On lower and upper bounds for Asian-type options: a unified approach

In the context of dealing with financial risk management problems it is desirable to have accurate bounds for option prices in situations when pricing formulae do not exist in the closed form. A unified approach for obtaining upper and lower bounds for Asian-type options, including options on VWAP, is proposed in this paper. The bounds obtained are applicable to the continuous and discrete-time frameworks for the case of time-dependent interest rates. Numerical examples are provided to illustrate the accuracy of the bounds

Pitman estimators: An asymptotic variance revisited

We provide an analytic expression for the variance of ratio of integral functionals of fractional Brownian motion which arises as an asymptotic variance of Pitman estimators for a location parameter of independent identically distributed observations. The expression is obtained in terms of derivatives of a logarithmic moment of the integral functional of limit likelihood ratio process (LLRP). In the particular case when the LLRP is a geometric Brownian motion, we show that the established expression leads to the known representation of the asymptotic variance of Pitman estimator in terms of Riemann zeta-function. © by SIAM

First Passage Time of Filtered Poisson Process with Exponential Shape Function

Solving some integro-differential equation we find the Laplace transformation of the first passage time for Filtered Poisson Process generated by pulses with uniform or exponential distributions. Also, the martingale technique is applied for approximations of expectations accuracy is veryfying with the help of Monte-Carlo simulations.first passage times; laplace transformation; martingales; integro-differential equations; filtered poisson process; ornstein-uhlenbeck process

On moments of pitman estimators: The case of fractional Brownian motion

© 2014 Society for Industrial and Applied Mathematics. In some nonregular statistical estimation problems, the limiting likelihood processes are functionals of fractional Brownian motion (fBm) with Hurst’s parameter H, 0 < H ≦ 1. In this paper we present several analytical and numerical results on the moments of Pitman estimators represented in the form of integral functionals of fBm. We also provide Monte Carlo simulation results for variances of Pitman and asymptotic maximum likelihood estimators

Approximations for weighted Kolmogorov–Smirnov distributions via boundary crossing probabilities

© 2016, The Author(s). A statistical application to Gene Set Enrichment Analysis implies calculating the distribution of the maximum of a certain Gaussian process, which is a modification of the standard Brownian bridge. Using the transformation into a boundary crossing problem for the Brownian motion and a piecewise linear boundary, it is proved that the desired distribution can be approximated by an n-dimensional Gaussian integral. Fast approximations are defined and validated by Monte Carlo simulation. The performance of the method for the genomics application is discussed

Pricing of asian-type and basket options via bounds

© 2017 Society for Industrial and Applied Mathematics. This paper sets out to provide a general framework for the pricing of average-type options via lower and upper bounds. This class of options includes Asian, basket, and options on the volume-weighted average price. The use of lower and upper bounds is proposed in response to the inherent difficulty in finding analytical representations for the true price of these options and the requirement for numerical procedures to be fast and efficient. We demonstrate that in some cases lower bounds allow for the dimensionality of the problem to be reduced and that these methods provide reasonable approximations to the price of the option

Estimation of cusp location of stochastic processes: a survey

© 2018, Springer Science+Business Media B.V., part of Springer Nature. We present a review of some recent results on estimation of location parameter for several models of observations with cusp-type singularity at the change point. We suppose that the cusp-type models fit better to the real phenomena described usually by change point models. The list of models includes Gaussian, inhomogeneous Poisson, ergodic diffusion processes, time series and the classical case of i.i.d. observations. We describe the properties of the maximum likelihood and Bayes estimators under some asymptotic assumptions. The asymptotic efficiency of estimators are discussed as well and the results of some numerical simulations are presented. We provide some heuristic arguments which demonstrate the convergence of log-likelihood ratios in the models under consideration to the fractional Brownian motion

Combining Molecular Sieve and Complexing Properties of the Column Packing in Gas Chromatography

Influence of modifying pentasyl group synthetic zeolites (Silicalite-1 and Silicalite-2) with metal cations capable to specific interactions on the separation ability of the chromatographic column has been studied.Вивчено вплив модифікування синтетичних цеолітів сімейства пентасілів (Сілікаліту-1 і Сілікаліту-2) катіонами металів, здатних до специфічних взаємодій, на розділювальну здатність хроматографічної колонки.Изучено влияние модифицирования синтетических цеолитов семейства пентасилов (Силикалита-1 и Силикалита-2) катионами металлов, способными к специфическим взаимодействиям, на разделительную способность хроматографической колонки

Bounds and approximations for distributions of weighted Kolmogorov-Smirnov tests

© Springer International Publishing AG 2017. The paper is motivated by the use of weighted Kolmogorov-Smirnov (wKS) tests in Gene Set Enrichment Analysis where the key requirements are speed and accuracy of computations. We reduce the problem of finding of distributions of one-and two-sided wKS statistics to the nonlinear boundary crossing problem for a Brownian motion. Theoretical estimates of accuracy of the approximations using piecewise linear boundaries are derived. The approximations with 2-knot piecewise linear boundaries are discussed for the one-sided wKS. In the numerical example the estimates of tail probabilities obtained with the use of upper and lower bounds were validated using Monte-Carlo simulation

Martingales and first passage times of AR(1) sequences

Using the martingale approach we find sufficient conditions for exponential boundedness of first passage times over a level for ergodic first order autoregressive sequences. Further, we prove a martingale identity to be used in obtaining explicit bounds for the expectation of first passage times