Distribution of linear functions of ordered rectangular variates

Abstract

The derivations of the distributions of functions of ordered statistics are complicated due to the fact that the integrals occurring in the derivation theory are to be evaluated over ordered ranges of variables of integration. The manipulations in such cases are tedious and involved. This difficulty is sometimes partially or completely obviated by transforming the ordered variates to the unordered ones as done by Kabe [2, 3], amongst several authors. Also the given distribution problem may be transformed to an equivalent one in some other variates which are easy to handle, see e.g., Laurent [5]. Since we are adopting Laurent's procedure in this paper we outline it briefly.     

Pricing discretely monitored Asian options under Lévy processes

We present methodologies to price discretely monitored Asian options when the underlying evolves according to a generic Lévy process. For geometric Asian options we provide closed-form solutions in terms of the Fourier transform and we study in particular these formulas in the Lévy-stable case. For arithmetic Asian options we solve the valuation problem by recursive integration and derive a recursive theoretical formula for the moments to check the accuracy of the results. We compare the implementation of our method to Monte Carlo simulation implemented with control variates and using different parametric Lévy processes. We also discuss model risk issues.     

The relationship between expected utility and higher moments for distributions captured by the Gram–Charlier class

In this letter we derive the closed form solution for expected utility in terms of higher moments of the distribution of wealth when expected utility takes the CARA form and the distribution of wealth is captured by the Gram–Charlier class of distributions. We derive the condition under which positive skewness can be associated with a decrease in expected utility.     

Exact moments of order statistics from a power-function distribution

Summary

In this note a problem on exact moments of order statistics from a power-function distribution is considered. The characteristic function of the kth order statistic is obtained and moments about the origin of the kth order statistic are expressed in terms of gamma functions. An exact expression for the covariance of any two order statistics Y_i < Y_j is obtained in terms of beta and gamma functions. Various recurrence relations between the expected values of order statistics are also obtained.     

Inverse moments of order statistics from Weibull distribution

Abstract

A general expression for moments of order statistics of positive and negative orders from Weibull distribution has been obtained and the result has been utilized to establish two identities.     

On the Laplace Transform of the Aggregate Discounted Claims with Markovian Arrivals

Abstract

We present an explicit formula for the Laplace transform of the distribution of the aggregate discounted claims when interclaim times follow a Markovian arrival process. In addition, we derive explicit formulas for the first two moments and then show that the higher moments may be obtained by numerically solving a system of ordinary differential equations.     

Orthogonal expansions for VIX options under affine jump diffusions

In this work we derive new closed-form pricing formulas for VIX options in the jump-diffusion SVJJ model proposed by Duffie et al. [Econometrica, 2000, 68, 1343–1376]. Our approach is based on the classic methodology of approximating a density function with an orthogonal expansion of polynomials weighted by a kernel. Orthogonal expansions based on the Gaussian distribution, such as Edgeworth or Gram–Charlier expansions, have been successfully employed by a number of authors in the context of equity options. However, these expansions are not quite suitable for volatility or variance densities as they inherently assign positive mass to the negative real line. Here we approximate option prices via expansions that instead are based on kernels defined on the positive real line. Specifically, we consider a flexible family of distributions, which generalizes the gamma kernel associated with the classic Laguerre expansions. The method can be employed whenever the moments of the underlying variance distribution are known. It provides fast and accurate price computations, and therefore it represents a valid and possibly more robust alternative to pricing techniques based on Fourier transform inversions.     

Two-Sided Bounds for Tails of Compound Negative Binomial Distributions in the Exponential and Heavy-Tailed Cases

This paper derives two-sided bounds for tails of compound negative binomial distributions, both in the exponential and heavy-tailed cases. Two approaches are employed to derive the two-sided bounds in the case of exponential tails. One is the convolution technique, as in Willmot & Lin (1997). The other is based on an identity of compound negative binomial distributions; they can be represented as a compound Poisson distribution with a compound logarithmic distribution as the underlying claims distribution. This connection between the compound negative binomial, Poisson and logarithmic distributions results in two-sided bounds for the tails of the compound negative binomial distribution, which also generalize and improve a result of Willmot & Lin (1997). For the heavy-tailed case, we use the method developed by Cai & Garrido (1999b). In addition, we give two-sided bounds for stop-loss premiums of compound negative binomial distributions. Furthermore, we derive bounds for the stop-loss premiums of general compound distributions among the classes of HNBUE and HNWUE.     

Some results on moments and cumulants

Abstract

In the present paper we discuss various results related to moments and cumulants of probability distributions and approximations to probability distributions. As the approximations are not necessarily probability distributions themselves, we shall apply the concept of moments and cumulants to more general functions. Recursions are deduced for moments and cumulants of functions in the form R_k [a, b] as defined by Dhaene & Sundt (1996). We deduce a simple relation between the De Pril transform and the cumulants of a function. This relation is applied to some classes of approximations to probability distributions, in particular the approximations of Hipp and De Pril.     

The Tail Stein's Identity with Applications to Risk Measures

In this article, we examine a generalized version of an identity made famous by Stein, who constructed the so-called Stein's Lemma in the special case of a normal distribution. Other works followed to extend the lemma to the larger class of elliptical distributions. The lemma has had many applications in statistics, finance, insurance, and actuarial science. In an attempt to broaden the application of this generalized identity, we consider the version in the case where we investigate only the tail portion of the distribution of a random variable. Understanding the tails of a distribution is very important in actuarial science and insurance. Our article therefore introduces the concept of the “tail Stein's identity” to the case of any random variable defined on an appropriate probability space with a Lebesgue density function satisfying certain regularity conditions. We also examine this “tail Stein's identity” to the class of discrete distributions. This extended identity allows us to develop recursive formulas for generating tail conditional moments. As examples and illustrations, we consider several classes of distributions including the exponential family, and we apply this result to demonstrate how to generate tail conditional moments. This holds a large promise for applications in risk management.     

On Tail Value-at-Risk for sums of non-independent random variables with a generalized Pareto distribution

Recently in actuarial literature several authors have derived lower and upper bounds in the sense of convex order for sums of random variables with given marginal distributions and unknown dependency structure. In this paper, we derive convex bounds for sums of non-independent and identically distributed random variables when marginal distributions are mixture models. In particular, we examine some well-known risk measures and we find approximations for Tail Value-at-Risk of the sums considered when marginal distributions are generalized Pareto distributions. By numerical examples we illustrate the goodness of the presented approximations.      

Models for stock returns

Historically, the normal variance model has been used to describe stock return distributions. This model is based on taking the conditional stock return distribution to be normal with its variance itself being a random variable. The form of the actual stock return distribution will depend on the distribution for the variance. In practice, the distributions chosen for the variance appear to be very limited. In this note, we derive a comprehensive collection of formulas for the actual stock return distribution, covering some sixteen flexible families. The corresponding estimation procedures are derived by the method of moments and the method of maximum likelihood. We feel that this work could serve as a useful reference and lead to improved modelling with respect to stock market returns.     

Moments of Compound Mixed Poisson Distributions

Recursive formulae are derived for the evaluation of the moments and the descending factorial moments about a point n of mixed Poisson and compound mixed Poisson distributions, in the case where the derivative of the logarithm of the mixing density can be written as a ratio of polynomials. As byproduct, we also obtain recursive formulae for the evaluation of the moments about the origin, central moments, descending and ascending factorial moments of these distributions. Examples are also presented for a number of mixing densities.     

On sums and products of truncated exponential variates

Abstract

The distribution of any linear combination of a finite number of truncated exponential variates from possibly n distinct populations is obtained by using the Laplace transform. The distribution is demonstrated in a compact form which is quite suitable for computational purposes. The results are exemplified. Finally, a brief remark on the distribution of the product of truncated exponential variates is also added.     

On the Class of Erlang Mixtures with Risk Theoretic Applications

Abstract

A wide variety of distributions are shown to be of mixed-Erlang type. Useful computational formulas result for many quantities of interest in a risk-theoretic context when the claim size distribution is an Erlang mixture. In particular, the aggregate claims distribution and related quantities such as stop-loss moments are discussed, as well as ruin-theoretic quantities including infinitetime ruin probabilities and the distribution of the deficit at ruin. A very useful application of the results is the computation of finite-time ruin probabilities, with numerical examples given. Finally, extensions of the results to more general gamma mixtures are briefly examined.     

Recursions for Distribution Functions and Stop-Loss Transforms

For any function f on the non-negative integers, we can evaluate the cumulative function o f given by o f ( s )= ~ s x=0 f ( x ) from the values of f by the recursion o f ( s )= o f ( s -1)+ f ( s ). Analogously we can use this procedure t times to evaluate the t -th order cumulative function o t f . As an alternative, in the present paper we shall derive recursions for direct evaluation of o t f when f itself satisfies a certain sort of recursion. We shall also derive recursions for the t -th order tails v t f where v f ( s )= ~ X x=s+1 f ( x ). The recursions can be applied for exact and approximate evaluation of distribution functions and stop-loss transforms of probability distributions. The class of recursions for f includes the classes discussed by Sundt (1992), incorporating the class studied by Panjer's (1981). We discuss in particular convolutions and compound functions.     

A solvency study in non-life insurance

Abstract

This is the first of two papers which report on a solvency study. The study is based on statistical analyses of policy and claims data of a portfolio of single-family houses and dwellings. This paper deals mainly with analyses of fire, windstorm, and glass liabilities. Claim frequencies and claim size distributions are estimated, and the results are used to derive moments of the annual claim amounts and to provide examples of solvency margin requirements for different classes of business. The second paper is devoted to a broader discussion of solvency margin requirements in non-life insurance.     

Conditional higher order moments in metal asset returns

This study examines the role of higher order moments in the returns of four important metals, aluminium, copper, gold and silver, using the asymmetric GARCH (AGARCH) model with a conditional skewed generalized-t (SGT) distribution. Implications of higher order moments in metal returns are evaluated by comparing the performances of conditional value-at-risk measures obtained from the AGARCH models with SGT distributions to those obtained from the AGARCH models with normal and student-t distributions. With the exception of gold, the AGARCH model with the SGT distribution appears to have the best fit for all metals examined.     

On an inverse Gaussian process

Abstract

Tweedie [11] investigated properties of the Inverse Gaussian distribution. We define in this paper the Inverse Gaussian process. For the discrete case we find the density function of the functions of Inverse Gaussian variates. We look for covariance function and stochastic integral as well as conditional density functions of an Inverse Gaussian process.