Introduction to Time Series Analysis 111
Application to S&P 500 Index Returns As an example to illustrate equation
(5.4), consider the daily S&P 500 stock index prices between November 3,
2003, and December 31, 2003. The values are given in Table 5.2 along with
the daily price changes. The resulting plot is given in Figure 5.4. The intu-
ition given by the plot is roughly that, on each day, the information influenc-
ing the following day’s price is unpredictable and, hence, the price change
seems completely arbitrary. Hence, at first glance much in this figure seems
to support the concept of a random walk. Concerning the evolution of the
underlying price process, it looks reasonable to assume that the next day’s
price is determined by the previous day’s price plus some random change.
From Figure 5.4, it looks as if the changes occur independently of each other
and in a manner common to all changes (i.e., with identical distribution).
Error Correction
We next present a price model that builds on the relationship between spot
and forward markets. Suppose we extend the random walk model slightly
by introducing some forward price for the same underlying stock S. That is,
at time t, we agree by contract to purchase the stock at t + 1 for some price
determined at t. We denote this price by F(t). At time t + 1, we purchase the
stock for F(t). The stock, however, is worth St+ 1 at that time and need not—
and most likely will not—be equal to F(t). It is different from the agreed
forward price by some random quantity εt+ 1. If this disturbance has zero
mean, as defined in the random walk model, then the price is fair. Based on
this assumption, the reasonable forward price would equal^9
Ft()==ES[|tt+ 1 tE][St+=εtt|]SSo, on average, the difference between S and F should fulfill the following
condition:
∆≡SFt+ 1 − ()t ≈ 0If, however, the price process permits some constant terms such as some
upward trend, for example, the following period’s price will no longer be
equal to this period’s price plus some random shock. The trend will spoil
the fair price, and the forward price designed as the expected value of the
(^9) Note that we employ expected values conditional on time t to express that we base
our forecast on all information available at time t.