Chapter 3
Conditional Heteroscedastic Models
The objective of this chapter is to study some methods and econometric models available in the literature for modeling the volatility of an asset return. The models are referred to as conditional heteroscedastic models.
Volatility is an important factor in options trading. Here volatility means the conditional standard deviation of the underlying asset return. Consider, for example, the price of a European call option, which is a contract giving its holder the right, but not the obligation, to buy a fixed number of shares of a specified common stock at a fixed price on a given date. The fixed price is called the strike price and is commonly denoted by K. The given date is called the expiration date. The important time duration here is the time to expiration (measured in years), and we denote it by ℓ. The well-known Black–Scholes option pricing formula states that the price of such a call option is
where Pt is the current price of the underlying stock, r is the continuously compounded risk-free interest rate, σt is the annualized conditional standard deviation of the log return of the specified stock, and Φ(x) is the cumulative distribution function of the standard normal random variable evaluated at x. A derivation of the formula is given in Chapter 6. The formula has several nice interpretations, but it suffices to say here that the conditional standard deviation of the log return of the underlying stock plays an important role. This volatility evolves over time. If the holder can exercise her right any time on or before the expiration date, then the option is called an American call option.
Volatility has many other financial applications. As discussed in Chapter 7, volatility modeling provides a simple approach to calculating value at risk of a financial position in risk management. It plays an important role in asset allocation under the mean-variance framework. Furthermore, modeling the volatility of a time series can improve the efficiency in parameter estimation and the accuracy in interval forecast. Finally, the volatility index of a market has recently become a financial instrument. The VIX volatility index compiled by the Chicago Board of Option Exchange (CBOE) started to trade in futures on March 26, 2004.
The univariate volatility models discussed in this chapter include the autoregressive conditional heteroscedastic (ARCH) model of Engle (1982), the generalized ARCH (GARCH) model of Bollerslev (1986), the exponential GARCH (EGARCH) model of Nelson (1991), the threshold GARCH (TGARCH) model of Glosten, Jagannathan, and Runkle (1993) and Zakoian (1994), the conditional heteroscedastic autoregressive moving-average (CHARMA) model of Tsay (1987), the random coefficient autoregressive (RCA) model of Nicholls and Quinn (1982), and the stochastic volatility (SV) models of Melino and Turnbull (1990), Taylor (1994), Harvey, Ruiz, and Shephard (1994), and Jacquier, Polson, and Rossi (1994). We also discuss advantages and weaknesses of each volatility model and show some applications of the models. Multivariate volatility models, including those with time-varying correlations, are discussed in Chapter 10. The chapter also discusses some alternative approaches to volatility modeling in Section 3.15, including use of daily high and low prices of an asset.