The Seducing Simplicity of Gaussian Models

This is a great illustration from Paul De Grauwe, Leonardo Iania and Pablo Rovira Kaltwasser of the dangers of using the wrong probability distributions in a statistical model. In eurointelligence.com (ht Yves Smith, emphasis is mine):

October 2008 was certainly a spectacular month in the stock markets. Large daily changes occurred that surprised most investors. Yet, although many investors had not seen such wild gyrations of stock prices for a long time, there was a general sense that this had happened before.

Those of us who studied modern finance theory, however, were truly astonished by the sheer improbability of the events occurring in the stock markets during that fateful month. One of the basic assumptions used in almost all our finance models is that returns are normally distributed. These models are widely used to price derivatives and other complex financial products. What do these models tell us about the probabilities of the events that occurred in October?

The following table gives an answer. We selected the six largest daily percentage changes in the Dow Jones Industrial Average during October, and asked the question of how frequent these changes occur assuming that, as is commonly done in finance models, these events are normally distributed. The results are truly astonishing. There were two daily changes of more than 10% during the month. With a standard deviation of daily changes of 1.032% (computed over the period 1971-2008) movements of such a magnitude can occur only once every 73 to 603 trillion billion years. Since our universe, according to most physicists, exists a mere 20 billion years we, finance theorists, would have had to wait for another trillion universes before one such change could be observed. Yet it happened twice during the same month. A truly miraculous event. The other four changes during the same month of October have a somewhat higher frequency, but surely we did not expect these to happen in our lifetimes.

Trying to explain these deviations as just a few outliers in an otherwise well-behaving Gaussian process just doesn't cut it. A simple visual comparison of these time series is enough:

Figure 1: Dow Jones Industrial Average 1928-2008

Figure 2: Random Normal Process