Conditional Heteroscedasticity and Applications in Finance

In 1900, Louis Bachelier published the findings of his doctoral research on stock prices; his empirical results indicated that stock prices behaved like a random walk. However, this study was overlooked for the next 50 years. Then, in 1953, Maurice Kendall published his analysis of stock market prices in which he suggested that price changes were essentially random. Such a claim ran counter to the perceived wisdom of the times, but the empirical studies that followed confirmed Kendall’s claim and ultimately led to the path-breaking work of Black and Scholes (1973) and Merton (1973) on the Efficient Market Hypothesis. In essence, the Black-Scholes theory states that prices will move randomly in an efficient market. Intuitively, we may argue that if prices were predictable, trading would quickly take place to erode the implied advantage. Of course, the theory does not apply to insider knowledge exploited by the few!

We discuss the Black–Scholes model briefly in Sect. 19.1 and relate it to our analysis of discrete time processes. This development leads naturally to conditionally heteroscedastic processes, which is the subject of Sect. 19.2. Then, in Sect. 19.3 we examine time series that evolve over time in both their conditional mean and conditional variance structures. We conclude with a re-analysis of the US gasoline price data considered earlier in Chap. 9, which illustrates the value of conditionally heteroscedastic models in the construction of prediction intervals.