Conditional Characteristic Functions Research Articles

Identification and estimation of panel semiparametric conditional heteroskedastic frontiers with dynamic inefficiency

We study a semiparametric panel stochastic frontier model with nonseparable unobserved heterogeneity, which allows for time-varying conditional heteroskedastic productivity components. It does not require distributional assumptions on random noise except conditional symmetry. We utilize conditional characteristic functions from Kotlarski’s Lemma to derive new moment conditions that yield the identification of the heteroskedastic variance parameters of inefficiency and random noise. Identification only requires a panel with three periods for serially correlated inefficiency. A nonparametric estimation procedure is also developed for the conditional variance of inefficiency, and its convergence rate is established. Monte Carlo simulation shows that the estimator is robust to misspecification of inefficiency distributions.

A Stochastically Correlated Bivariate Square-Root Model

We introduce a novel stochastically correlated two-factor (i.e., bivariate) diffusion process under the square-root format, for which we analytically obtain the corresponding solutions for the conditional moment-generating functions and conditional characteristic functions. Such solutions recover verbatim those of the uncorrelated case which encompasses a range of processes similar to those produced by a bivariate square-root process in which entries are correlated in the standard way, that is, via a constant correlation coefficient. Note that closed-form solutions for the conditional characteristic and moment-generating functions are not available for the latter. We focus on the financial scenario of obtaining closed-form expressions for the exact price of a zero-coupon bond and Asian option prices using a Fourier cosine series method.

Foreign exchange option pricing under the 4/2 stochastic volatility model with CIR interest rates.

. Foreign exchange (FX) options are studied in a new stochastic volatility model (the 4/2 stochastic volatility model), which includes, as special instances, the Heston model and the 3/2 model, for the exchange rate in combination with the Cox-Ingersoll-Ross (CIR) dynamics for the domestic and foreign stochastic interest rates. We allow the correlation between the instantaneous volatility and the dynamics of the exchange rate, whereas the interest rates in both domestic and foreign markets are assumed to be independent of the dynamics of the exchange rate. We provide a semi-analytical formula for the price of the European FX call option via Fourier inversion. Through the change of measure technique, we derived explicit expressions for conditional characteristic functions. Our model is computationally tractable, thereby the pricing procedure is accurate and illustrates more efficiency in comparison to the Monte Carlo simulation method. Finally, we perform calibrations on market data, demonstrating that our model fits the implied volatilities well and outperforms the Heston model with CIR interest rates.

Test of conditional independence in factor models via Hilbert–Schmidt independence criterion

This work is concerned with testing conditional independence under a factor model setting. We propose a novel multivariate test for non-Gaussian data based on the Hilbert–Schmidt independence criterion (HSIC). Theoretically, we investigate the convergence of our test statistic under both the null and the alternative hypotheses, and devise a bootstrap scheme to approximate its null distribution, showing that its consistency is justified. Methodologically, we generalize the HSIC-based independence test approach to a situation where data follow a factor model structure. Our test requires no nonparametric smoothing estimation of functional forms including conditional probability density functions, conditional cumulative distribution functions and conditional characteristic functions under the null or alternative, is computationally efficient and is dimension-free in the sense that the dimension of the conditioning variable is allowed to be large but finite. Further extension to nonlinear, non-Gaussian structure equation models is also described in detail and asymptotic properties are rigorously justified. Numerical studies demonstrate the effectiveness of our proposed test relative to that of several existing tests.

TESTING FOR ANTICIPATED CHANGES IN SPOT VOLATILITY AT EVENT TIMES

We propose a test for anticipated changes in spot volatility, either due to continuous or discontinuous price moves, at the times of realization of event risk in the form of pre-scheduled releases of economic information such as earnings announcements by firms and macroeconomic news announcements. These events can generate nontrivial volatility in asset returns, which does not scale even locally in time. Our test is based on short-dated options written on an underlying asset subject to event risk, which takes place after the options’ observation time and prior to or after their expiration. We use options with different tenors to estimate the conditional (risk-neutral) characteristic functions of the underlying asset log-returns over the horizons of the options. Using these estimates and a relationship between the conditional characteristic functions with three different tenors, which holds true if and only if continuous and discontinuous spot volatility does not change at the event time, we design a test for this hypothesis. In an empirical application, we study anticipated individual stocks’ volatility changes following earnings announcements for a set of stocks with good option coverage.

Property analysis of multivariate conditional linear random processes in the problems of mathematical modelling of signals

The object of research is the process of mathematical modelling of a multidimensional random signal, which in the structure of its generation is the sum of a large number of random impulses that occur at random times. Examples of stochastic signals of this type can be, in particular, electroencephalographic and cardiographic signals, photoplethysmography signals, resource consumption processes (electricity, gas, water consumption), radar signals, vibrations of bearings of electric machines and others. A common mathematical model (especially in the multidimensional case) of this type of signal is a linear random process that allows the signal to be represented as the sum of a large number of stochastically independent random impulses that occur at Poisson moments. If the impulses are stochastically dependent (or the moments of time of their appearance are not Poisson), then the mathematical model is a conditional linear random process. The definition and analysis of the probabilistic properties of such processes for the multidimensional case have not been conducted. The paper defines a multidimensional conditional linear random process, each component of which is represented as a stochastic integral of a random kernel driven by a process with independent increments. Expressions for the characteristic function and moment functions of the specified process are obtained. The approach used was to use the mathematical apparatus of conditional characteristic functions, as well as the known representation of an infinitely divisible characteristic function of a linear random process as a functional of a process with independent increments. The obtained results provide a possibility for theoretical analysis of probabilistic properties of multichannel stochastic signals, the mathematical model of which is a multidimensional conditional linear random process. Justification of their properties of stationarity or cyclostationarity, which are the consequence of corresponding properties of the kernel and process with independent increments, can be carried out.

Martingales and the fixation time of evolutionary graphs with arbitrary dimensionality.

Evolutionary graph theory (EGT) investigates the Moran birth–death process constrained by graphs. Its two principal goals are to find the fixation probability and time for some initial population of mutants on the graph. The fixation probability of graphs has received considerable attention. Less is known about the distribution of fixation time. We derive clean, exact expressions for the full conditional characteristic functions (CCFs) of a close proxy to fixation and extinction times. That proxy is the number of times that the mutant population size changes before fixation or extinction. We derive these CCFs from a product martingale that we identify for an evolutionary graph with any number of partitions. The existence of that martingale only requires that the connections between those partitions are of a certain type. Our results are the first expressions for the CCFs of any proxy to fixation time on a graph with any number of partitions. The parameter dependence of our CCFs is explicit, so we can explore how they depend on graph structure. Martingales are a powerful approach to study principal problems of EGT. Their applicability is invariant to the number of partitions in a graph, so we can study entire families of graphs simultaneously.

Martingales and the characteristic functions of absorption time on bipartite graphs.

Evolutionary graph theory investigates how spatial constraints affect processes that model evolutionary selection, e.g. the Moran process. Its principal goals are to find the fixation probability and the conditional distributions of fixation time, and show how they are affected by different graphs that impose spatial constraints. Fixation probabilities have generated significant attention, but much less is known about the conditional time distributions, even for simple graphs. Those conditional time distributions are difficult to calculate, so we consider a close proxy to it: the number of times the mutant population size changes before absorption. We employ martingales to obtain the conditional characteristic functions (CCFs) of that proxy for the Moran process on the complete bipartite graph. We consider the Moran process on the complete bipartite graph as an absorbing random walk in two dimensions. We then extend Wald’s martingale approach to sequential analysis from one dimension to two. Our expressions for the CCFs are novel, compact, exact, and their parameter dependence is explicit. We show that our CCFs closely approximate those of absorption time. Martingales provide an elegant framework to solve principal problems of evolutionary graph theory. It should be possible to extend our analysis to more complex graphs than we show here.

A regression-based Monte Carlo method to solve two-dimensional forward backward stochastic differential equations

The purpose of this paper is to investigate the numerical solutions to two-dimensional forward backward stochastic differential equations(FBSDEs). Based on the Fourier cos-cos transform, the approximations of conditional expectations and their errors are studied with conditional characteristic functions. A new numerical scheme is proposed by using the least-squares regression-based Monte Carlo method to solve the initial value of FBSDEs. Finally, a numerical experiment in European option pricing is implemented to test the efficiency and stability of this scheme.

Wald’s martingale and the conditional distributions of absorption time in the Moran process

Many models of evolution are stochastic processes, where some quantity of interest fluctuates randomly in time. One classic example is the Moranbirth–death process, where that quantity is the number of mutants in a population. In such processes, we are often interested in their absorption (i.e. fixation) probabilities and the conditional distributions of absorption time. Those conditional time distributions can be very difficult to calculate, even for relatively simple processes like the Moran birth–death model. Instead of considering the time to absorption, we consider a closely related quantity: the number of mutant population size changes before absorption. We use Wald’s martingale to obtain the conditional characteristic functions of that quantity in the Moran process. Our expressions are novel, analytical and exact, and their parameter dependence is explicit. We use our results to approximate the conditional characteristic functions of absorption time. We state the conditions under which that approximation is particularly accurate. Martingales are an elegant framework to solve principal problems of evolutionary stochastic processes. They do not require us to evaluate recursion relations, so when they are applicable, we can quickly and tractably obtain absorption probabilities and times of evolutionary models.

The mean‐reverting 4/2 stochastic volatility model: Properties and financial applications

AbstractThis article defines and studies a stochastic process that combines two important stylized facts of financial data: reversion to the mean, and a flexible generalized stochastic volatility process: the 4/2 process. This work is motivated by the modeling of at least two financial asset classes: commodities and volatility indexes. We provide analytical expressions for the conditional characteristic functions and closed‐form approximations to relevant cases, in particular a mean‐reverting Heston stochastic volatility model. Our results describe feasible changes of measure with the final aim of pricing financial derivatives. The empirical analysis and the estimation methodology confirm the need of such a model in several examples from the targeted asset classes. Applications to option pricing corroborate the substantial impact on the implied volatility surfaces of the new parameters.

Distributional Properties of Continuous Time Processes: From CIR to Bates

We focus on returns defined as log-price differentials and generated by a diffusion process which incorporates stochastic volatility and jumps in prices. The jumps are properly compensated for this model. The stochastic volatility follows the well-known CIR process. We present general conditional and unconditional (co-)moment formulas for the solution of this process. By identifying these moments with those of a non-central chi-squared distribution, we derive distributional properties in a way that significantly differs from the historic approaches. Next, we derive the conditional and unconditional characteristic functions of log-returns which allows us to generate conditional and unconditional moments. We provide closed form expressions for the first four unconditional moments of log-returns.

On the identification of models with conditional characteristic functions

In this paper, we revisit a potential identification failure of estimation of models based on conditional characteristic functions. An arbitrary choice of moment conditions does not guarantee the parameter identifiability. We show that moment conditions based on the conditional characteristic functions and an exponential instrument satisfy the identification condition. We also provide a consistent GMM estimator that is computationally tractable and its asymptotic properties. This paper could provide useful guidelines to empirical researchers estimating this class of models.

Pricing of FX options in the MPT/CIR jump-diffusion model with approximative fractional stochastic volatility

In this paper, the pricing of foreign exchange (FX) options is studied under the Moretto–Pasquali–Trivellato (MPT) stochastic volatility model by introducing an approximative fractional stochastic volatility and jumps, in which the FX rate has log-normal jump amplitudes, the volatility has asymmetric double exponential jump amplitudes, and the domestic and foreign interest rates are governed by Cox–Ingersoll–Ross (CIR) dynamics. By employing a suitable version of the Fourier inversion technique for corresponding conditional characteristic functions, a semi-analytical formula for the price of FX European call options is obtained under mild conditions. The behavior of the newly derived pricing formula is further demonstrated through some numerical experiments.

Linearized filtering of affine processes using stochastic Riccati equations

We consider an affine process X which is only observed up to an additive white noise, and we ask for the law of Xt, for some t>0, conditional on all observations up to time t. This is a general, possibly high dimensional filtering problem which is not even locally approximately Gaussian, whence essentially only particle filtering methods remain as solution techniques. In this work we present an efficient numerical solution by introducing an approximate filter for which conditional characteristic functions can be calculated by solving a system of generalized Riccati differential equations depending on the observation and the process characteristics of X. The quality of the approximation can be controlled by easily observable quantities in terms of a macro location of the signal in state space. Asymptotic techniques as well as maximization techniques can be directly applied to the solutions of the Riccati equations leading to novel very tractable filtering formulas. The efficiency of the method is illustrated with numerical experiments for Cox–Ingersoll–Ross and Wishart processes, for which Gaussian approximations usually fail.

A stochastic volatility factor model of heston type. Statistical properties and estimation

In this paper we study the properties of a new multidimensional continuous-time stochastic covariance process, the Stochastic Volatility Factor model. Two helpful conditional characteristic functions, one needed for estimation and the other used in the pricing of financial derivatives are provided, together with the properties of the instantaneous dependence structure. Conditions for stationarity, ergodicity and mixing properties of the increments are studied. The estimation of the model is performed using the Continuum-Generalized Method of Moments (CGMM). A simulation exercise is included showing parameters are recovered, confirming identifiability. The model is also calibrated to exemplary financial data.

Asymptotic Relation for the Transition Density of the Three-Dimensional Markov Random Flight on Small Time Intervals

We consider the Markov random flight $\bold X(t), \; t>0,$ in the three-dimensional Euclidean space $\Bbb R^3$ with constant finite speed $c>0$ and the uniform choice of the initial and each new direction at random time instants that form a homogeneous Poisson flow of rate $\lambda>0$. Series representations for the conditional characteristic functions of $\bold X(t)$ corresponding to two and three changes of direction, are obtained. Based on these results, an asymptotic formula, as $t\to 0$, for the unconditional characteristic function of $\bold X(t)$ is derived. By inverting it, we obtain an asymptotic relation for the transition density of the process. We show that the error in this formula has the order $o(t^3)$ and, therefore, it gives a good approximation on small time intervals whose lengths depend on $\lambda$. Estimate of the accuracy of the approximation is analysed.

Parameter estimation for multivariate diffusion processes with the time inhomogeneously positive semidefinite diffusion matrix

ABSTRACTStatistical inference for the diffusion coefficients of multivariate diffusion processes has been well established in recent years; however, it is not the case for the drift coefficients. Furthermore, most existing estimation methods for the drift coefficients are proposed under the assumption that the diffusion matrix is positive definite and time homogeneous. In this article, we put forward two estimation approaches for estimating the drift coefficients of the multivariate diffusion models with the time inhomogeneously positive semidefinite diffusion matrix. They are maximum likelihood estimation methods based on both the martingale representation theorem and conditional characteristic functions and the generalized method of moments based on conditional characteristic functions, respectively. Consistency and asymptotic normality of the generalized method of moments estimation are also proved in this article. Simulation results demonstrate that these methods work well.

Some results following from conditional characteristic functions

ABSTRACTThe purpose of this article is to derive rigorously some results following from conditional characteristic functions and anticipate that they will prove to be of significant applicability. Specifically, Khintchine weak law of large numbers, Lévy central limit theorem, inversion theorem, uniqueness theorem, characterization of identical distribution, and characterization of independence are all generalized to conditional setup.

On discrete-time stable multiple Markov processes

ABSTRACTIn this paper, we provide necessary conditions for discrete-time symmetric α-stable processes to be linear 2-ple Markov. The aim of this paper is to extend the results given by Adler et al. (1990) to general multiple Markov processes, called linear multiple Markov processes. A necessary and sufficient condition based on the covariation for SαS processes to be linear multiple Markov is provided. A complete description of this class of covariation functions including the stationary case is given. We show that the sum of two independent time-changed Levy motions is linear multiple Markov of order 2. The SαS conditional distributions are also studied, and the conditional characteristic functions of linear 2-ple Markov SαS processes are formed. Finally, we study two famous classes of SαS processes, namely, sub-Gaussian and harmonizable processes, and discuss their Markov properties.