’. A wild day in
the markets is as likely to be followed by a quiet day as by another wild day.
An asset price following geometric Brownian motion can be thought of as having
an urge to wander away from any starting point, but not in any particular direction.
The volatility parameter can be thought of as a scaling factor for that urge to wander.
Figure 1.1 illustrates its properties with six possible time paths over a year of an
asset price, the sterling–dollar exchange rate, with a starting value of USD1.60, an
annual volatility of 12%, and an expected rate of return of zero.
Empirical research on asset price behavior
While the random walk is a perfectly serviceable first approximation to the behavior
of asset prices, in reality, it is only an approximation. Even though most widely
traded cash asset returns are close to normal, they display small but important ‘nonnormalities’.
In particular, the frequency and direction of large moves in asset prices,
which are very important in risk management, can be quite different in real-life
markets than the random walk model predicts. Moreover, a few cash assets behave
very differently from a random walk.
The random walk hypothesis on which the Black–Scholes model is based is a good
first approximation to the behavior of most asset prices most of the time. However,
even nominal asset returns that are quite close to normally distributed display small
but important deviations from normality. The option price patterns discussed below
reveal how market participants perceive the distribution of future asset prices.
Empirical studies of the stochastic properties of nominal returns focus on the
behavior of realized asset prices. The two approaches largely agree.