"Frontmatter". In: Analysis of Financial Time Series

12 FINANCIAL TIME SERIES AND THEIR CHARACTERISTICSthere is no lower bound for r t , and the lower bound for R t is satisfied using 1 + R t =exp(r t ). However, the lognormal assumption is not consistent with all the propertiesof historical stock returns. In particular, many stock returns exhibit a positive excesskurtosis.Stable DistributionThe stable distributions are a natural generalization of normal in that they are stableunder addition, which meets the need of continuously compounded returns r t .Furthermore, stable distributions are capable of capturing excess kurtosis shown byhistorical stock returns. However, non-normal stable distributions do not have a finitevariance, which is in conflict with most finance theories. In addition, statistical modelingusing non-normal stable distributions is difficult. An example of non-normalstable distributions is the Cauchy distribution, which is symmetric with respect to itsmedian, but has infinite variance.Scale Mixture of Normal DistributionsRecent studies of stock returns tend to use scale mixture or finite mixture of normaldistributions. Under the assumption of scale mixture of normal distributions, the logreturn r t is normally distributed with mean µ and variance σ 2 [i.e., r t ∼ N(µ, σ 2 )].However, σ 2 is a random variable that follows a positive distribution (e.g., σ −2 followsa Gamma distribution). An example of finite mixture of normal distributionsisr t ∼ (1 − X)N(µ, σ 2 1 ) + XN(µ, σ 2 2 ),where 0 ≤ α ≤ 1, σ1 2 is small and σ 2 2 is relatively large. For instance, with α =0.05, the finite mixture says that 95% of the returns follow N(µ, σ1 2 ) and 5% followN(µ, σ2 2). The large value of σ 2 2 enables the mixture to put more mass at the tails ofits distribution. The low percentage of returns that are from N(µ, σ2 2 ) says that themajority of the returns follow a simple normal distribution. Advantages of mixturesof normal include that they maintain the tractability of normal, have finite higherorder moments, and can capture the excess kurtosis. Yet it is hard to estimate themixture parameters (e.g., the α in the finite-mixture case).Figure 1.1 shows the probability density functions of a finite mixture of normal,Cauchy, and standard normal random variable. The finite mixture of normalis 0.95N(0, 1) + 0.05N(0, 16) and the density function of Cauchy isf (x) =1π(1 + x 2 , −∞ < x < ∞.)It is seen that Cauchy distribution has fatter tails than the finite mixture of normal,which in turn has fatter tails than the standard normal.