Nino Kordzakhia

Robust estimation in the logistic regression model

This paper presents robust M-estimates based on the influence function approach for the multiple logistic regression model. Under the assumption that the sequence of distributions corresponding to the contaminated models is contiguous to the pure model, the asymptotic normality of these estimators is determined. The optimal influence function is found as the analytical solution of the minimax problem, that is by minimizing the mean-squared deviance for worst-case contamination. A numerical implementation is given with the performance of the proposed robust estimators evaluated both in a simulation study and with two real datasets.

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.

Pricing of Volume-Weighted Average Options: Analytical Approximations and Numerical Results

The volume weighted average price (VWAP) over rolling number of days in the averaging period is used as a benchmark price by market participants and can be regarded as an estimate for the price that a passive trader will pay to purchase securities in a market. The VWAP is commonly used in brokerage houses as a quantitative trading tool and also appears in Australian taxation law to specify the price of share-buybacks of publically-listed companies. Most of the existing literature on VWAP focuses on strategies and algorithms to acquire market securities at a price as close as possible to VWAP. In our setup the volume process is modeled via a shifted squared Ornstein-Uhlenbeck process and a geometric Brownian motion is used to model the asset price. We derive the analytical formulae for moments of VWAP and then use the moment matching approach to approximate a distribution of VWAP. Numerical results for moments of VWAP and call-option prices have been verified by Monte Carlo simulations.

ON MOMENTS OF PITMAN ESTIMATORS: THE CASE OF FRACTIONAL BROWNIAN MOTION ∗

In some non-regular statistical estimation problems, the limiting likelihood processes are functionals of fractional Brownian motion (fBm) with Hursts 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.

Lower and upper bounds for prices of Asian-type options

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 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.

Bounds on prices for Asian options via Fourier methods

The problem of pricing arithmetic Asian options is nontrivial, and has attracted much interest over the last two decades. This paper provides a method for calculating bounds on option prices and approximations to option deltas in a market where the underlying asset follows a geometric Levy process. The core idea is to find a highly correlated, yet more tractable proxy to the event that the option finishes in-the-money. The paper provides a means for calculating the joint characteristic function of the underlying asset and proxy processes, and relies on Fourier methods to compute prices and deltas. Numerical studies show that the lower bound provides accurate approximations to prices and deltas, while the upper bound provides good though less accurate results. doi:10.1017/S1446181115000231

Estimation of cusp location of stochastic processes: a survey

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.

Bounds and Approximations for Distributions of Weighted Kolmogorov-Smirnov Tests

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.

Multi-Factor Statistical Modelling of Demand and Spot Price of Electricity

Abstract The deregulation of electricity markets in different parts of the world has exposed consumers to irregularities in electricity prices driven by the principle of supply and demand. Development of accurate statistical models contributes to establishing protective mechanisms and risk measurement policies for both suppliers, consumers. In this paper multi-factor modelling methodology, solely applied to the spot price of electricity or demand for electricity in earlier studies, is extended to a bivariate process of spot price of electricity and demand for electricity. The suggested model accommodates common idiosyncrasies observed in deregulated electricity markets such as cyclical trends in price and demand for electricity, occurrence of extreme spikes in prices, and a mean-reversion effect seen in the settling of prices from extreme values to the mean level over a short period of time. A time series model for de-seasonalised demand for electricity is used in combination with a linear regression model developed for logarithms of deseasonalised daily averages of electricity spot prices. The spiky behaviour of prices occurring in clusters, interpreted as ‘a post-spike’ effect, is addressed by a filtered Poisson (i.e. shot noise) factor of the model. The demand for electricity is found to be the primary stochastic factor driving the electricity prices. In the linear regression model for ‘de-seasonalised’ and ‘de-spiked’ spot prices the back-shifted variables play the role of exogenous variables. These variables capture the ‘price’ and ‘demand’ inter-dependence observed in practice. The historical data is obtained from the NSW section of Australian Energy Markets.