Chapter 15. Volatility, Implied Volatility, ARCH, and GARCH

In finance, we know that risk is defined as uncertainty since we are unable to predict the future more accurately. Based on the assumption that prices follow a lognormal distribution and returns follow a normal distribution, we could define risk as standard deviation or variance of the returns of a security. We call this our conventional definition of volatility (uncertainty). Since a normal distribution is symmetric, it will treat a positive deviation from a mean in the same manner as it would a negative deviation. This is against our conventional wisdom since we treat them differently. To overcome this, Sortino (1983) suggests a lower partial standard deviation. Most of the time, it is assumed that the volatility of a time series is a constant. Obviously this is not true. Another observation is volatility clustering, which means that high volatility is usually followed by a high-volatility period, and this is true for...