Empirical Characteristic Function Research Articles

A data-driven framework for consistent financial valuation and risk measurement

In this paper, we propose a general data-driven framework that unifies the valuation and risk measurement of financial derivatives, which is especially useful in markets with thinly-traded derivatives. We first extract the empirical characteristic function from market-observable time series for the underlying asset prices, and then utilize Fourier techniques to obtain the physical nonparametric density and cumulative distribution function for the log-returns process, based on which we compute risk measures. Then we risk-neutralize the nonparametric density and distribution functions to model-independently valuate a variety of financial derivatives, including path-independent European options and path-dependent exotic contracts. By estimating the state-price density explicitly, and utilizing a convenient basis representation, we are able to greatly simplify the pricing of exotic options all within a consistent model-free framework. Numerical examples, and an empirical example using real market data (Brent crude oil prices) illustrate the accuracy and versatility of the proposed method in handling pricing and risk management of multiple financial contracts based solely on observable time series data.

A Data-Driven Framework for Consistent Financial Valuation and Risk Measurement

In this paper, we propose a general data-driven framework that unifies the valuation and risk measurement of financial derivatives, which is especially useful in markets with thinly-traded derivatives. We first extract the empirical characteristic function from market-observable time series for the underlying asset prices, and then utilize Fourier techniques to obtain the physical non-parametric density and cumulative distribution function for the log-returns process, based on which we compute risk measures. Then we risk-neutralize the non-parametric density and distribution functions to model-independently valuate a variety of financial derivatives, including path-independent European options and path-dependent exotic contracts. By estimating the state-price density explicitly, and utilizing a convenient basis representation, we are able to greatly simplify the pricing of exotic options all within a consistent model-free framework. Numerical examples, and an empirical example using real market data (Brent crude oil prices) illustrate the accuracy and versatility of the proposed method in handling pricing and risk management of multiple financial contracts based solely on observable time series data.

A minimal contrast estimator for the linear fractional stable motion

In this paper we present an estimator for the three-dimensional parameter $$(\sigma , \alpha , H)$$ of the linear fractional stable motion, where H represents the self-similarity parameter, and $$(\sigma , \alpha )$$ are the scaling and stability parameters of the driving symmetric Levy process L. Our approach is based upon a minimal contrast method associated with the empirical characteristic function combined with a ratio type estimator for the self-similarity parameter H. The main result investigates the strong consistency and weak limit theorems for the resulting estimator. Furthermore, we propose several ideas to obtain feasible confidence regions in various parameter settings. Our work is mainly related to Ljungdahl and Podolskij (A note on parametric estimation of Levy moving average processes, p 294, 2019) and Mazur et al. (Bernoulli 26(1): 226–252, 2020) in which parameter estimation for the linear fractional stable motion and related Levy moving average processes has been studied.

Estimating the distribution of a stochastic sum of IID random variables

Abstract Suggested is a non-parametric method and algorithm for estimating the probability distribution of a stochastic sum of independent identically distributed continuous random variables, based on combining and numerically inverting the associated empirical characteristic function (CF) derived from the observed data. This is motivated by classical problems in financial risk management, actuarial science, and hydrological modelling. This approach can be naturally generalized to more complex semi-parametric modelling and estimating approaches, e.g., by incorporating the generalized Pareto distribution fit for modelling heavy tails of the considered continuous random variables, or by considering the weighted mixture of the parametric CFs (used to incorporate the expert knowledge) and the empirical CFs (used to incorporate the knowledge based on the observed or historical data). The suggested numerical approach is based on combination of the Gil-Pelaez inversion formulae for deriving the probability distribution (PDF and CDF) from the associated CF and the trapezoidal quadrature rule used for the required numerical integration. The presented non-parametric estimation method is related to the bootstrap estimation approach, and thus, it shares similar properties. Applicability of the proposed estimation procedure is illustrated by estimating the aggregate loss distribution from the well-known Danish fire losses data.

Testing distributional assumptions using a continuum of moments

We propose specification tests for parametric distributions that compare the potentially complex theoretical and empirical characteristic functions using the continuum of moment conditions analogue to an overidentifying restrictions test, which takes into account the correlation between influence functions for different argument values. We derive its asymptotic distribution for fixed regularization parameter and when this vanishes with the sample size. We show its consistency against any deviation from the null, study its local power and compare it with existing tests. An extensive Monte Carlo exercise confirms that our proposed tests display good power in finite samples against a variety of alternatives.

Change-point methods for multivariate time-series: paired vectorial observations

We consider paired and two-sample break-detection procedures for vectorial observations and multivariate time series. The new methods involve L2-type criteria based on empirical characteristic functions and are easy to compute regardless of dimension. We obtain asymptotic results that allow for application of the methods to a wide range of settings involving on-line as well as retrospective circumstances with dependence between the two time series as well as with dependence within each series. In the ensuing Monte Carlo study the new detection methods are implemented by means of resampling procedures which are properly adapted to the type of data at hand, be it independent or paired, autoregressive or GARCH structured, medium or heavy-tailed. The new methods are also applied on a real dataset from the financial sector over a time period which includes the Brexit referendum.

Correlated Squared Returns

Joint densities for a sequential pair of returns with weak autocorrelation and strong correlation in squared returns are formulated. The marginal return densities are either variance gamma or bilateral gamma. Two dimensional matching of empirical characteristic functions to its theoretical counterpart is employed for dependency parameter estimation. Estimations are reported for 3920 daily return sequences of a thousand days. Path simulation is done using conditional distribution functions. The paths display levels of squared return correlation and decay rates for the squared return autocorrelation function that are comparable to these magnitudes in daily return data. Regressions of log characteristic functions at different time points are used to estimate time scaling coefficients. Regressions of these time scaling coefficients on squared return correlations support the view that autocorrelation in squared returns slows the rate of passage of economic time.

Stieltjes Sample Characteristic Function: Singular Continuous Distributions Parameters Estimation

In this paper, the Steiltjes sample characteristic function is introduced and studied. It is applied to establish a new parameter estimation method. The method is useful to estimate parameters of singular continuous distributions, where other estimation procedures such as maximum likelihood and empirical characteristic function fail to be applied. We fit well the Cantor's type distribution with the parameters estimated by the proposed method on the log returns of the daily price of Brazilian coffee. Effectiveness of the estimation approach is also numerically demonstrated in estimating the parameters of the standard two-sided power and the asymmetric Laplace distributions, using simulated and a real financial data set. Certain theoretical derivations concerning the Stieltjes sample characteristic function are given as well.

Sketched Clustering via Hybrid Approximate Message Passing

In sketched clustering, a dataset of T samples is first sketched down to a vector of modest size, from which the centroids are subsequently extracted. Its advantages include 1) reduced storage complexity and 2) centroid extraction complexity independent of T. For the sketching methodology recently proposed by Keriven et al., which can be interpreted as a random sampling of the empirical characteristic function, we propose a sketched clustering algorithm based on approximate message passing. Numerical experiments suggest that our approach is more efficient than the state-of-the-art sketched clustering algorithm “CL-OMPR” (in both computational and sample complexity) and more efficient than k-means++ when T is large.

Empirical Characteristic Functions‐Based Estimation and Distance Correlation for Locally Stationary Processes

In this article, we propose a kernel‐type estimator for the local characteristic function of locally stationary processes. Under weak moment conditions, we prove joint asymptotic normality for local empirical characteristic functions. For time‐varying linear processes, we establish a central limit theorem under the assumption of finite absolute first moments of the process. Additionally, we prove weak convergence of the local empirical characteristic process. We apply our asymptotic results to parameter estimation. Furthermore, by extending the notion of distance correlation to locally stationary processes, we are able to provide asymptotic theory for local empirical distance correlations. Finally, we provide a simulation study on minimum distance estimation for α‐stable distributions and illustrate the pairwise dependence structure over time of log returns of German stock prices via local empirical distance correlations.

A two-sample test for the equality of univariate marginal distributions for high-dimensional data

A recurring theme in modern statistics is dealing with high-dimensional data whose main feature is a large number, p, of variables but a small sample size. In this context our aim is to address the problem of testing the null hypothesis that the marginal distributions of p variables are the same for two groups. We propose a test statistic motivated by the simple idea of comparing, for each of the p variables, the empirical characteristic functions computed from the two samples. The asymptotic normality of the test statistic is derived under mixing conditions. In our asymptotic analysis the number of variables tends to infinity, while the size of individual samples remains fixed. In order to obtain a practical test several estimators of the variance are proposed, leading to three somewhat different versions of the test. An alternative global test based on the P-values derived from permutation tests is also proposed. A simulation study to investigate the finite sample properties of the proposed tests is carried out, and a practical illustration involving microarray data is provided.

Goodness-of-fit tests in the Cox proportional hazards model

We consider a variety of tests for testing goodness-of-fit in a parametric Cox proportional hazards (PH) model and compare their performance. Aspects of the model under test include the baseline distribution and time-invariance of covariates. We also test for the PH model itself against a certain generalization. This is done through an extensive Monte Carlo study where we simulate the performance of the tests for these three paired hypotheses. The results show that the tests based on the empirical characteristic function and those based on the empirical Laplace transform have the best overall power performance. It is also found that the distributions of the considered test statistics do not depend on the specific functional form of the covariate function.

The STNRW-ECF estimator for stable distributions with application in ECG beat

The α-stable distribution can describe a degree of concentration of the observations around the mean as well as their asymmetry, regardless of sample size. As the α stability index is the most interesting parameter in the applications, several estimators have been proposed for α. Here we develop an estimator for α (called STNRW-ECF) based on the empirical characteristic function and the Seismic Trace Noise Reduction by Wavelets and Double Threshold Estimation method (STNRW). We analyze the proposed estimator using Monte Carlo simulations and prove its asymptotic Gaussian distribution. Electrocardiography (ECG) is the process of recording the electrical activity of the heart over a period of time using electrodes placed on the skin. The time intervals between its various peaks, may contain useful information about the nature of disease afflicting the heart. To analyze this kind of data can be tiring and more prone to errors when interpreted by human beings, since there is a huge amount information to be processed. Here we propose the STNRW-ECF estimator to be an additional diagnostic tool that may provide an indication of cardiac arrhythmia, and we also propose a test based on the principal wavelet shrinkage to confirm whether or not the STNRW-ECF estimator should be used for this purpose.

Using degradation-with-jump measures to estimate life characteristics of lithium-ion battery

Degradation-with-jump measures are time series data sets containing the information of both continuous and randomly jumping degradation evolution of a system. Traditional maximum likelihood estimation and Bayesian estimation are not convenient for such general jump processes without closed-form distributions. Based on general degradation models derived using Lévy driven non-Gaussian Ornstein-Uhlenbeck (OU) processes, we propose a systematic statistical method using linear programing estimators and empirical characteristic functions. The point estimates of reliability function and lifetime moments are obtained by deriving their explicit expressions. We also construct bootstrap procedures for the confidence intervals. Simulation studies for a stable process and a stable driven OU process are performed. In the case study, we use a general Lévy process to fit the Li-ion battery life data, and then estimate the reliability and lifetime moments of the battery. By integrally analyzing degradation data series embedded with jump measures, our work provides the efficient and precise estimation for life characteristics.

Analytic performance investigation of signal level estimator based on empirical characteristic function in impulsive noise

In this paper, we evaluate the mean square error (MSE) performance of empirical characteristic function (ECF) based signal level estimator in a binary communication system. By calculating Cramér-Rao lower bound (CRLB) we investigate the performance of the ECF based estimator in the presence of Laplace and Gaussian mixture noises. We have derived an analytic expression for the variance of the ECF based estimator which shows that it is asymptotically unbiased and consistent. Simulation and analytic results indicate that the ECF based level estimator outperforms the previously proposed estimators in some signal to noise ratio (SNR) regions when the observation noise distribution is unknown.

Time-Varying Periodicity in Intraday Volatility

ABSTRACT We develop a nonparametric test for whether return volatility exhibits time-varying intraday periodicity using a long time series of high-frequency data. Our null hypothesis, commonly adopted in work on volatility modeling, is that volatility follows a stationary process combined with a constant time-of-day periodic component. We construct time-of-day volatility estimates and studentize the high-frequency returns with these periodic components. If the intraday periodicity is invariant, then the distribution of the studentized returns should be identical across the trading day. Consequently, the test compares the empirical characteristic function of the studentized returns across the trading day. The limit distribution of the test depends on the error in recovering volatility from discrete return data and the empirical process error associated with estimating volatility moments through their sample counterparts. Critical values are computed via easy-to-implement simulation. In an empirical application to S&P 500 index returns, we find strong evidence for variation in the intraday volatility pattern driven in part by the current level of volatility. When volatility is elevated, the period preceding the market close constitutes a significantly higher fraction of the total daily integrated volatility than during low volatility regimes. Supplementary materials for this article are available online.

Empirical cumulant function based parameter estimation in stable laws

Stable distributions are a subclass of infinitely divisible distributions that form the only family of possible limiting distributions for sums of independent identically distributed random variables. A challenging problem is estimating their parameters because many have densities with no explicit form and infinite moments. To address this problem, a class of closed-form estimators, called cumulant estimators, has been introduced. Cumulant estimators are derived from the logarithm of empirical characteristic function at two arbitrary distinct positive real arguments. This paper extends cumulant estimators in two directions: (i) it is proved that they are asymptotically normal and (ii) a sample based rule for selecting the two arguments is proposed. Extensive simulations show that under the provided selection rule, the closed-form cumulant estimators generally outperform the well-known algorithmic methods.

A test for equality of two distributions via integrating characteristic functions

In this paper, we investigate the problem of testing the equality of two distributions by integrating the squared norm of the difference between two corresponding empirical characteristic functions. This results in a linear combination of three different U-statistics. So the original testing problem is reduced to testing whether this linear combination is zero or not. We apply the Jackknife Empirical Likelihood (JEL) method to the new hypothesis testing problem. It has been proved that, under multivariate case, the log JEL statistic after scaling tends to a chi-square distribution with degree of freedom 1. Simulation studies are presented to assess the finite-sample performance of our method.

Function-based hypothesis testing in censored two-sample location-scale models.

Function-based hypothesis testing in two-sample location-scale models has been addressed for uncensored data using the empirical characteristic function. A test of adequacy in censored two-sample location-scale models is lacking, however. A plug-in empirical likelihood approach is used to introduce a test statistic, which, asymptotically, is not distribution free. Hence for practical situations bootstrap is necessary for performing the test. A multiplier bootstrap and a model appropriate resampling procedure are given to approximate critical values from the null asymptotic distribution. Although minimum distance estimators of the location and scale are deployed for the plug-in, any consistent estimators can be used. Numerical studies are carried out that validate the proposed testing method, and real example illustrations are given.

Applications of distance correlation to time series

The use of empirical characteristic functions for inference problems, including estimation in some special parametric settings and testing for goodness of fit, has a long history dating back to the 70s (see for example, Feuerverger and Mureika (1977), Csorgo (1981a,1981b,1981c), Feuerverger (1993)). More recently, there has been renewed interest in using empirical characteristic functions in other inference settings. The distance covariance and correlation, developed by Szekely and Rizzo (2009) for measuring dependence and testing independence between two random vectors, are perhaps the best known illustrations of this. We apply these ideas to stationary univariate and multivariate time series to measure lagged auto- and cross-dependence in a time series. Assuming strong mixing, we establish the relevant asymptotic theory for the sample auto- and cross-distance correlation functions. We also apply the auto-distance correlation function (ADCF) to the residuals of an autoregressive processes as a test of goodness of fit. Under the null that an autoregressive model is true, the limit distribution of the empirical ADCF can differ markedly from the corresponding one based on an iid sequence. We illustrate the use of the empirical auto- and cross-distance correlation functions for testing dependence and cross-dependence of time series in a variety of different contexts.