Analysis of Financial Time Series, Third Edition

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

(3.1)

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 ...