APPLICATION: JUMP-DIFFUSION

In the GARCH model, volatility changes gradually over time. In financial markets we do observe this sort of behavior, but we also see extreme events that seem to come out of nowhere. For example, on February 27, 2007, in the midst of otherwise calm markets, rumors that the Chinese central bank might raise interest rates, along with some bad economic news in the United States, contributed to what, by some measures, was a –8 standard deviation move in U.S. equity markets. A move of this many standard deviations would be extremely rare in most standard parametric distributions.

One popular way to generate this type of extreme return is to add a so-called jump term to our standard time series model. This can be done by adding a second disturbance term: