1.5 Financial Applications of Stable Laws

Many techniques in modern finance rely heavily on the assumption that the random variables under investigation follow a Gaussian distribution. However, time series observed in finance - but also in other applications - often deviate from the Gaussian model, in that their marginal distributions are heavy-tailed and, possibly, asymmetric. In such situations, the appropriateness of the commonly adopted normal assumption is highly questionable.

It is often argued that financial asset returns are the cumulative outcome of
a vast number of pieces of information and individual decisions arriving almost continuously in time. Hence, in the presence of heavy-tails it is natural to assume that they are approximately governed by a stable non-Gaussian distribution. Other leptokurtic distributions, including Student's $ t$, Weibull, and hyperbolic, lack the attractive central limit property.


Table 1.1: Fits to 2000 Dow Jones Industrial Average (DJIA) index returns from the period February 2, 1987 - December 29, 1994. Test statistics and the corresponding $ p$-values based on 1000 simulated samples (in parentheses) are also given.
Parameters: $ \alpha $ $ \sigma$ $ \beta$ $ \mu$
         
$ \alpha $-stable fit 1.6411 0.0050 -0.0126 0.0005
Gaussian fit   0.0111   0.0003
         
Tests: Anderson-Darling Kolmogorov
         
$ \alpha $-stable fit 0.6441 0.5583
  (0.020) (0.500)
Gaussian fit +$ \infty$ 4.6353
  ($ <$0.005) ($ <$0.005)

3559 STFstab06.xpl

Stable distributions have been successfully fit to stock returns, excess bond returns, foreign exchange rates, commodity price returns and real estate returns (McCulloch; 1996; Rachev and Mittnik; 2000). In recent years, however, several studies have found, what appears to be strong evidence against the stable model (Gopikrishnan et al.; 1999; McCulloch; 1997). These studies have estimated the tail exponent directly from the tail observations and commonly have found $ \alpha $ that appears to be significantly greater than 2, well outside the stable domain. Recall, however, that in Section 1.4.1 we have shown that estimating $ \alpha $ only from the tail observations may be strongly misleading and for samples of typical size the rejection of the $ \alpha $-stable regime unfounded. Let us see ourselves how well the stable law describes financial asset returns.

In this section we want to apply the discussed techniques to financial data. Due to limited space we chose only one estimation method - the regression approach of Koutrouvelis (1980), as it offers high accuracy at moderate computational time. We start the empirical analysis with the most prominent example - the Dow Jones Industrial Average (DJIA) index, see Table 1.1. The data set covers the period February 2, 1987 - December 29, 1994 and comprises 2000 daily returns. Recall, that it includes the largest crash in Wall Street history - the Black Monday of October 19, 1987. Clearly the 1.64-stable law offers a much better fit to the DJIA returns than the Gaussian distribution. Its superiority, especially in the tails of the distribution, is even better visible in Figure 1.6.

Figure 1.6: Stable (cyan) and Gaussian (dashed red) fits to the DJIA returns (black circles) empirical cdf from the period February 2, 1987 - December 29, 1994. Right panel is a magnification of the left tail fit on a double logarithmic scale clearly showing the superiority of the 1.64-stable law.
\includegraphics[width=.7\defpicwidth]{STFstab06a.ps} \includegraphics[width=.7\defpicwidth]{STFstab06b.ps}

Figure 1.7: Stable (cyan) and Gaussian (dashed red) fits to the Boeing stock returns (black circles) empirical cdf from the period July 1, 1997 - December 31, 2003. Right panel is a magnification of the left tail fit on a double logarithmic scale clearly showing the superiority of the 1.78-stable law.
\includegraphics[width=.7\defpicwidth]{STFstab07a.ps} \includegraphics[width=.7\defpicwidth]{STFstab07b.ps}


Table 1.2: Fits to 1635 Boeing stock price returns from the period July 1, 1997 - December 31, 2003. Test statistics and the corresponding $ p$-values based on 1000 simulated samples (in parentheses) are also given.
Parameters: $ \alpha $ $ \sigma$ $ \beta$ $ \mu$
         
$ \alpha $-stable fit 1.7811 0.0141 0.2834 0.0009
Gaussian fit   0.0244   0.0001
         
Tests: Anderson-Darling Kolmogorov
         
$ \alpha $-stable fit 0.3756 0.4522
  (0.18) (0.80)
Gaussian fit 9.6606 2.1361
  ($ <$0.005) ($ <$0.005)

3581 STFstab07.xpl

To make our statistical analysis more sound, we also compare both fits through Anderson-Darling and Kolmogorov test statistics (D'Agostino and Stephens; 1986). The former may be treated as a weighted Kolmogorov statistics which puts more weight to the differences in the tails of the distributions. Although no asymptotic results are known for the stable laws, approximate $ p$-values for these goodness-of-fit tests can be obtained via the Monte Carlo technique, for details see Chapter 13. First the parameter vector is estimated for a given sample of size $ n$, yielding $ \hat\theta$, and the test statistics is calculated assuming that the sample is distributed according to $ F(x;\hat\theta)$, returning a value of $ d$. Next, a sample of size $ n$ of $ F(x;\hat\theta)$-distributed variates is generated. The parameter vector is estimated for this simulated sample, yielding $ \hat\theta_1$, and the test statistics is calculated assuming that the sample is distributed according to $ F(x;\hat\theta_1)$. The simulation is repeated as many times as required to achieve a certain level of accuracy. The estimate of the $ p$-value is obtained as the proportion of times that the test quantity is at least as large as $ d$.

For the $ \alpha $-stable fit of the DJIA returns the values of the Anderson-Darling and Kolmogorov statistics are 0.6441 and 0.5583, respectively. The corresponding approximate $ p$-values based on 1000 simulated samples are 0.02 and 0.5 allowing us to accept the $ \alpha $-stable law as a model of DJIA returns. The values of the test statistics for the Gaussian fit yield $ p$-values of less than 0.005 forcing us to reject the Gaussian law, see Table 1.1.

Next, we apply the same technique to 1635 daily returns of Boeing stock prices from the period July 1, 1997 - December 31, 2003. The $ 1.78$-stable distribution fits the data very well, yielding small values of the Anderson-Darling (0.3756) and Kolmogorov (0.4522) test statistics, see Figure 1.7 and Table 1.2. The approximate $ p$-values based, as in the previous example, on 1000 simulated samples are 0.18 and 0.8, respectively, allowing us to accept the $ \alpha $-stable law as a model of Boeing returns. On the other hand, the values of the test statistics for the Gaussian fit yield $ p$-values of less than 0.005 forcing us to reject the Gaussian distribution.

The stable law seems to be tailor-cut for the DJIA index and Boeing stock price returns. But does it fit other asset returns as well? Unfortunately, not. Although, for most asset returns it does provide a better fit than the Gaussian law, in many cases the test statistics and $ p$-values suggest that the fit is not as good as for these two data sets. This can be seen in Figure 1.8 and Table 1.3, where the calibration results for 4444 daily returns of the Japanese yen against the US dollar (JPY/USD) exchange rate from December 1, 1978 to January 31, 1991 are presented. The empirical distribution does not exhibit power-law tails and the extreme tails are largely overestimated by the stable distribution. For a risk manager who likes to play safe this may not be a bad idea, as the stable law overestimates the risks and thus provides an upper limit of losses. However, from a calibration perspective other distributions, like the hyperbolic or truncated stable, may be more appropriate for many data sets (Weron; 2004).


Table 1.3: Fits to 4444 JPY/USD exchange rate returns from the period December 1, 1978 - January 31, 1991. Test statistics and the corresponding $ p$-values (in parentheses) are also given.
Parameters: $ \alpha $ $ \sigma$ $ \beta$ $ \mu$
         
$ \alpha $-stable fit 1.3274 0.0020 -0.1393 -0.0003
Gaussian fit   0.0049   -0.0001
         
Tests: Anderson-Darling Kolmogorov
         
$ \alpha $-stable fit 4.7833 1.4520
  ($ <$0.005) ($ <$0.005)
Gaussian fit 91.7226 6.7574
  ($ <$0.005) ($ <$0.005)

3587 STFstab08.xpl

Figure 1.8: Stable (cyan) and Gaussian (dashed red) fits to the JPY/USD exchange rate returns (black circles) empirical cdf from the period December 1, 1978 - January 31, 1991. Right panel is a magnification of the left tail fit on a double logarithmic scale. The extreme returns are largely overestimated by the stable law.
\includegraphics[width=.7\defpicwidth]{STFstab08a.ps} \includegraphics[width=.7\defpicwidth]{STFstab08b.ps}