Generalized hyperbolic secant distributions. With applications to finance (Q2437607)

The aim of the monograph is to summarize the author's work on the generalized hyperbolic secant distribution with applications to finance. The last chapter (Chapter 6) contains applications to finance from the flexibility of the hyperbolic secant distribution to different financial returns, stock indices and exchanges. By using four criteria (log-likelihood value, Akaike criterion, Kolmogorov-Smirnov distance and the Anderson-Darling statistics), Tables 6.2--6.5 illustrate the poor fitting of the usually used stable distributions and goodness-of-fit for the skewed generalized hyperbolic secant classes by the example of the weekly returns of the Nikkei from July 31, 1983 to April 9, 1995 and the daily exchange rates of EUR-USD from Jan 1, 2002 to April 30, 2012. Chapter 1 provides definitions and summaries of the hyperbolic secant distribution. A random variable, given by the logarithm of the absolute value of the ratio of two independent standard normal variables, has a hyperbolic secant distribution. Its probabilistic properties are summarized in Table 1.1: characteristic function, moment-generating function, the cumulative distribution function, moments, tail behavior, infinite divisibility, maximum domain of attraction, self-reciprocality, entropy (\(\ln (2 \pi)\)), and relation to other distributions. Chapter 2 starts with Perk's distribution family as a general hyperbolic secant distribution family, to show the skewness by splitting a parameter \(\gamma\) (symmetric if \(\gamma =1\), skew to the right if \(\gamma > 1\), skew to the left if \(0< \gamma < 1\)). Then, the author constructs asymmetric densities by the Esscher transformation, a tool in actuarial science. Vaughan advocated skew-extended generalized hyperbolic secant distribution families as a natural generalization of the generalized secant hyperbolic representative. Chapter 3 further discusses the generalization of the hyperbolic secant distribution with respect to its tail behavior as Harkness and Harkness' \(\rho\)-th convolution from the characteristic function. Devroye's algorithm on acceptance-rejection works for \(\rho \geq 1\) is presented, and the natural exponential characterization and the Meixner-Pollaczek polynomials characterization of the generalized hyperbolic secant distributions are listed. Chapter 4 focuses on the beta-hyperbolic secant distribution. In contrast to the beta-normal distribution and to the beta-Student-\(t\) distribution, the beta-hyperbolic secant densities are always unimodal and all moments exist, and they are more flexible to range skewness and leptokurtosis combinations. Chapter 5 considers the variable transformation as Johnson's inverse hyperbolic sine transformation, Jones-Pewsey's sinh-arcsinh transformation, as well as their basic properties of the so transformed variables and their distributions. Appendix A gives R-code on fitting a beta-hyperbolic secant distribution for both conditional and unconditional cases. It is nice to see how hyperbolic secant random variables are used in finance for pricing derivatives like call options and its comparison with the classical Black-Scholes-Merton formula.