166 CHAPTER 5. BROWNIAN MOTION
agreement. The Central Limit Theorem provides a reason to believe
the agreement, assuming the requirements of the Central Limit The-
orem are met, including independence. Unfortunately, although the
figure shows what appears to be reasonable agreement a more rigorous
statistical analysis shows that the data distribution does not match
normality.
Another good reason for still using the assumption of normality for the
increments is that the normal distribution is easy to work with. The
probability density is easy to work with, the cumulative distribution is
tabulated, the moment-generating function is easy to use, and the sum
of independent normal distributions is again normal. A substitution
of another distribution is possible but makes the resulting stochastic
process models very difficult to work with, and beyond the scope of
this treatment.
Moreover, this assumption ignores the small possibility that negative
stock prices could result from a large negative change. This is not
reasonable (and the log normal distribution from geometric Brownian
motion which avoids this possibility is a better model).
Moreover, the assumption of a constant variance on different intervals
of the same length is not a good assumption since stock volatility itself
seems to be volatile. That is, the variance of a stock price changes and
need not be proportional to the length of the time interval.
- The assumption of independent increments seems to be a reasonable as-
sumption, at least on a long enough term. From second to second, price
increments are probably correlated. From day to day, price increments
are probably independent. Of course, the assumption of independent
increments in stock prices is the essence of what economists call the
Efficient Market Hypothesis, or the Random Walk Hypothesis, which
we take as a given in order to apply elementary probability theory.
- The assumption ofW(0) = 0 is simply a normalizing assumption and
needs no discussion.
- The assumption of continuity is a mathematical abstraction, but it
makes sense, particularly if securities are traded minute by minute, or
hour-by hour where prices could jump discretely, but then examined
