Laplace transform approach to option pricing for time-changed Brownian models

In this article, we consider European option pricing for time-changed Brownian models using Laplace transform. We obtain a general formula for the option price as the integral of a real-valued function involving the Laplace transform of the random time change. Unlike the usual Fourier transform technique, our method does not suffer from difficulties specific to complex integration, such as the evaluation of multiple-valued functions, and allows for a model-independent analysis of the truncation error. In the numerical analysis part, we compare option prices in variance gamma (VG), normal inverse Gaussian (NIG), and generalized hyperbolic (GH) models obtained by Laplace transform with those obtained by the Fourier transform method introduced by Carr and Madan in 1999. The results show that our method converges faster than the Fourier approach when the Laplace transforms of the subordinators decay exponentially, for examples like NIG and GH models.     

General affine transform families: why is the Pareto an exponential transform?

A slight extension of the transform method by Efron (1982) is used to analyze general affine transforms of arbitrary random variables. Our application concerns the simple question “Why is the Pareto an exponential transform?”. This result, already known to Gumbel (1958), finds a satisfactory mathematical answer in the framework of general affine transforms, and holds for several other distributions including heavy-tailed ones.     

Stationary distributions for a class of Markov-modulated tandem fluid queues

ABSTRACT

We consider a model consisting of two fluid queues driven by the same background continuous-time Markov chain, such that the rates of change of the fluid in the second queue depend on whether the first queue is empty or not: when the first queue is nonempty, the content of the second queue increases, and when the first queue is empty, the content of the second queue decreases.

We analyze the stationary distribution of this tandem model using operator-analytic methods. The various densities (or Laplace–Stieltjes transforms thereof) and probability masses involved in this stationary distribution are expressed in terms of the stationary distribution of some embedded process. To find the latter from the (known) transition kernel, we propose a numerical procedure based on discretization and truncation. For some examples we show the method works well, although its performance is clearly affected by the quality of these approximations, both in terms of accuracy and run time.     

On the probability of ruin in a continuous risk model with two types of delayed claims

ABSTRACT

This article studies a risk model involving one type of main claims and two types of by-claims, which is an extension of the general risk model with delayed claims. We suppose that every main claim may not induce any by-claims or may induce one by-claim belonging to one of the two types of by-claims with a certain probability. In addition, assume that the by-claim and its associated main claim may occur at the same time and that the occurrence of the by-claim may be delayed. An integro-differential equation system for survival probabilities is derived by using two auxiliary risk models. The expression of the survival probability is obtained by applying Laplace transforms and Rouché theorem. Furthermore, we provide a method for solving the survival probability when the two by-claim amounts satisfy different exponential distributions. As a special case, an explicit expression of survival probability is given when all the claim amounts obey the same exponential distribution. Finally, numerical results are provided to examine the proposed method.     

Selection of the Transformation Variable in the Laplace Transform Method of Estimation

The work of this paper is based on the innovative approach of Feigin et al. (1983), who estimate parameters of lifetime distributions by equating empirical and theoretical Laplace transforms. We show that the optimal choice of the transform variable depends critically upon the number of sampling times, the way they are spaced, and how the empirical transform is formed. Two new approaches for choosing the transform variable, viz. using cross-validation or constrained optimisation, are introduced and shown to have potential for wide-ranging use.     

Semiparametric regression of clustered current status data

This paper discusses regression analysis of clustered current status data under semiparametric additive hazards models. In particular, we consider the situation when cluster sizes can be informative about correlated failure times from the same cluster. To address the problem, we present estimating equation-based estimation procedures and establish asymptotic properties of the resulting estimates. Finite sample performance of the proposed method is assessed through an extensive simulation study, which indicates the procedure works well. The method is applied to a motivating data set from a lung tumorigenicity study.     

A new orthogonality-based estimation for varying-coefficient partially linear models

Varying coefficient partially linear models are usually used for longitudinal data analysis, and an interest is mainly to improve efficiency of regression coefficients. By the orthogonality estimation technology and the quadratic inference function method, we propose a new orthogonality-based estimation method to estimate parameter and nonparametric components in varying coefficient partially linear models with longitudinal data. The proposed procedure can separately estimate the parametric and nonparametric components, and the resulting estimators do not affect each other. Under some mild conditions, we establish some asymptotic properties of the resulting estimators. Furthermore, the finite sample performance of the proposed procedure is assessed by some simulation experiments.     

Approximate wavelet-based simulation of long memory processes

In this article, we investigate an algorithm for the fast O(N) and approximate simulation of long memory (LM) processes of length N using the discrete wavelet transform. The algorithm generates stationary processes and is based on the notion that we can improve standard wavelet-based simulation schemes by noting that the decorrelation property of wavelet transforms is not perfect for certain LM process. The method involves the simulation of circular autoregressive process of order one. We demonstrate some of the statistical properties of the processes generated, with some focus on four commonly used LM processes. We compare this simulation method with the white noise wavelet simulation scheme of Percival and Walden [Percival, D. and Walden, A., 2000, Wavelet Methods for Time Series Analysis (Cambridge: Cambridge University Press).].     

A wavelet-based approach to the analysis and modelling of financial time series exhibiting strong long-range dependence: the case of Southeast Europe

This paper demonstrates the utilization of wavelet-based tools for the analysis and prediction of financial time series exhibiting strong long-range dependence (LRD). Commonly emerging markets' stock returns are characterized by LRD. Therefore, we track the LRD evolvement for the return series of six Southeast European stock indices through the application of a wavelet-based semi-parametric method. We further engage the á trous wavelet transform in order to extract deeper knowledge on the returns term structure and utilize it for prediction purposes. In particular, a multiscale autoregressive (MAR) model is fitted and its out-of-sample forecast performance is benchmarked to that of ARMA. Additionally, a data-driven MAR feature selection procedure is outlined. We find that the wavelet-based method captures adequately LRD dynamics both in calm as well as in turmoil periods detecting the presence of transitional changes. At the same time, the MAR model handles with the complicated autocorrelation structure implied by the LRD in a parsimonious way achieving better performance.     

Smooth Semi‐nonparametric Analysis for Mixture Cure Models and Its Application to Breast Cancer

Mixture cure models are widely used when a proportion of patients are cured. The proportional hazards mixture cure model and the accelerated failure time mixture cure model are the most popular models in practice. Usually the expectation–maximisation (EM) algorithm is applied to both models for parameter estimation. Bootstrap methods are used for variance estimation. In this paper we propose a smooth semi‐nonparametric (SNP) approach in which maximum likelihood is applied directly to mixture cure models for parameter estimation. The variance can be estimated by the inverse of the second derivative of the SNP likelihood. A comprehensive simulation study indicates good performance of the proposed method. We investigate stage effects in breast cancer by applying the proposed method to breast cancer data from the South Carolina Cancer Registry.     

Temporal Disaggregation: Methods,Information Loss,and Diagnostics

This research provides a generalized framework to disaggregate lower-frequency time series and evaluate the disaggregation performance. The proposed framework combines two models in separate stages: a linear regression model to exploit related independent variables in the first stage and a state–space model to disaggregate the residual from the regression in the second stage. For the purpose of providing a set of practical criteria for assessing the disaggregation performance, we measure the information loss that occurs during temporal aggregation while examining what effects take place when aggregating data. To validate the proposed framework, we implement Monte Carlo simulations and provide two empirical studies. Supplementary materials for this article are available online.     

A ‘nondecimated’ lifting transform

Classical nondecimated wavelet transforms are attractive for many applications. When the data comes from complex or irregular designs, the use of second generation wavelets in nonparametric regression has proved superior to that of classical wavelets. However, the construction of a nondecimated second generation wavelet transform is not obvious. In this paper we propose a new ‘nondecimated’ lifting transform, based on the lifting algorithm which removes one coefficient at a time, and explore its behavior. Our approach also allows for embedding adaptivity in the transform, i.e. wavelet functions can be constructed such that their smoothness adjusts to the local properties of the signal. We address the problem of nonparametric regression and propose an (averaged) estimator obtained by using our nondecimated lifting technique teamed with empirical Bayes shrinkage. Simulations show that our proposed method has higher performance than competing techniques able to work on irregular data. Our construction also opens avenues for generating a ‘best’ representation, which we shall explore.     

Exploring Parameter Relations for Multi‐Stage Models in Stage‐Wise Constant and Time Dependent Hazard Rates

Single cohort stage‐frequency data are considered when assessing the stage reached by individuals through destructive sampling. For this type of data, when all hazard rates are assumed constant and equal, Laplace transform methods have been applied in the past to estimate the parameters in each stage‐duration distribution and the overall hazard rates. If hazard rates are not all equal, estimating stage‐duration parameters using Laplace transform methods becomes complex. In this paper, two new models are proposed to estimate stage‐dependent maturation parameters using Laplace transform methods where non‐trivial hazard rates apply. The first model encompasses hazard rates that are constant within each stage but vary between stages. The second model encompasses time‐dependent hazard rates within stages. Moreover, this paper introduces a method for estimating the hazard rate in each stage for the stage‐wise constant hazard rates model. This work presents methods that could be used in specific types of laboratory studies, but the main motivation is to explore the relationships between stage maturation parameters that, in future work, could be exploited in applying Bayesian approaches. The application of the methodology in each model is evaluated using simulated data in order to illustrate the structure of these models.     

Nonparametric rank-based test procedures for non-additive models in the two-way layout I. no replications

One of the major unresolved problems in the area of nonparametric statistics is the need for satisfactory rank-based test procedures for non-additive models in the two-way layout, especially when there is only one observation on each combination of the levels of the experimental factors. In this paper we consider an arbitrary non-additive model for the two-way layout with n levels of each factor. We utilize both alignment and ranking of the data together with basic properties of Latin squares to develop rank tests for interaction (non-additivity). Our technique involves first aligning within one of the main effects, ranking within the other main effects (columns and rows) and then adding the resulting ranks within “interaction bands” corresponding to orthogonal partitions of the interaction for the model, as denoted by the letters of an n × n Latin square. A Friedman-type statistic is then computed on the resulting sums. This is repeated for each of (n?1) mutually orthogonal Latin squares (thus accounting for all the interaction degrees of freedom). The resulting (n?1) Friedman-type statistics are finally combined to obtain an overall test statistic. The necessary null distribution tables for applying the proposed test for non-additivity are presented and we discuss the results of a Monte Carlo simulation study of the relative powers of this new procedure and other (parametric and nonparametric) procedures designed to detect interaction in a two-way layout with one observation per cell.     

A K-S Type Test for Exponentiality Based on Empirical Hankel Transforms

A uniqueness theorem for a recently introduced special Hankel transform of probability distributions on the non negative half-line motivates a K-S type test statistic based on empirical Hankel transforms for testing the hypothesis of exponentiality. This article deals with the asymptotic behavior of the new test.     

Instrumental variable based variable selection for generalized linear models with endogenous covariates

We study the variable selection problem for a class of generalized linear models with endogenous covariates. Based on the instrumental variable adjustment technology and the smooth-threshold estimating equation (SEE) method, we propose an instrumental variable based variable selection procedure. The proposed variable selection method can attenuate the effect of endogeneity in covariates, and is easy for application in practice. Some theoretical results are also derived such as the consistency of the proposed variable selection procedure and the convergence rate of the resulting estimator. Further, some simulation studies and a real data analysis are conducted to evaluate the performance of the proposed method, and simulation results show that the proposed method is workable.     

Numerical method for means of linear Hawkes processes

Abstract

Linear Hawkes processes are widely used in many fields and means are the basic and critical information of them. However, there is little research on linear Hawkes processes’ means. In this paper, we present a numerical method based on the Laplace transform and inverse Laplace transform for means of linear Hawkes processes. The advantage of this method is that whatever the kernel function is, we can always obtain the numerical solutions of means for a linear Hawkes process. In addition, this numerical method provides the basic information of linear Hawkes processes by means. As an application, the numerical method is applied in a WeChat network model.     

灰色成分数据模型在中国产业结构分析预测中的应用

针对成分数据这种特殊类型的统计数据,提出一种新的预测建模方法:对于一列按照时间顺序收集的成分数据,先运用对数变换使成分数据降维,然后对降维后的数据运用GM(1,1)模型进行预测,最后再将预测值进行反对数变换,从而得到了各成分的预测值.根据提出的方法,建立了中国产业结构的预测模型,并分析了中国产业结构的发展趋势和未来状况.经检验,运用该方法预测出的数据与实际值十分吻合.     

Additive Transformation Models for Recurrent Events

In this article, we propose a class of additive transformation models for recurrent event data, which includes the additive rates model as a special case. The new models offer great flexibility in formulating the effects of covariates on the mean function of recurrent events. Estimating equation approaches are developed for the model parameters, and asymptotic properties of the resulting estimators are established. In addition, a model checking procedure is presented to assess the adequacy of the model. The finite sample performance of the proposed estimators is examined through simulation studies, and an application to a bladder cancer study is presented.     

A new stochastic order based upon Laplace transform with applications

This paper deals with nonnegative random variables having Laplace transforms as their reliability functions. We study a new stochastic order based upon Laplace transform. Some applications in actuarial science, frailty models and reliability are presented as well.