It is important for a financial institution to monitor the volatilities of the market variables (interest rates, exchange rates, equity prices, commodity prices, etc.) on which the value of its portfolio depends. This chapter describes the procedures it can use to do this.

 

The chapter starts by explaining how volatility is defined. It then examines the common assumption that percentage returns from market variables are normally distributed and presents the power law as an alternative. After that it moves on to consider models with imposing names such as exponentially weighted moving average (EWMA), autoregressive conditional heteroscedasticity (ARCH), and generalized autoregressive conditional heteroscedasticity (GARCH). The distinctive feature of these models is that they recognize that volatility is not constant. During some periods, volatility is relatively low, while during other periods it is relatively high. The models attempt to keep track of variations in volatility through time.

 

A variable’s volatility, σ, is defined as the standard deviation of the return provided by the variable per unit of time when the return is expressed using continuous compounding. (See Appendix A for a discussion of compounding frequencies.) When volatility is used for option pricing, the unit of time is usually one year, so that volatility is the standard deviation of the continuously compounded return per year. When volatility is used for risk management, the unit of time is usually one day so that volatility is the standard deviation of the continuously compounded return per day.

 

Define Si as the value of a variable at the end of day i. The continuously compounded return per day for the variable on day i is

 

Ln (Si/ Si−1)

 

This is almost exactly the same as

 

(Si  -  Si−1) / Si−1

 

An alternative definition of daily volatility of a variable is therefore the standard deviation of the proportional change in the variable during a day. This is the definition that is usually used in risk management.