Parametric pricing of higher order moments in S&P500 options

A general parametric framework based on the generalized Student t‐distribution is developed for pricing S&P500 options. Higher order moments in stock returns as well as time‐varying volatility are priced. An important computational advantage of the proposed framework over Monte Carlo‐based pricing methods is that options can be priced using one‐dimensional quadrature integration. The empirical application is based on S&P500 options traded on select days in April 1995, a total sample of over 100,000 observations. A range of performance criteria are used to evaluate the proposed model, as well as a number of alternative models. The empirical results show that pricing higher order moments and time‐varying volatility yields improvements in the pricing of options, as well as correcting the volatility skew associated with the Black–Scholes model. Copyright © 2004 John Wiley & Sons, Ltd.     

On the existence of moments of partially restricted reduced form coefficients

In this paper ridgelike Bayesian estimators of structural coefficients have been used to form the partially restricted reduced form estimators. These partially restricted reduced form estimators are simple in form and possess finite sampling moments and risk in contrast to other restricted reduced form estimators that possess no finite moments and have infinite risk relative to quadratic loss functions. The usual k-class implied partially restricted reduced form estimators with 0≦k≦1 do not posses finite moments unless the degree of overidentification (or the excess of sample size over the number of coefficients) of the structural equation being estimated is suitably restricted.     

On multivariate higher order von Mises expansions

In this paper we review the von Mises calculations of higher order for statistical functionals of one variable. For the functionals of several variables, the higher order Gateaux differentials are defined leading to the corresponding multivariate higher order von Mises expansions. These expansions are used to analyze the bias of the corresponding statistical functionals. The second and third order derivatives are computed for M-estimates. Applications of these expansions to study the bias of M-estimates of location and to simultaneous M-estimates of location and scale are also given. Received: July 2000     

On the existence of moments of the partially restricted reduced-form estimators from a simultaneous-equation model

This article proves the existence of all moments of the partially restricted reduced-form estimator. It highlights this estimation method as it appears to be the only reduced-form estimator to possess finite moments, and is thus a valid alternative to restricted reduced-form estimation (where the moments do not exist). The estimation method is described briefly and then the existence proof is formulated, first, for the case of two included endogenous variables in the structural equation and then, we extend the result for any number of included endogenous variables.     

An identity for the product moments of order statistics

A general identity for the product moments of successive order statistics is given, which is valid in a class of probability distributions including Weibull, Pareto, exponential and Burr distributions.     

Stationarity and the existence of moments of a family of GARCH processes

This paper investigates some structural properties of a family of GARCH processes. A simple sufficient condition for the existence of the αδ-order stationary solution of the processes is derived, where α∈(0,1] and δ>0. The solution is strictly stationary and ergodic, and the causal expansion of the family of GARCH processes is also established. Furthermore, the necessary and sufficient condition for the existence of the moments is obtained. The technique used in this paper for the moment conditions is different from that used in He and Terasvirta (J. Econom. 92 (1999a) 173), and avoids the assumption that the process started at some finite value infinitely many periods ago. Moreover, the conditions for the strict stationarity of the model and the existence of its moments are simple to check and should prove useful in practice.     

Sharp bounds on moments of generalized order statistics

Sharp lower and upper bounds on expected values of generalized order statistics are proven by the use of rearranged Moriguti's inequality. The method yields improvements of known quantile and moment bounds for expectations of order and record statistics based on independent identically distributed random variables. The bounds are attainable providing new characterizations of two-point distributions. Received: January 1999     

On a recent characterization of the bivariate normal distribution

Summary A recent characterization of the bivariate normal distribution is recalled and a few remarks regarding this characterization and its extension to the multivariate case are given.     

A characterization of the exponential distribution by higher order gap

Summary SupposeX is a non-negative random variable with an absolutely continuous (with respect to Lebesgue measure) distribution functionF (x) and the corresponding probability density functionf(x). LetX ₁,X ₂,...,X _n be a random sample of sizen fromF andX _i,n is thei-th smallest order statistics. We define thej-th order gapg _i,j(n) asg _i,j(n)=X _i+j,n–X_i,n 1i<n, 1nn–i. In this paper a characterization of the exponential distribution is given by considering a distribution property ofg _i,j(n).     

On the existence of locally mvu estimators

It is proved that there exists an unbiased estimator for some real parameter of a class of distributions, which has minimal variance for some fixed distribution among all corresponding unbiased estimators, if and. only if the corresponding minimal variances for all related unbiased estimation problems concerning finite subsets of the underlying family of distributions are bounded. As an application it is shown that there does not exist some unbiased estimator for θ^k+c(ε≥0) with minimal variance for θ =0 among all corresponding unbiased estimators on the base of k i.i.d. random variables with a Cauchy-distribution, where θ denotes some location parameter.     

On testing the correlation coefficient of a bivariate normal distribution

Summary Here we propose two tests for testing in the bivariate normal population assuming that the ratio of the variances in it is known. The first test (U.M.P.U.) is derived by using the Neyman-Pearson lemma, whereas the second test is obtained through testing the scale parameter of the Cauchy distribution. The powers of the first and second tests are compared with a well-known test, based on the sample correlation coefficient for small and large samples respectively.     

On the existence of random measure preserving bijections

This paper establishes the existence of an invariant probability measure on the set Π of measure-preserving bijections of the unit interval. In fact, the measure may be further restricted to be a Baire-measure. This result is compared with Aumann's (1967) and Aumann and Shapley's (1974) impossibility theorems.     

On the inclusion of bivariate marked point processes in graphical models

Graphical models use graphs to represent conditional independence relationships among random variables of a multivariate probability distribution. This paper introduces a new kind of chain graph models in which nodes also represent marked point processes. This is relevant to the analysis of event history data, i.e. data consisting of random sequences of events or time durations of states. Survival analysis and duration models are particular cases. This article considers the case of two marked point processes. The idea consists of representing a whole process by a single node and a conditional independence statement by a lack of connection. We refer to the resulting models as graphical duration models.     

The existence of moments of some simple Bayes estimators of coefficients in a simultaneous equation model

In this paper a simple modification of the usual k-class estimators has been suggested so that for 0 ≦ k ≦ 1 the problem of the non-existence of moments disappears. These modified estimators can be interpreted either as Bayes estimators or as constrained estimators subject to the restriction that the squared length of the coefficient vector is less than or equal to a given number.     

On the convergence of moments of geometric and harmonic means

The moments of the geometric mean of n independent and identically distributed random variables are shown to converge as n→∞. Rates of convergence are determined for the first moment and the variance. The results relate to recent work on long term investment returns when yearly rates of return are randomly varying. Application is made to moments of the harmonic mean.     

On the existence of walrasian equilibrium in large-square economies

An existence theorem for Walrasian equilibrium is demonstrated for an economy with a continuum of consumers and an infinite-dimensional commodity space, such as l₁ or c₀, having an ‘order-compatible’ basis.     

The higher order likelihood method for the common mean of several log-normal distributions

In this paper, we discuss interval estimation for the common mean of several heterogeneous log-normal (LN) populations. The proposed procedure is based on a higher order likelihood method. The merits of our proposed method are numerically compared with other three methods with respect to their expected lengths and coverage probabilities. Numerical studies have shown that the coverage probabilities of the proposed method are very accurate even for very small samples. The methods are also illustrated with an example.     

On a bivariate Markov process arising in the theory of single-server retrial queues

We introduce a bivariate Markov process which can be seen as the joint process of the channel state and the number of customers in orbit of a Markovian single-server retrial queue with state dependent intensities. We obtain a necessary and sufficient condition for the process to be regular, and necessary and sufficient conditions for ergodicity and recurrence. A product-form formula for the stationary distribution is obtained. Besides, we study the busy period, the number of served customers and other related quantities. We show that for all the above problems there exist "equivalent" birth-and-death processes. However, a "uniformly equivalent" birth-and-death process does not exist.     

On the existence of equilibrium in social systems with coordination

Social systems with coordination were introduced by Vind (1983) as a general model of economic institutions. The present paper gives a general existence theorem for equilibria in social systems with coordination extending Vind's result.