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FIGURE 7.1 Distribution of returns for SPY (ETF that mimics S&P 500) for the period 1997 to 2009 in comparison with normal distribution and Cauchy distribution.

Another important finding is that squared (or absolute) returns are clearly correlated. This implies that while returns are practically uncorrelated, they are not independent. Volatility of returns may vary over time. Volatility can also be clustered, that is, large returns (of any sign) follow large returns. These facts lead to even greater relaxation of the strict random walk model.

In fact, distributions of returns may differ significantly from the Gaussians. In particular, these distributions may have high kurtosis, fat distribution tails (which implies a decay slower than the exponential one), and asymmetrical shape (see the distribution of S&P 500 returns in Figure 7.1).

Importance of long-range dependencies in the financial time series was greatly emphasized by Mandelbrot (1997). Early research of long-range behavior in the financial time series was based on the stable distributions called Levy flights (see Appendix A). This approach, however, has an important drawback: Levi flights have infinite variance. Such uncertainty of the volatility measure is generally unacceptable for financial applications. The truncated Levy flights that satisfy the requirement for finite volatility are sometimes used as a way around (Mantegna & Stanley 2000; Bouchaud & Potters 2000).

Recent findings by Gabaix et al. (2003) indicate that the asymptotic distributions of returns in equity markets for the period 1984 to 1996, at the

timescales varying from one minute to several days, follow the power law with an index close to three. At longer timescales, distributions of returns slowly approach the Gaussian form. Power-law distributions were also found in the FX market (Ohnishi et al. 2008). The presence of fat tails means that financial crises are less improbable than one might expect from the normal distribution. This certainly has important implications for risk management.

In general, power-law distributions describe scale-free processes. Scale invariance implies that the distribution has similar shape on different scales of the independent variable. Namely, function f(x) is scale-invariant to the transformation x!ax if there is such a parameter L that

where n ¼ –ln(L)/ln(a). The function f(x) is scale-free since the ratio f(ax)/ f(x) ¼ L does not depend on x.

Several theoretical models offered in explanation of the power-law distributions in finance are reviewed by Schmidt (2004) and Gabaix (2009). In particular, Gabaix et al. (2003) have derived a model that is based on two observations:

1.Distribution of the trading volumes obeys the power law with an index of about 1.5.

2.Distribution of the number of trades is a power law with an index of about 3 (in fact, it is close to 3.4).

Two other assumptions were made in this model:

1.Price movements are caused primarily by activity of large mutual funds whose size distribution is the power law with an index of 1 (so-called

Zipf’s law).

2.Mutual fund managers trade in an optimal way.

The mathematical transformations based on these rules yield a powerlaw return distribution with an index of 3.

In another approach, LeBaron (2001) has shown that power-law distributions can be generated by a mix of normal distributions with different timescales. Namely, fitting a combination of Gaussians with a half-life of about 2.7 years, 2.5 weeks, and 1 day to the Dow returns for 100 years (from 1900 to 2000), yields the power-law index in the range between 2.98

and 3.33 for the timescales in the range between 1 and 20 days. The rationale for three different Gaussians reflects various time horizons for investors: from buy-and-hold strategy to daily trading.

While studying the power-law distributions, it is important to remember the comments on spurious scaling laws made by Lux (2001a). There may be a problem with extracting the scaling index from finite (and not sufficiently long) financial data samples if the true distribution has the form

where L(x) is a slowly varying function that determines the behavior of the distribution in the short-range region. In other words, the power law may be valid only asymptotically. Obviously, the universal scaling exponent a ¼ –log[f(x)]/log(x) is as accurate as L(x) is close to a constant.

F R A C T A L S I N F I N A N C E

Now, let’s examine the basic notions of the fractal theory, as it has become popular in analysis of financial market data.5 In natural sciences, fractals denote the geometric objects whose defining property is self-similarity; that is, they can be constructed by repeating some geometric pattern at a smaller and smaller scale. The original fractal theory and its applications in finance were developed by Mandelbrot (1997). Self-similarity of an object implies its isotropic properties. This is not applicable for time series in which dependent variables (price, return) and independent variable (time) are measured with different units. In this case, the notion of self-affinity is used instead.

A self-affine random process X(t) satisfies the following scaling rule:

where the parameter H is called the Hurst exponent.

Now, let’s take a look at the fractional Brownian motion BH(t). This random process satisfies the following condition for all t and T:

The fractional Brownian motion is reduced to the regular Brownian motion when H ¼ 1/2. It can be shown that the correlation between the

past average E[BH(t) – BH(t – T)]/T and the future average E[BH(t þ T) – BH(t)]/T equals

Note that this correlation does not depend on T. If 1/2 < H < 1, then C > 0 and it is said that BH(t) is a persistent process. Namely, if BH(t) grew (fell) in the past, most probably it will grow (fall) in the immediate future. In other words, persistent processes maintain trend. In the opposite case (0 < H < 1/2, C < 0), the process is named anti-persistent. It is also said that anti-persistent processes are mean reverting; that is, if the current process innovation is positive, then the next one will most probably be negative, and vice versa.

The Hurst exponent of a time series sample can be estimated using the rescaled range (R/S) analysis. Consider a data set xi (i ¼ 1, . . . N) with mean mN and the standard deviation sN. First, the partial sums Sk must be calculated:

The value of R/S is always greater than zero since max(Sk) > 0 and min (Sk) < 0. For a given R/S, the Hurst exponent can be estimated using the relation

where a is a constant. Namely, when log(R/S) and log(N) are plotted against each other, the slope of the line determines H.

The R/S analysis is superior to many other methods of determining the long-range dependencies and hence can be useful in trading. Obviously, confirming current trend is important for choosing trend-based trading strategies (see Chapter 10). Still, it should be noted that the R/S analysis has a shortcoming: high sensitivity to the short-range memory (Campbell et al. 1997).

A more complex multifractal theory has recently drawn the attention of economists. In a nutshell, the multifractal theory describes the objects defined with a generic relationship

Application of the multifractal theory in finance can be found in Lux (2001b) and Calvet and Fisher (2002).

SUMMARY

&EMH states that markets instantly incorporate all new information in the asset prices. EMH is based on the RWH and on the concept of rational investors. Both RWH and rational investor behavior have been questioned in recent empirical research.

&Predictability of returns is determined by the properties of innovations in the time series. The random walk with strict white noise (RW1) is completely unpredictable.

&When innovations have independent but not identical increments, the process (RW2) is a martingale. While the process expectations are not forecastable, higher moments of RW2 may be predictable.

&In RW3 (white noise), innovations are uncorrelated yet higher moments, for example variance, can be dependent and hence forecastable. RW3 exhibits conditional heteroskedasticity, which is well supported with empirical data on volatility. RW3 does not permit linear forecasting of expectations, but non-linear forecasting may be possible.

&Empirical distributions of returns may differ significantly from the Gaussians. Namely, they may have high kurtosis, power-law asymptotes, and asymmetrical shape.

&Fractal theory has proven to be a promising framework for describing empirical properties of returns.

older data. Then, only the last n data points are used for calculating current volatility:

Poon & Granger (2003) describe several practical approaches used for forecasting volatility. The simplest forecast in the spirit of the martingale hypothesis is referred to as the random walk forecast: s^t ¼ st 1. Other popular linear forecasting methods are:

& Simple moving average (SMA):

&Exponential smoothing average (often called exponential moving average or EMA):

Note that the value of s^1 in (8.3) is not defined. Often it is assumed that s^t ¼ s1; sometimes SMA for a few initial values of st is used instead.

&Exponentially weighed moving average (EWMA), which is a truncated version of EMA:

n. n

XX

It can be shown by comparing EMA and EWMA in the limit of high n that the smoothing parameter equals

EMA with relation (8.5) is widely used in technical analysis (see Chapter 10).

As indicated previously, the concept of conditional volatility appears in the random walk RW3, in which variance is uncorrelated but not independent. Forecasting volatility with conditional heteroskedasticity models is widely used in financial research. This topic is addressed in the following section.

Traditionally, estimates of volatility are made using daily returns. However, with the proliferation of publicly available high-frequency market data, there has been a growing interest in the analysis of high-frequency effects on realized volatility (see details below).

The notion of implied volatility is based on the Black-Scholes theory of option pricing (see, e.g., Hull 2006). In this theory, volatility is one of parameters that determines the price of an option. Hence, reverse calculation of volatility is possible when the option price is available. There are several volatility indices that are used to gauge market uncertainty. Arguably the most famous index, VIX (the so-called fear index) represents an estimate of expected volatility for the next 30 days using prices of several S&P 500 index options.

Finally, there is the notion of stochastic volatility. This implies that price dynamics is not a weakly stationary process, and that volatility follows its own stochastic process. The current state of art in this field can be found in Shephard (2005) and Taylor (2005).

C O N D I T I O N A L H E T E R O S K E D A S T I C I T Y

As was indicated in Chapter 7, there may be noticeable autocorrelations in squared asset returns. This means that large returns (either positive or negative) follow large returns. In this case, it is said that the volatility of returns is clustered.1 The effect of volatility clustering is named also autoregressive conditional heteroskedasticity (ARCH). Several models in which past shocks contribute to the current volatility have been developed. Generally, these models are rooted in the ARCH(m) model, where the conditional variance is a weighed sum of m squared lagged returns:

In (8.6), eðtÞ ¼ Nð0; s2ðtÞÞ; v > 0; a1; . . . ; 0. Unfortunately, application of the ARCH(m) process to modeling the financial time series often requires polynomials with high order m.

A more efficient model is the generalized ARCH (GARCH) process. The GARCH(m, n) process combines the ARCH(m) process with the AR(n) process2 for lagged variance:

s2ðtÞ ¼ v þ a1e2ðt 1Þ þ a2e2ðt 2Þ þ . . . þ ame2ðt mÞ þ b1s2ðt 1Þ þ b2s2ðt 2Þ þ . . . þ bns2ðt nÞ

The simple GARCH(1, 1) model is widely used in financial applications:

Equation (8.7) can be transformed into

The last term in (8.8) being conditioned on information available at time (t – 1) has zero mean and can be treated as a shock to volatility. Therefore, the unconditional expectation of volatility for the GARCH(1, 1) model equals

This implies that the GARCH(1, 1) process is weakly stationary when a þ b < 1. The advantage of the stationary GARCH(1, 1) model is that it can be easily used for forecasting. Namely, the conditional expectation of volatility at time (t þ k) equals

E½s2ðt þ kÞ& ¼ ða þ bÞk½s2ðtÞ v=ð1 a bÞ& þ v=ð1 a bÞ ð8:10Þ The GARCH(1, 1) model (8.8) can be rewritten as

s2ðtÞ ¼ v=ð1 bÞ þ aðe2ðt 1Þ þ be2ðt 2Þ þ b2e2ðt 3Þ þ . . .Þ ð8:11Þ

Equation (8.11) implies that the GARCH(1, 1) model is equivalent to the infinite ARCH model with exponentially weighed coefficients. This explains why the GARCH models are more efficient than the ARCH models.

Several other GARCH models have been derived for addressing specifics of various economic and financial time series. One popular GARCH(1, 1) model, in which a þ b ¼ 1, is named integrated GARCH (IGARCH). This model has the autoregressive unit root; that is, volatility in this case follows the random walk. For IGARCH, the expectation (8.10) is reduced to

Then, IGARCH can be presented in the form