If financial time series exhibits persistence or long-memory, then their unconditional probability distribution may not be normal. This has important implications for many areas in finance, especially asset pricing, option pricing, portfolio allocation and risk management. Furthermore, if the random walk does not apply, a wide range of results obtained by quantitative analysis may be inappropriate. The capital asset pricing model, the Black-Scholes option pricing formula, the concept of risk as standard deviation or volatility, and the use of Sharpe, Treynor, and other performance measures are not consistent with nonnormal distributions. Unfortunately, nonnormality is common among distributions of financial time series according to observations from empirical studies of financial series.
Strict assumptions have to be imposed on the returns of the financial asset to yield an explicit formula for practical applications. For example, in one of the strictest forms, we have to assert that the returns are statistically independent through time and identical across time for a cross-section of returns. Under the assumptions, we can derive a simple and yet elegant relationship between risk and return, as in the case of the security market line. These assumptions can be relaxed, and skewness as well as excess kurtosis can be easily accommodated using other distributions. For example, we can be more flexible in the specification of the distribution function using log-normal or stable class such as Pareto-Levy or stable Paretian distributions (e.g. Cauchy and Bernoulli), of which the normal distribution is a special case.
Models that take into account the asymmetric and fatter tails empirical distribution
have been used to model financial time series behaviour. Recent studies concentrated
on models that assume returns are drawn from a fat-tailed distribution with finite
higher moments. These include -distribution, mix-normal or conditionally normal.
Closed-form expressions that give meaningful relationship are rare and in most cases,
the results are not easy to manipulate mathematically or empirically implemented.
Furthermore, once nonlinearity is introduced, the possibility is infinite. It becomes
difficult analytically and intuitively. In most cases, the mode of analysing and solving
the problem is computational.
However, observations suggest that many aspects of financial behaviour may be nonlinear, Attitudes towards risk and expected return are evidently nonlinear, contrary to what unconditional CAPM and other linear models suggest. Derivatives pricing is also inherently nonlinear. Therefore, it is naturally to model such behaviour using nonlinear models.
Once we abandon the random walk hypothesis and without more specific theoretical structure, it is difficult to infer much about phenomena that spans a significant portion of the entire dataset. One area that can yield important insights and addresses some of the violations is long range dependence or the phenomena of persistence in time series. In this chapter, our efforts are focused on exploring persistence in financial time series.
A time series persists in the sense that observations in the past are correlated with observations in the distant future and the relationship may be nontrivial. In the frequency domain, this is characterised by high power at low frequencies, especially near the origin. Detection of long range dependence or persistence has importance implications for short-term trading and long range investment strategies. Transaction costs are not negligible for tactical asset allocation based on short-term strategies and long-horizon predictability may be a more genuine and appropriate form of exploiting profit opportunities. Allocation decisions will be sensitive to the time horizon and may be dependent on the degree of long-term memory.
Empirically, most results of the study of long-memory are focused on financial markets from the developed economies. Here, we look at the stock indices and exchange rates of markets in Asia. We have obtained results on indices and currencies of 10 countries using XploRe . Our discussions below rely heavily on the materials in Chapter 14.