The theory and practice of finance draws heavily on probability theory. All
MBA programs prepare finance majors for their career in the profession by
requiring one generalist course in probability theory and statistics attended by
all business majors. While several probability distributions are covered in the
course, the primary focus is on the normal or Gaussian distribution.
Students find it easy to understand and apply the normal distribution:
Give them the expected value and standard deviation and probability
statements about outcomes can be easily made. Moreover, even if
a random variable of interest is not normally distribution, students are
told that a theorem in statistics called the
Central Limit Theorem
proves
that under certain conditions the sum of independent random variables
will be asymptotically normally distributed. Loosely speaking, this
means that as the number of random variables are summed, the sum
will approach a normal distribution.
Armed with this rudimentary knowledge of probability theory, finance
students march into their elective courses in finance that introduce them to
the quantitative measures of risk (the standard deviation) and the quantitative
inputs needed to implement modern portfolio theory (the expected
value or mean and the standard deviation). In listing assumptions for most
theories of finance, the first assumption on the list is often: “Assume asset
returns are normally distributed.” The problem, however, is that empirical
evidence does not support the assumption that many important variables in
finance follow a normal distribution. The application of the Central Limit
Theorem to such instances is often inappropriate because the conditions
necessary for its application are not satisfied.
And this brings us to the purpose of this book. Our purpose is fourfold.
First, we explain alternative probability distributions to the normal
distributions for describing asset returns as well as defaults. We
focus on the stable Paretian (or alpha stable) distribution because of the
strong support for that distribution that dates back four decades to the
seminal work of Benoit Mandelbrot. Second, we explain how to estimate
distributions. Third, we present empirical evidence rejecting the
hypothesis that returns for stocks and bonds are normally distributed
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and instead show that they exhibit fat tails and skewness. Finally, we
explain the implications of fat tails and skewness to portfolio selection,
risk management, and option pricing.
We must admit that our intent at the outset was to provide a “nontechnical”
treatment of the topic. However, we could not do so. Rather, we
believe that we have provided a less technical treatment than is provided in
the many excellent books and scholarly articles that deal with probability
and statistics applied to finance and risk management. The book is not simple
reading. It must be studied to appreciate the pitfalls that result from the
application of the normal distribution to real-world financial problems.
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