Chapter 3Important Probability Distributions

To model financial market risks, one needs to (1) understand how financial data behave and (2) select probability distributions that incorporate the important characteristics of the data. Chapter 2 reviewed useful statistical terminology, described the normal distribution, and introduced the concept of risk measures. This chapter presents further examples of probability distributions and continues the discussion with an emphasis on their use for simulation and risk management.

The normal distribution was and continues to be one of the most widely used distributions for modeling financial risk. It implicitly underlies Modern Portfolio Theory (MPT), which we discuss in Chapter 8, and continues to be used even in the calculation of tail-risk measures such as value-at-risk and conditional value-at-risk.1 However, it has been known for a long time (at least as far back as the influential papers of Mandelbrot (1963) and Fama (1963)) that extreme events in the financial markets happen a lot more frequently than the assumption that financial returns follow a normal distribution would lead one to believe. Observed characteristics of financial data include:2

  • The empirical distributions of financial returns are leptokurtic,3 or in other words they have “fat tails”4 compared to the tails of the normal distribution.
  • The empirical distribution of returns is left-skewed, that is, large negative returns are possible.
  • There is a difference in the ...

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