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Estimation of Affine Asset Pricing Models Using the Empirical Characteristic Function
Abstract
The known functional form of the conditional characteristic function (CCF) of discretely sampled observations from an affine diffusion is used to develop computationally tractable and asymptotically efficient estimators of the parameters of affine diffusions, and of asset pricing models in which the state vectors follow affine diffusions. Both ‘time-domain’ estimators, based on Fourier inversion of the CCF, and ‘frequency-domain’ estimators, based directly on the CCF, are constructed. A method-of-moments estimator based on the CCF is shown to approximate the efficiency of maximum likelihood for affine diffusion and asset pricing models.
... Since HFD is measured using an accelerometer, the resulting value is displayed in terms of acceleration due to gravity {g}. Tis method provides an early warning of bearing problems [68][69][70]. ...
... Te measured signal value is expressed in {gSE} (SE acceleration units). SE measurement reveals early signs of rolling bearing failure [68,70,71]. ...
... Te technical condition of rolling bearings is estimated by the magnitude of the peaks of the values of individual impulses and expressed in decibels [69,72]. Te SPM method makes it possible to detect the deterioration of lubrication conditions and the appearance of defects in rolling bearings at an early stage [69,70]. Signal spectrum analysis reveals the cause of bearing condition changes. ...
The perfection of methods and means of nondestructive testing and technical diagnostics is determined by the level of development of science and modern industrial technologies. The desire to develop technologies determines the extent and degree to which the monitoring of the state of substances, materials, products—and now the state of the natural environment—are becoming increasingly relevant. The methods and means of condition monitoring and the diagnostics of rolling bearings have been in development for more than 60 years. Despite some successes, however, there is currently no information concerning the veracity of means to completely resolve the bearing diagnostics problem. This paper provides a fairly brief overview of methods and means for monitoring the condition and diagnosis of rolling bearings and also describes one of the newest trends in this field—the analysis of the properties of the characteristic function of vibroacoustic (VA) signals in order to determine the condition of the objects of control and, in particular, rolling bearings. It is shown that the magnitude of the module and the area of the characteristic function of the VA signal are very effective criteria for assessing the technical condition of a rolling bearing.
... Thus, the EL estimator cannot achieve high efficiency in a finite sample. Second, previous studies [4,45,48] point out that the weight matrix of the optimal GMM estimator tends to be singular as the number of moment conditions increases. Therefore, the GMM-type estimators are infeasible when the number of moment conditions is relatively large compared to the sample size. ...
... Some studies pointed out that the GMM estimator is infeasible when the grid of (u 1 , . . . , u k ) is too fine, because the weight matrix becomes singular [2,4,45,48]. There might be some confusion regarding this argument. ...
... Remark 2.3. The models considered by [45] and [2] are different from ours. They treat more complicated financial models such as affine price models. ...
A trigonometrically approximated maximum likelihood estimation for -stable laws is proposed. The estimator solves the approximated likelihood equation, which is obtained by projecting a true score function on the space spanned by trigonometric functions. The projected score is expressed only by real and imaginary parts of the characteristic function and their derivatives, so that we can explicitly construct the targeting estimating equation. We study the asymptotic properties of the proposed estimator and show consistency and asymptotic normality. Furthermore, as the number of trigonometric functions increases, the estimator converges to the exact maximum likelihood estimator, in the sense that they have the same asymptotic law. Simulation studies show that our estimator outperforms other moment-type estimators, and its standard deviation almost achieves the Cram\'er--Rao lower bound. We apply our method to the estimation problem for -stable Ornstein--Uhlenbeck processes in a high-frequency setting. The obtained result demonstrates the theory of asymptotic mixed normality.
... It operationalizes this model by providing and analyzing feasible methods for computing its likelihood and its maximum likelihood estimator. Singleton (2001) developed similar methods for a different class of models, discretely sampled affine diffusions. He noted that the density of an observation of such a diffusion conditional on the previous observation is not known explicitly, but that its characteristic function is. ...
... Singleton (2001) developed a similar GMM estimator for discretely sampled diffusions, based on their characteristic function. In that context, censoring is not important and such a GMM estimator is a natural alternative to maximum likelihood. ...
... This is howSingleton (2001) handled his maximum likelihood estimator of a discretely sampled affine diffusion, which, like our estimator, required numerical Fourier inversion. He expressed some worries about the computational burden of his Fourier inversion procedure, but only for the multivariate case. ...
We present a method for computing the likelihood of a mixed hitting-time model that specifies durations as the first time a latent L\'evy process crosses a heterogeneous threshold. This likelihood is not generally known in closed form, but its Laplace transform is. Our approach to its computation relies on numerical methods for inverting Laplace transforms that exploit special properties of the first passage times of L\'evy processes. We use our method to implement a maximum likelihood estimator of the mixed hitting-time model in MATLAB. We illustrate the application of this estimator with an analysis of Kennan's (1985) strike data.
... The moment-based method is a case in point but has yet to be recognized fully. A few important contributions include the simulated MM (SMM); see Duffie and Singleton (1993)), efficient MM (EMM); see Bansal et al. (1994) and Gallant and Tauchen (1996), and generalized MM (GMM); see Singleton (2001), Bollerslev and Zhou (2002), Jiang and Knight (2002), and Chacko and Viceira (2003). SMM simulates sequences from the target models and estimates parameters by matching simulated data moments with actual data moments numerically. ...
... It is notable that these affine SV models do not possess closed-form solution for their transition density functions. However, for the general affine SV model, closed-form expressions for their conditional Characteristic Functions (CFs) can be derived (Duffie et al., 2000;Singleton, 2001;Chacko and Viceira, 2003). Jiang and Knight (2002) extended this work by developing a closed-form CF for a simplified version of the Heston model. ...
We develop moment estimators for the parameters of affine stochastic volatility models. We first address the challenge of calculating moments for the models by introducing a recursive equation for deriving closed-form expressions for moments of any order. Consequently, we propose our moment estimators. We then establish a central limit theorem for our estimators and derive the explicit formulas for the asymptotic covariance matrix. Finally, we provide numerical results to validate our method.
... In general, estimation strategies based on the transform space of conditional characteristic functions are, of course, not new to the literature. For instance, Carrasco and Florens (2000) develop a generalized method of moments (GMM) estimator with a continuum of moment conditions based on the CCF; see also Singleton (2001), Carrasco, Chernov, Florens, and Ghysels (2007). In applications to option prices, Boswijk, Laeven, and Lalu (2015) and Boswijk, Laeven, Lalu, and Vladimirov (2021) propose to imply the latent state vector from a panel of options and then estimate the model via GMM with a continuum of moments. ...
... This is because in practice we can have different expiration periods for different days. 7 See Singleton (2001) and Chacko and Viceira (2003), who use this approach in a GMM estimation setting based on the empirical characteristic function. 8 The transition matrix Tt will not be time-varying in stationary AJD processes with equidistant observations, but we do not impose this time-constancy in the notation, also to avoid confusion with the sample size T . ...
We develop a novel filtering and estimation procedure for parametric option pricing models driven by general affine jump-diffusions. Our procedure is based on the comparison between an option-implied, model-free representation of the conditional log-characteristic function and the model-implied conditional log-characteristic function, which is functionally affine in the model's state vector. We formally derive an associated linear state space representation and establish the asymptotic properties of the corresponding measurement errors. The state space representation allows us to use a suitably modified Kalman filtering technique to learn about the latent state vector and a quasi-maximum likelihood estimator of the model parameters, which brings important computational advantages. We analyze the finite-sample behavior of our procedure in Monte Carlo simulations. The applicability of our procedure is illustrated in two case studies that analyze S&P 500 option prices and the impact of exogenous state variables capturing Covid-19 reproduction and economic policy uncertainty.
... The existing literature on the estimation of the models with CCF has mainly focused on the efficiency of an estimator by assuming the global identification condition (e.g., Singleton (2001) E-mail address: hyojin_han@xmu.edu.cn. and Carrasco et al. (2007)). ...
... may fail to identify θ 0 because this uses information from the marginal distribution of X t only and cannot capture the correlation between X t and X t+1 . Although several choices of moment conditions have been proposed in the existing literature such as Singleton (2001) and Carrasco et al. (2007), they focus on the moment conditions that lead to efficient estimation only while assuming the identification condition. In order to ensure parameter identifiability, we consider the instrument in the exponential form ...
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.
... Since the Heston model belongs to the class of affine stochastic volatility models, its characteristic function can be derived from the affine model structure and the corresponding density can be obtained by classic Fourier inversion techniques from the characteristic function (see e.g. Bates [13]), which is however a time-consuming numerical task. Alternatively, estimation in the Heston model can solely be based on empirical estimates of the characteristic function as for example in Jiang and Knight [72] or Singleton [103], or Fourier inversion-based methods can be applied to calibrate the model parameters directly to observed option prices. The latter approach however determines the model parameters under an equivalent martingale measure, a so-called risk-neutral measure, see for example Eberlein and Kallsen [44,Section 11.2.3], while estimation routines based on observed returns like maximum likelihood or least squares minimisation procedures fit the model parameters under the true physical measure driving the returns. ...
This paper is devoted to parameter estimation for partially observed polynomial state space models. This class includes discretely observed affine or more generally polynomial Markov processes. The polynomial structure allows for the explicit computation of a Gaussian quasi-likelihood estimator and its asymptotic covariance matrix. We show consistency and asymptotic normality of the estimating sequence and provide explicitly computable expressions for the corresponding asymptotic covariance matrix.
... After that, several authors (cf. Singleton, 2001; described implementation of the ECF method in econometric analysis and finance. On the other hand, several other new theoretical extensions of the CF-based estimators can be found, for instance in Balakrishnan et al. (2013) and Kotchoni (2012Kotchoni ( , 2014. ...
A general type of a Split-BREAK process with Gaussian innovations (henceforth, Gaussian Split-BREAK or GSB process) is considered. The basic stochastic properties of the model are studied and its characteristic function derived. A procedure to estimate the parameter of the GSB model based on the Empirical Characteristic Function (ECF) is proposed. Our simulations suggest that the proposed method performs well compared to a Method of Moment procedure used as benchmark. The empirical use of the GSB model is illustrated with an application to the time series of total values of shares traded at Belgrade Stock Exchange.
... Statistical inference based on the characteristic functions was proposed by Feuerverger and Mureika [20], Feuerverger and McDunnough [19] for independent observations and Feuerverger [18] for discrete time series. Singleton [32] introduced the approach to inference for parametric continuous-time Markov processes and showed that estimation can be carried out based on the CCF without having to carry out the the Fourier inversion. Chacko and Viceira [8] proposed a generalized method of moment estimator (GMM) for parameters at a finite number of frequencies of the CCF. ...
Markov processes are used in a wide range of disciplines, including finance. The transition densities of these processes are often unknown. However, the conditional characteristic functions are more likely to be available, especially for L\'{e}vy-driven processes. We propose an empirical likelihood approach, for both parameter estimation and model specification testing, based on the conditional characteristic function for processes with either continuous or discontinuous sample paths. Theoretical properties of the empirical likelihood estimator for parameters and a smoothed empirical likelihood ratio test for a parametric specification of the process are provided. Simulations and empirical case studies are carried out to confirm the effectiveness of the proposed estimator and test.
... Central to the affine asset-pricing models is the "extended transform" of the underlying AJD process, which has applications to a wide variety of valuation problems: affine defaultable term structure models (Dai and Singleton (2000), Duffie and Singleton (1999), Pan and Singleton (2008), Dai and Singleton (2002), Duffee (2002)), maximum likelihood estimation of affine asset pricing models (Singleton (2001)), affine option pricing models (Heston (1993), Bakshi et al. (1997)), reduced-form credit risk models (Duffie (2005)), and computation of credit value adjustments (Ballotta et al. (2019)). In the subsequent literature, Leippold and Wu (2002) proposed a quadratic term structure model where the bond yield are quadratic functions of the state vector, and laid out its transform and specification analysis. ...
We establish a novel duality relationship between continuous and discrete non-negative additive functionals of stochastic (not necessarily Markovian) processes and their right inverses. For general Markov processes, we further extend and develop a theoretical and computational framework for the transform analysis via an operator-based approach, i.e. through the infinitesimal generators. More precisely, we characterize the joint double transforms of additive functionals of Markov processes and the terminal values in both discrete and continuous time. In particular, under the continuous-time Markov chain (CTMC) setting, we obtain single Laplace transforms for continuous/discrete additive functionals and their inverses. Lastly, we discuss the potential applications of the proposed transform methodology in contextual areas in finance and queuing theory within operations research.
... The trend of financial market development, from financial engineering to model risk management, is of strategic significance to financial institutions and regulators [26]. Teaching reform of financial engineering majors can enhance discipline strength, which refers to the comprehensive strength of a discipline in academic research, teaching level, talent cultivation, and other aspects; the reform should conform to the development of the times, incorporate financial technology factors, and cultivate innovative financial talents [27]. ...
In the post-epidemic era, the labor market has become increasingly complex, making it even more crucial to incorporate sustainability into employment demand. As we enter the post-pandemic era, a globalization trend has become more apparent. It is crucial to modernize employability through educational reform in order to assist employees in enhancing their professional skills. This study began by analyzing the importance of financial engineering practice instruction and graduate employability in the post-epidemic era. Second, the study proposed the content and a plan for inter-disciplinary teaching reform to address talent cultivation needs based on labor market requirements. Third, a face-to-face survey and interview were conducted with students affected by changes in teaching, and the results were analyzed and summarized. On this basis, the impact of education reform was evaluated using both the expert scoring method and the analytic hierarchy approach. The results indicated that the suggested financial engineering teaching reform program improved the school’s discipline strength, enrollment rate, employment rate, and competition awards, especially discipline strength. This research can be used to inform the teaching of financial engineering majors in various countries, assist job candidates in enhancing their professional skills, and build a formidable talent pool for the labor market.
... This was achieved by applying the generalized method of moments (GMM) as a robustness check. GMM estimation has the benefit that it generates consistent and efficient results (Singleton, 2001). Further, GMM is useful in that it addresses the issue of endogeneity (Blundell and Bond, 1998). ...
... The additional dependency parameters C, ν may be estimated by least squares matching of the empirical characteristic function and the theoretical characteristic function (Feuerverger and McDunnough 1981;Singleton 2001). Here we estimate C, ν by matching simulated model tail probabilities to observed tail probabilities in multiple dimensions by a procedure described later in the paper. ...
Two price economy principles motivate measuring risk by the cost of acquiring the opposite of the centered or pure risk position at its upper price. Asymmetry in returns leads to differences in risk charges for short and long positions. Short risk charges dominate long ones when the upper tail dominates the comparable lower tail for charges based on distorted expectations. Positive mean return targets acquire long positions with negative mean return targets taking short positions. In each case the appropriate risk charge is minimized to construct two frontiers, one for the positive, and the other for negative, mean return targets. Multivariate return distributions reflect limit laws given by Q self-decomposable laws displaying decay rates in skewness and excess kurtosis slower than those for processes of independent and identically distributed returns. Frontiers at longer horizons display greater efficiency reflected by lower risk charges for comparable mean return targets. The short side frontiers also display greater risk charges than their long side counterparts. All efficient portfolios deliver asset pricing equations whereby required returns in excess of a reference rate are a market price of risk times a risk gradient evaluated at the efficient portfolio. Variations in frontiers and points on the frontier induce differences in reference rates, risk gradients, and the market prices of risk that can yet lead to comparable required returns.
... α ν x 2t−1, x 2t t = 1, · · · N /2 N For the dependency parameter in VGCTC and in BGCTC, we consider estimation via the matching of empirical characteristic functions Feuerverger and McDunnough (1981) and Singleton (2001). First, form adjacent pairs of returns for for an even number of daily return observations. ...
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 one 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. An analysis of financial markets for 2020 in comparison with 2019 displays a post-COVID slowdown in financial markets.
... Another approaches contain Monte-Carlo Markov Chain methods and Metropolis-Hastings algorithm [7]. It can be also estimated by characterictic function [8] or indirect estimation [9]. Methods, proposed here require more computer resources and error is very difficult to control. ...
The article considers parameter estimation constructing such as quasi-maximum likelyhood estimation and one step estimation in statistical models generated by solution of stochastic differential equation. It has been developed a software for parameter estimating and has been presented correspondent testing and comparing.
... Our results will help investors better diversify their portfolio by adding this cryptocurrency and this was clearly approved, especially after the "covid 19" crisis, where the volume of transactions has risen sharply in this type of market. This paper has made certain contributions, but several extensions are still possible, and it can find the best results if you opt for extensions of SVOL like the model of Singleton (2001Singleton ( ), knight et al. (2002 and others. ...
The purpose of this article is to find a better technique for estimating the volatility of the price of bitcoin on the one hand and to check if this special kind of asset called cryptocurrency behaves like other stock market indices. We include five stock market indexes for different countries such as Standard and Poor’s 500 composite Index (S&P), Nasdaq, Nikkei, Stoxx, and DowJones. Using daily data over the period 2010–2019. We examine two asymmetric stochastic volatility models used to describe the volatility dependencies found in most financial returns. Two models are compared, the first is the autoregressive stochastic volatility model with Student’s t-distribution (ARSV-t), and the second is the basic SVOL. To estimate these models, our analysis is based on the Markov Chain Monte-Carlo method. Therefore, the technique used is a Metropolis–Hastings (Hastings in Biometrika 57:97–109, 1970), and the Gibbs sampler (Casella and George in Am Stat 46:167–174, 1992; Gelfand and Smith in J Am Stat Assoc 85:398–409, 1990; Gilks and Wild in 41:337–348, 1992). Model comparisons illustrate that the ARSV-t model performs better performances. We conclude that this model is better than the SVOL model on the MSE and AIC function. This result concerns bitcoin as well as the other stock market indices. Without forgetting that our finding proves the efficiency of Markov Chain for our sample and the convergence and stability for all parameters to a certain level. On the whole, it seems that permanent shocks have an effect on the volatility of the price of the bitcoin and also on the other stock market. Our results will help investors better diversify their portfolio by adding this cryptocurrency.
... , Singleton [11] , Jiang Knight [12] , Chacko Viceira [13] , GMM . ...
... Monte Carlo evidence suggests that the finite sample efficiency of the proposed GMM estimator is comparable to the asymptotically efficient MLE, the 3 Recently there is a growing literature on jump-diffusion interest rate modeling (see Chacko and Das, 1999;Johannes, 1999;Piazzesi, 2000, among many others), which ranges from short-rate dynamics to fixedincome derivatives, from market-implied jumps to macroeconomic announcements, and from parametric to nonparametric specifications. 4 Alternatively, an equivalent spectral method of moments is developed by exploiting the closed-form conditional characteristic functions for the affine jump-diffusion model (Chacko and Viceira, 1999;Singleton, 2001;Jiang and Knight, 2002;Carrasco et al., 2002). However, the selection of spectral moments remains as a difficult challenge, whereas in the classical method of moments, a natural choice is the lower-order moments. ...
This paper exploits the Itô's formula to derive the conditional moments vector for the class of interest rate models that allow for nonlinear volatility and flexible jump specifications. Such a characterization of continuous-time processes by the Ito Conditional Moment Generator noticeably enlarges the admissible set beyond the affine jump-diffusion class. A simple GMM estimator can be constructed based on the analytical solution to the lower order moments, with natural diagnostics of the conditional mean, variance, skewness, and kurtosis. Monte Carlo evidence suggests that the proposed estimator has desirable finite sample properties, relative to the asymptotically efficient MLE. The empirical application singles out the nonlinear quadratic variance as the key feature of the U.S. short rate dynamics.
... To inverse the characteristic function ψ t (u), we truncate it at u = ±50, as it largely damps out after these values of u [see Singleton [20]]. In implementing the AS-D method, we selected the bandwidth according to the small sample method of Fan and Gijbels [10]. ...
... A particularly fruitful development in dynamic asset pricing models is the affine 31 jump diffusions (AJD) by Duffie et al. (2000). Central to the affine asset-pricing models is the 32 "extended transform" of the underlying AJD process, which has applications to a wide variety of 33 valuation problems: affine defaultable term structure models (Dai and Singleton (2000), Duffie and 34 Singleton (1999), Pan and Singleton (2008), Dai and Singleton (2002), Duffee (2002)), maximum 35 likelihood estimation of affine asset pricing models (Singleton (2001)), affine option pricing models 36 ( Heston (1993), Bakshi et al. (1997)), reduced-form credit risk models ( Duffie (2005)), and computation 37 of credit value adjustments ( Ballotta et al. (2019)). In the subsequent literature, Leippold and Wu 38 (2002) proposed a quadratic term structure model where the bond yield are quadratic functions of the 39 state vector, and laid out its transform and specification analysis. ...
... A particularly fruitful development in dynamic asset pricing models is the affine jump diffusions (AJD) by Duffie et al. (2000). Central to the affine asset-pricing models is the "extended transform" of the underlying AJD process, which has applications to a wide variety of valuation problems: affine defaultable term structure models (Dai and Singleton (2000), Duffie and Singleton (1999), Pan and Singleton (2008), Dai and Singleton (2002), Duffee (2002)), maximum likelihood estimation of affine asset pricing models (Singleton (2001)), affine option pricing models (Heston (1993), Bakshi et al. (1997)), reduced-form credit risk models (Duffie (2005)), and computation of credit value adjustments (Ballotta et al. (2019)). In the subsequent literature, Leippold and Wu (2002) proposed a quadratic term structure model where the bond yield are quadratic functions of the state vector, and laid out its transform and specification analysis. ...
We establish a novel duality relationship between continuous and discrete non-negative additive functionals of stochastic (not necessarily Markovian) processes and their right inverses. For general Markov processes, we further extend and develop a theoretical and computational framework for the transform analysis via an operator-based approach, i.e. through the infinitestimal generators. More precisely, we characterize the joint double transforms of additive functionals of Markov processes and the terminal values in both discrete and continuous time. In particular, under the continuous-time Markov chain (CTMC) setting, we obtain single Laplace transforms for continuous/discrete additive functionals and their inverses. Lastly, we discuss the potential applications of the proposed transform methodology in various contextual areas in finance and queuing theory within operations research.
This paper studies the asymptotic behavior of the Fisher information for a Levy process discretely sampled at an increasing frequency. We show that it is possible to distinguish not only the continuous part of the process from its jumps part, but also different types of jumps, and derive the rates of convergence of efficient estimators.
We introduce ajdmom, a Python package designed for automatically deriving moment formulas for the well-established affine jump diffusion (AJD) processes. ajdmom can produce explicit closed-form expressions for moments or conditional moments of any order, significantly enhancing the usability of AJD models. Additionally, ajdmom can compute partial derivatives of these moments with respect to the model parameters, offering a valuable tool for sensitivity analysis. The package's modular architecture makes it easy for adaptation and extension by researchers. ajdmom is open-source and readily available for installation from GitHub or the Python package index (PyPI).
The perfection of methods and means of non-destructive testing and technical diagnostics is determined by the level of development of science and modern industrial technologies. The desire to develop technologies determines the growing needs for monitoring the state of substances, materials, products, and now the control of the state of the natural environment is becoming more and more relevant. Methods and means of condition monitoring and diagnostics of rolling bearings have been developed and developed for more than 60 years. Despite certain successes, today there is no information about the veracity of ways to solve this problem. The paper provides a fairly brief overview of methods and methods for monitoring the condition and diagnosis of rolling bearings, and also describes one of the newest trends in this field – the analysis of the properties of the characteristic function of vibroacoustic signals in order to determine the condition of the objects of control and, in particular, rolling bearings. It is shown that the magnitude of the module and the area of the characteristic function of the vibroacoustic signal are very effective criteria for assessing the technical condition of a rolling bearing.
We examine time-varying jump risk for modeling stock price dynamics and cross-sectional option prices. We explore jump-diffusion specifications with two independently evolving processes for stochastic volatility and jump intensity, respectively. We explicitly impose time-series consistency in model estimation using a Markov Chain Monte Carlo (MCMC) method. We find that both the jump size and standard deviation of jump size premia are more prominent under time-varying jump risk. simultaneous jumps in returns and volatility help reconcile the time series of returns, volatility, and jump intensities. Finally, independent time-varying jump intensities improve the cross-sectional fit of option prices, especially at longer maturities.
Transform inversion is a popular approach to pricing many types of path-dependent options under various dynamic models. However, it is usually difficult to guarantee the accuracy due to the lack of error control, which often depends heavily on the asymptotic property of the transform function. This article characterizes the asymptotic behaviors of all the complex roots of the exponent equation under the mixed-exponential jump diffusion model (MEM), by which one can output the error bounds and achieve any pre-specified error tolerance in the transform inversion algorithm when pricing various path-dependent options under the MEM. Numerical examples indicate that the resulting transform inversion algorithm with our computable error bounds are reliably accurate and efficient for pricing lookback options and barrier options, and evaluating a joint distribution under the MEM.
Transform inversion algorithm is a popular method in the areas of Statistics and Financial Engineering. However, it is often difficult to guarantee its accuracy due to the lack of error control, which depends heavily on the asymptotic feature of the transform function. This article characterizes the asymptotic behavior of the Heston stochastic volatility model, by which one can derive computable error bounds and achieve any pre‐specified error tolerance in the transform inversion algorithm. Moreover, a universal method is proposed to specify a closed‐form upper bound of the discretization error. Numerical examples indicate that the resulting inversion algorithm with our computable error bounds is reliably accurate and efficient, for parameter estimation and option pricing under the Heston model.
This paper provides tractable expressions for broad classes of standard transforms for the case of affine jump diffusions. Extending and unifying existing results, these include expressions for exponential, polynomial-log-linear, and polynomial transforms of a state vector in both conditional and unconditional form. We cover the usual single-period setting, but also extensions to multi-period settings, non-stationarity, and time-inhomogeneous state dynamics. Using these standard transforms, we discuss applications in asset pricing and parameter estimation.
We estimate and test for multiple structural breaks in distribution via an empirical characteristic function approach. By minimizing the sum of squared generalized residuals, we can consistently estimate the break fractions. We propose a sup- F type test for structural breaks in distribution as well as an information criterion and a sequential testing procedure to determine the number of breaks. We further construct a class of derivative tests to gauge possible sources of structural breaks, which is asymptotically more powerful than the smoothed nonparametric tests for structural breaks. Simulation studies show that our method performs well in determining the appropriate number of breaks and estimating the unknown breaks. Furthermore, the proposed tests have reasonable size and excellent power in finite samples. In an application to exchange rate returns, our tests are able to detect structural breaks in distribution and locate the break dates. Our tests also indicate that the documented breaks appear to occur in variance and higher-order moments, but not so often in mean.
In this article, we develop an identifiable multi-factor stochastic volatility model for international portfolio choice problems in complete and incomplete markets. Allowing for stochastic covariance between financial asset returns and foreign exchange rates, optimal investment strategies are derived in closed form and welfare losses arising from suboptimal investment strategies are analysed. Moreover, we provide a two-step procedure for estimating as well as calibrating the model parameters and use this ansatz to illustrate optimal investment decisions for the S&P 500, the German blue chip index DAX, and the USD/EUR foreign exchange rate. We find, both theoretically and empirically, that the model satisfies various well-known stylized facts of equity and foreign exchange rate markets and that investors who invest myopically or ignore derivative assets can incur substantial welfare losses implying strong evidence for significant welfare benefits from international diversification across different asset classes.
Errors in variables in linear regression continue to be a significant empirical issue in financial econometrics. We propose using the characteristic function (CF) to obtain estimates for linear models with errors in the variables. By assuming that the explanatory variable follows a flexible double gamma distribution, we obtain closed-form expressions for the analytic CF of the data generating process. We show that our method performs well relative to existing techniques that address error-in-variables (EIVs) through simulations. We further extend our CF technique to a multivariate setting where it continues to produce accurate estimates. We illustrate the performance of our procedure by estimating the capital asset pricing model and a two-factor model.
Correlation graphs are introduced to delineate the levels observed in data and models for return and squared return correlations. A sample of 2048 representative pairs of equity assets is selected from a possible collection of [Formula: see text] pairs by quantization. Five copulas are estimated and simulated on these pairs of returns, the Gaussian, t-copula, Clayton, Gumbel and Frank. Additionally, the multivariate bilateral gamma (MBG) model that introduces dependence via common time changes is also fit and simulated. Results of fit statistics on returns, CoSkew and CoKurtosis pairs are reported. The general ordering of the models is MBG, t-copula, followed by the Gaussian, Frank, Gumbel and Clayton copulas.
Derivatives, especially equity and volatility options, contain valuable and oftentimes essential information for estimating stochastic volatility models. Absent strong assumptions, their typically highly nonlinear pricing dependence on the state vector prevents or at least severely impedes their inclusion into standard estimation approaches. This paper develops a novel and unified methodology to incorporate moments involving derivatives prices into a GMM estimation procedure. Invoking new results from generalized transform analysis, we derive analytically tractable expressions for exact moments and devise a computationally attractive approximation procedure. We exemplify our methodology with an estimation problem that jointly accounts for stock returns as well as prices of equity and volatility options. Finally, we provide numerical results that support the effectiveness of our methodology.
We present a method for computing the likelihood of a mixed hitting-time model that specifies durations as the first time a latent Lévy process crosses a heterogeneous threshold. This likelihood is not generally known in closed form, but its Laplace transform is. Our approach to its computation relies on numerical methods for inverting Laplace transforms that exploit special properties of the first passage times of Lévy processes. We use our method to implement a maximum likelihood estimator of the mixed hitting-time model in MATLAB. We illustrate the application of this estimator with an analysis of Kennan’s (1985) strike data.
In this paper, we extend the existing generalized transform analysis in a way that allows us to propose a novel GMM approach for estimating asset pricing models. Our methodology is capable of incorporating a broad class of assets within both reduced-form and structural models, while avoiding the drawbacks of competing approaches. To formulate moment conditions, we derive expressions for moments involving transform-based asset prices. Our theory yields analytically tractable expressions for exact moments and an additionally computationally efficient strategy to obtain approximate moments, whose convergence we establish under standard conditions. Exact and approximate moments induce exact and approximate GMM estimators, respectively, for which we discuss asymptotic properties. Finally, we exemplify and numerically support our methodology with a worked-out estimation problem involving equity options.
We propose a Wishart Affine Stochastic Correlation (WASC) model for the joint dynamics of the SDF in an international economy. We derive exchange rate dynamics and a quasi-closed-form solution for currency option pricing. This solution includes Heston’s stochastic volatility model as a special case. We benchmark our approach to a vector-based model inspired by Bakshi/Carr/Wu (2008, JFE). We estimate both models for the US, Europe, and Japan. Empirically, the WASC model is more robust with respect to the estimation period. In contrast to the benchmark model, estimated risk sharing indices seem to reflect the Euro crisis (2011/12) in the WASC model. Moreover, the explanatory power of filtered Sharpe ratios for stock market returns and volatilities is higher (both in- and out-of-sample).
This paper develops estimators of the transition density, filters, and parameters of multivariate jump–diffusion models. The drift, volatility, jump intensity, and jump magnitude are allowed to be state-dependent and non-affine. It is not necessary to diagonalize the volatility matrix. Our density and filter estimators converge at the canonical rate typically associated with exact Monte Carlo estimation. Our parameter estimators have the same asymptotic distribution as maximum likelihood estimators, which are often intractable for the class of models we consider. The results of this paper enable the empirical analysis of previously intractable models of asset prices and economic time series.
Multivariate return distributions consistent with bilateral gamma marginals are formulated and termed multivariate bilateral gamma (MBG). Tail probability distances and Wasserstein distances between return data, model simulations and their squares evaluate the model performance. A full Gaussian copula [Formula: see text] is taken as an alternate test model and the MBG delivers a comparatively better performance for equity pairs. The MBG is however inadequate for the S&P 500 index return when paired with the VIX returns. Applying MBG to the S&P 500 the index and regression residuals of VIX on the S&P 500 index return is successful. This model is termed MBGR. The residual taken as an independent bilateral gamma, delivers the model MBGIR. Characteristic function estimations are employed to develop asset-specific VIX levels and their joint returns with the asset return are studied. The CBOE SKEW index is generalized to be asset-specific and triples of returns for the asset, its VIX and its SKEW are studied using all four models and performance statistics. The model MBGR continues to deliver a good performance.
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.
This paper develops an unbiased Monte Carlo approximation to the transition density of a jump–diffusion process with state-dependent drift, volatility, jump intensity, and jump magnitude. The approximation is used to construct a likelihood estimator of the parameters of a jump–diffusion observed at fixed time intervals that need not be short. The estimator is asymptotically unbiased for any sample size. It has the same large-sample asymptotic properties as the true but uncomputable likelihood estimator. Numerical results illustrate its properties.
We consider a multi-factor Cox-Ingersoll-Ross (CIR) model of the term structure of interest rates with weak mean-reversion effect. We use perturbation theory to analyze its conditional characteristic function illustrated by a system of Riccati equations and derive the error bounds for the perturbation approximations. Using the Fourier inversion theorem, we clarify that the perturbation approximation of the conditional characteristic function can be applied to estimate the transition density and likelihood function. We provide their error bounds and accuracy orders. Finally, we discuss the performance of the perturbation approximation in estimating the transition density via simulation.
The asymptotic normality and arbitrarily high efficiency of some new statistical procedures based on the empirical characteristic function is established under general conditions.
Common statistical procedures such as maximum likelihood and M-estimation admit generalized representations in the Fourier domain. The Fourier domain provides fertile ground for approaching a number of difficult problems in inference. In particular, the empirical characteristic function and its extension for stationary time series are shown to be fundamental tools which support numerically simple inference procedures having arbitrarily high asymptotic efficiency and certain robustness features as well. A numerical illustration involving the symmetric stable laws is given.
The paper demonstrates that consistent and efficient estimation of an empirically relevant, but highly non-linear, three-factor model for the interest rate diffusion is feasible using the Efficient Method of Moments (EMM) procedure developed by Gallant and Tauchen. The model is shown to capture all structural features of the short rate process, including the rather extreme conditional heteroskedasticity. Moreover, the factors are quite intuitive, representing shocks to the short rate itself, the stochastic volatility and a time-varying mean level (the central tendency). We show that these factors may be identified with forces that induce shifts in the steepness, level and curvature of the yield curve. Thus the properties of the model are consistent with the set of stylized facts established by Litterman and Scheinkman in the late 1980's about the number of factors in the yield curve.
We describe an intuitive, simple, and systematic approach to generating moment conditions for generalized method of moments (GMM) estimation of the parameters of a structural model. The idea is to use the score of a density that has an analytic expression to define the GMM criterion. The auxiliary model that generates the score should closely approximate the distribution' of the observed data but is not required to nest it. If the auxiliary model nests the structural model then the estimator is as efficient as maximum likelihood. The estimator is advantageous when expectations under a structural model can be computed by simulation, by quadrature, or by analytic expressions but the likelihood cannot be computed easily.
Because the empirical characteristic function (ECF) is the Fourier transform of the empirical distribution function, it retains all the information in the sample but can overcome difficulties arising from the likelihood. This paper discusses an estimation method via the ECF for strictly stationary processes. Under some regularity conditions, the resulting estimators are shown to be consistent and asymptotically normal. The method is applied to estimate the stable autoregressive moving average (ARMA) models. For the general stable ARMA model for which the maximum likelihood approach is not feasible, Monte Carlo evidence shows that the ECF method is a viable estimation method for all the parameters of interest. For the Gaussian ARMA model, a particular stable ARMA model, the optimal weight functions and estimating equations are given. Monte Carlo studies highlight the finite sample performances of the ECF method relative to the exact and conditional maximum likelihood methods.
A general approach to testing serial dependence restrictions implied from financial models is developed. In particular, we
discuss joint serial dependence restrictions imposed by random walk, market microstructure, and rational expectations models
recently examined in the literature. This approach incorporates more information from the data by explicitly modeling dependencies
induced by the use of overlapping observations. Because the estimation problem is sufficiently simple in this framework, the
test statistics have simple representations in terms of only a few unknown parameters. As a result, relatively good size properties
are attained in small samples. In addition, the benefit to overlapping observations and the advantage of examining multiperiod
time series are explicitly quantified.
This paper uses an intertemporal general equilibrium asset pricing model to study the term structure of interest rates. In this model, anticipations, risk aversion, investment alternatives, and preferences about the timing of consumption all play a role in determining bond prices. Many of the factors traditionally mentioned as influencing the term structure are thus included in a way which is fully consistent with maximizing behavior and rational expectations. The model leads to specific formulas for bond prices which are well suited for empirical testing. © 2005 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.
We consider maximum likelihood estimation for stochastic differential equations based on discrete observations when the likelihood function is unknown. A sequence of approximations to the likelihood function is derived, and convergence results for the sequence are proven. Estimation by means of the approximate likelihood functions is easy and very generally applicable. The performance of the suggested estimators is studied in two examples, and they are compared with other estimators.
An extensive empirical literature in finance has documented not only the presence of anomalies in the Black-Scholes model, but also the term structures of these anomalies (for instance, the behavior of the volatility smile or of unconditional returns at different maturities). Theoretical efforts in the literature at addressing these anomalies have largely focused on two extensions of the Black-Scholes model: introducing jumps into the return process, and allowing volatility to be stochastic. We employ commonly used versions of these two classes of models to examine the extent to which the models are theoretically capable of resolving the observed anomalies. We find that each model exhibits some term structure patterns that are fundamentally inconsistent with those observed in the data. As a consequence, neither class of models constitutes an adequate explanation of the empirical evidence, although stochastic volatility models fare somewhat better than jumps.
Five parametric classes of symmetric distributions with expressions for their characteristic functions, including upto 3 parameters, are employed in a simulation study of parameter estimation using the empirical characteristic function. Parameters are estimated by minimizing a quadratic form in the empirical characteristic function evaluated at p equally spaced points. The estimation of single parameters, with others specified, is successful in all cases. Estimation of all parameters simultaneously fails for three of the classes studied at p = 3. Invertibility problems arise when p is increased to 20. Circumventing these by a principal components procedure, one further class is estimated, leaving 2 inestimable. Comparisons with the performance of maximum likelihood are indicative of a low relative efficiency of the quadratic form minimization procedure. /// Cinq classes de distributions symétriques, avec les fonctions caractéristiques connues et contenant au maximum trois paramètres, sont utilisées dans une étude par simulation d'estimateurs de paramètres en utilisant la fonction caractéristique empirique. Les paramètres sont estimés par minimisation d'une forme quadratique de la fonction caractéristique empirique calculée en p points équispacés. L'estimation d'un seul paramètre, avec les autres spécifiés, réussit en tous les cas. L'estimation simultanée pour tous les paramètres ne réussit pas pour trois classes étudiées avec p = 3. Les problèmes d'invertibilité apparaissent quand p est augmenté jusqu'à 20. Une procédure par les composantes principales donne l'estimation pour une classe en plus, laissant deux classes nonestimables. Le maximum de vraisemblance indique une relativement petite efficacité de la procédure de minimisation de la forme quadratique.
This paper examines the optimal consumption and portfolio-choice problem of long-horizon investors who have access to a riskless
asset with constant return and a risky asset (“stocks”) with constant expected return and time-varying precision—the reciprocal
of volatility. Markets are incomplete, and investors have recursive preferences defined over intermediate consumption. The
paper obtains a solution to this problem which is exact for investors with unit elasticity of intertemporal substitution of
consumption and approximate otherwise. The optimal portfolio demand for stocks includes an intertemporal hedging component
that is negative when investors have coefficients of relative risk aversion larger than one, and the instantaneous correlation
between volatility and stock returns is negative, as typically estimated from stock return data. Our estimates of the joint
process for stock returns and precision (or volatility) using U.S. data confirm this finding. But we also find that stock
return volatility does not appear to be variable and persistent enough to generate large intertemporal hedging demands.
SUMMARY In testing an hypothesis that a population has a specified distribution against the alternative that its distribution belongs
to a separate family it is well known that the likelihood ratio test does not apply. We propose a test based on the difference
between the sample moment generating function and its theoretical counterpart under Ho. The argument of the moment generating function is chosen to maximize the test's large-sample power. Small-sample properties
of the test in two applications are studied by simulation.
The empirical characteristic function φn(t) and its asymptotic distribution are utilized to find a chi-squared goodness-of-fit test for simple null hypotheses. In addition the optimum number and location of points t at which φn(t) has to be evaluated is investigated for specified alternative hypotheses.
This paper explores the structural differences and relative goodness-of-fits of affine term structure models ~ATSMs!. Within the family of ATSMs there is a tradeoff between f lexibility in modeling the conditional correlations and volatilities of the risk factors. This trade-off is formalized by our classification of N-factor affine family into N 1 1 non-nested subfamilies of models. Specializing to three-factor ATSMs, our analysis suggests, based on theoretical considerations and empirical evidence, that some subfamilies of ATSMs are better suited than others to explaining historical interest rate behavior. IN SPECIFYING A DYNAMIC TERM STRUCTURE MODEL—one that describes the comovement over time of short- and long-term bond yields—researchers are inevitably confronted with trade-offs between the richness of econometric representations of the state variables and the computational burdens of pricing and estimation. It is perhaps not surprising then that virtually all of the empirical implementations of multifactor term structure models that use time series data on long- and short-term bond yields simultaneously have focused on special cases of “affine” term structure models ~ATSMs! .A n ATSM accommodates time-varying means and volatilities of the state variables through affine specifications of the risk-neutral drift and volatility coefficients. At the same time, ATSMs yield essentially closed-form expressions for zero-coupon-bond prices ~Duffie and Kan ~1996!!, which greatly facilitates pricing and econometric implementation. The focus on ATSMs extends back at least to the pathbreaking studies by Vasicek ~1977! and Cox, Ingersoll, and Ross ~1985!, who presumed that the instantaneous short rate r~t! was an affine function of an N-dimensional state vector Y~t!, r~t! 5 d 0 1 d y ’ Y~t!, and that Y~t! followed Gaussian and square-root diffusions, respectively. More recently, researchers have explored formulations of ATSMs that extend the one-factor Markov represen
Estimation of the parameters of the stable laws is considered for samples of size at least 50. The modulus of the difference
between the empirical and theoretical characteristic functions is weighted and integrated over the real line; this function
of the sample values and the parameters is then minimized with respect to the parameters via a gradient search routine nested
within a sequential search. Extensive simulation experiments validate the effectiveness of the estimation procedure over the
entire parameter space. Application of the procedure to stock market price behaviour is made.
This paper examines the joint time series of the S&P 500 index and near-the-money short-dated option prices with an arbitrage-free model, capturing both stochastic volatility and jumps. Jump-risk premia uncovered from the joint data respond quickly to market volatility, becoming more prominent during volatile markets. This form of jump-risk premia is important not only in reconciling the dynamics implied by the joint data, but also in explaining the volatility "smirks" of cross-sectional options data. Further diagnostic tests suggest a stochastic-volatility model with two factors --- one strongly persistent, the other quickly mean-reverting and highly volatile.
In the stochastic volatility framework of Hull and White (1987), we characterize the so-called Black and Scholes implied volatility as a function of two arguments the ratio of the strike to the underlying asset price and the instantaneous value of the volatility By studying the variation m the first argument, we show that the usual hedging methods, through the Black and Scholes model, lead to an underhedged (resp. overhedged) position for in-the-money (resp out-of the-money) options, and a perfect partial hedged position for at the-money options These results are shown to be closely related to the smile effect, which is proved to be a natural consequence of the stochastic volatility feature the deterministic dependence of the implied volatility on the underlying volatility process suggests the use of implied volatility data for the estimation of the parameters of interest A statistical procedure of filtering (of the latent volatility process) and estimation (of its parameters) is shown to be strongly consistent and asymptotically normal.
It is shown under general conditions that arbitrarily high asymptotic efficiencies can be obtained when the parameters of a stationary time series are estimated by fitting the characteristic functions of the process to their empirical versions. A consistency and a central limit result are also given.On considère ľestimation des paramètres d'une série chronologique stationnaire via ľajustement de la fonction caractéristique du processus à sa version empirique. On montre que, sous des conditions non restrictives, une efficacité asymptotique arbitrairement grande peut ětre atteinte. Un résultat à propos de la convergence et un théorème de limite centrale sont aussi présentés.
This paper presents a consistent and arbitrage-free multifactor model of the term structure of interest rates in which yields at selected fixed maturities follow a parametric muitivariate Markov diffusion process with "stochastic volatility." the yield of any zero-coupon bond is taken to be a maturity-dependent affine combination of the selected "basis" set of yields. We provide necessary and sufficient conditions on the stochastic model for this affine representation. We include numerical techniques for solving the model, as well as numerical techniques for calculating the prices of term-structure derivative prices. the case of jump diffusions is also considered. Copyright 1996 Blackwell Publishers.
For many time series estimation problems, there is an infinite-dimensional class of generalized method of moments estimators that are consistent and asymptotically normal. This paper suggests a procedure for calculating the greatest lower bound for the asymptotic covariance matrices of such estimators. The analysis focuses on estimation problems in which the data are generated by a stochastic process that is stationary and ergodic. The calculation of the bound uses martingale difference approximations as suggested by Gordon (1969) and a matrix version of Hilbert space methods.
We present an econometric method for estimating the parameters of a diffusion model from discretely sampled data. The estimator is transparent, adaptive, and inherits the asymptotic properties of the generally unattainable maximum likelihood estimator. We use this method to estimate a new continuous-time model of the joint dynamics of interest rates in two countries and the exchange rate between the two currencies. The model allows financial markets to be incomplete and specifies the degree of incompleteness as a stochastic process. Our empirical results offer several new insights into the dynamics of exchange rates.
This paper derives a methodology for the estimation of continuous-time stochastic models based on the characteristic function. The estimation method does not require discretization of the stochastic process, and it is simple to apply in practice. The method is essentially generalized method of moments on the complex plane. Hence it shares the efficiency and distribution properties of GMM estimators. We illustrate the method with some applications to relevant estimation problems in continuous-time Finance. We estimate a model of stochastic volatility, a jump–diffusion model with constant volatility and a model that nests both the stochastic volatility model and the jump–diffusion model. We find that negative jumps are important to explain skewness and asymmetry in excess kurtosis of the stock return distribution, while stochastic volatility is important to capture the overall level of this kurtosis. Positive jumps are not statistically significant once we allow for stochastic volatility in the model. We also estimate a non-affine model of stochastic volatility, and find that the power of the diffusion coefficient appears to be between one and two, rather than the value of one-half that leads to the standard affine stochastic volatility model. However, we find that including jumps into this non-affine, stochastic volatility model reduces the power of the diffusion coefficient to one-half. Finally, we offer an explanation for the observation that the estimate of persistence in stochastic volatility increases dramatically as the frequency of the observed data falls based on a multiple factor stochastic volatility model.
This paper discusses the Monte Carlo maximum likelihood method of estimating stochastic volatility (SV) models. The basic SV model can be expressed as a linear state space model with log chi-square disturbances. The likelihood function can be approximated arbitrarily accurately by decomposing it into a Gaussian part, constructed by the Kalman filter, and a remainder function, whose expectation is evaluated by simulation. No modifications of this estimation procedure are required when the basic SV model is extended in a number of directions likely to arise in applied empirical research. This compares favorably with alternative approaches. The finite sample performance of the new estimator is shown to be comparable to the Monte Carlo Markov chain (MCMC) method.
Post-crash distributions inferred from S&P 500 future option prices have been strongly negatively skewed. This article examines two alternate explanations: stochastic volatility and jumps. The two option pricing models are nested, and are fitted to S&P 500 futures options data over 1988–1993. The stochastic volatility model requires extreme parameters (e.g., high volatility of volatility) that are implausible given the time series properties of option prices. The stochastic volatility/jump-diffusion model fits option prices better, and generates more plausible volatility process parameters. However, its implicit distributions are inconsistent with the absence of large stock index moves over 1988–93.
We examine Italian inflation rates and the Phillips curve with a very long-run perspective, one that covers the entire existence of the Italian lira from political unification (1861) to Italy's entry in the European Monetary Union (end of 1998). We first study the volatility, persistence and stationarity of the Italian inflation rate over the long run and across various exchange-rate regimes that have shaped Italian monetary history. Next, we estimate alternative Phillips equations and investigate whether nonlinearities, asymmetries and structural changes characterize the inflation-output trade-off in the long run. We capture the effects of structural changes and asymmetries on the estimated parameters of the inflation-output trade-off, relying partly on sub-sample estimates and partly on time-varying parameters estimated via the Kalman filter. Finally, we investigate causal relationships between inflation rates and output and extend the analysis to include the US and the UK for comparison purposes. The inference is that Italy has experienced a conventional inflation-output trade-off only during times of low inflation and stable aggregate supply.
Since the empirical characteristic function is the Fourier transformation of the emipirical distribution function, it retains all the information in the sample but can overcome difficulties arising from the likelihood. This paper discusses an estimation method using the empirical characteristic function for stationary processes. Under some regularity conditions, the resulting estimators are shown to be consistent and asymptotically normal. The method is applied to estimate Gaussion ARMA models. The optimal weight functions and estimating equations are given for in detail. Monte Carlo evidence shows that thc empirical characteristic function method can work as well as the exact maximum likelihood method and outperforms the conditional maximum likelihood method.
We propose a minimum chi-squared estimator for the parameters of an ergodic system of stochastic differential equations with partially observed state. We prove that the efficiency of the estimator approaches that of maximum likelihood as the number of moment functions entering the chi-squared criterion increases and as the number of past observations entering each moment function increases. The rninimised criterion is asymp totically chi-squared and can be used to test system adequacy. When a fitted system is rejected, inspecting studentised moments suggests how the fitted system might be modified to improve the fit. The method and diagnostic tests are applied to daily observations on the US. dollar to Deutschmark exchange rate from 1977 to 1992.
Substantial progress has been made in extending the Black-Scholes model to incorporate such features as stochastic volatility, stochastic interest rates and jumps.On the empirical front, however, it is not yet known whether and by how much each generalized feature will improve option pricing and hedging performance. This paper fills this gap by first developing an implementable option model in closed form that allows volatility, interest rates and jumps to bestochastic and that is parsimonious in the number of parameters. The model includes many known ones as special cases. Delta-neutral and single-instrument minimum-variance hedging strategies are derived analytically. Using S&P 500 options, we examine a set of alternative models from three perspectives: (1) internal consistency of implied parameters/volatility with relevant time-series data, (2)out-of-sample pricing and (3) hedging performance. The models of focus include the benchmark Black-Scholes formula and the ones that respectively allow for (i) stochastic volatility, (ii) both stochastic volatility and stochastic interest rates, and (iii) stochastic volatility and jumps.Overall, incorporating both stochastic volatility and random jumps produces the best pricing performance and the most internally-consistent implied-volatility process. Its implied volatility does not "smile" across moneyness. But, for hedging, adding either jumps or stochastic interest rates does not seem to improve performance any further once stochastic volatility is taken into account.
This paper proposes a version of the generalized method of moments procedure that handles both the case where the number of moment conditions is finite and the case where there is a continuum of moment conditions. Typically, the moment conditions are indexed by an index parameter that takes its values in an interval. The objective function to minimize is then the norm of the moment conditions in a Hilbert space. The estimator is shown to be consistent and asymptotically normal. The optimal estimator is obtained by minimizing the norm of the moment conditions in the reproducing kernel Hilbert space associated with the covariance. We show an easy way to calculate this estimator. Finally, we study properties of a specification test using overidentifying restrictions. Results of this paper are useful in many instances where a continuum of moment conditions arises. Examples include efficient estimation of continuous time regression models, cross-sectional models that satisfy conditional moment restrictions, and scalar diffusion processes.
An efficient method is developed for pricing American options on stochastic volatility/jump-diffusion processes under systematic jump and volatility risk. The parameters implicit in deutsche mark (DM) options of the model and various submodels are estimated over the period 1984 to 1991 via nonlinear generalized least squares, and are tested for consistency with $/DM futures prices and the implicit volatility sample path. The stochastic volatility submodel cannot explain the "volatility smile" evidence of implicit excess kurtosis, except under parameters implausible given the time series properties of implicit volatilities. Article published by Oxford University Press on behalf of the Society for Financial Studies in its journal, The Review of Financial Studies.
I use a new technique to derive a closed-form solution for the price of a European call option on an asset with stochastic
volatility. The model allows arbitrary correlation between volatility and spot-asset returns. I introduce stochastic interest
rates and show how to apply the model to bond options and foreign currency options. Simulations show that correlation between
volatility and the spot asset's price is important for explaining return skewness and strike-price biases in the Black-Scholes
(1973) model. The solution technique is based on characteristic functions and can be applied to other problems
This article investigates the existence of discontinuities in the sample path of exchange rates and of a stock market index.
Maximum-likelihood estimation of a mixed jump-diffusion process reveals that exchange rates exhibit systematic discontuinities,
even after allowing for conditional heteroskedasticity in the diffusion process. The results are much more significant in
the foreign exchange market than in the stock market, which suggests differences in the structure of these markets. Finally,
this jump component is shown to explain some of the empirically observed mispricings in the currency options market.
I modify the uniform-price auction rules in allowing the seller to ration bidders. This allows me to provide a strategic foundation for underpricing when the seller has an interest in ownership dispersion. Moreover, many of the so-called "collusive-seeming" equilibria disappear.
The primary aim of the paper is to place current methodological discussions in macroeconometric modeling contrasting the ‘theory first’ versus the ‘data first’ perspectives in the context of a broader methodological framework with a view to constructively appraise them. In particular, the paper focuses on Colander’s argument in his paper “Economists, Incentives, Judgement, and the European CVAR Approach to Macroeconometrics” contrasting two different perspectives in Europe and the US that are currently dominating empirical macroeconometric modeling and delves deeper into their methodological/philosophical underpinnings. It is argued that the key to establishing a constructive dialogue between them is provided by a better understanding of the role of data in modern statistical inference, and how that relates to the centuries old issue of the realisticness of economic theories.
When a continuous-time diffusion is observed only at discrete dates, not necessarily close together, the likelihood function of the observations is in most cases not explicitly computable. Researchers have relied on simulations of sample paths in between the observations points, or numerical solutions of partial differential equations, to obtain estimates of the function to be maximized. By contrast, we construct a sequence of fully explicit functions which we show converge under very general conditions, including non-ergodicity, to the true (but unknown) likelihood function of the discretely-sampled diffusion. We document that the rate of convergence of the sequence is extremely fast for a number of examples relevant in finance. We then show that maximizing the sequence instead of the true function results in an estimator which converges to the true maximum-likelihood estimator and shares its asymptotic properties of consistency, asymptotic normality and efficiency. Applications to the valuation of derivative securities are also discussed.
This article develops a multi-factor econometric model of the term structure of interest-rate swap yields. The model accommodates the possibility of counterparty default, and any differences in the liquidities of the Treasury and Swap markets. By parameterizing a model of swap rates directly, the authors are able to compute model-based estimates of the defaultable zero-coupon bond rates implicit in the swap market without having to specify a priori the dependence of these rates on default hazard or recovery rates. The time series analysis of spreads between zero-coupon swap and treasury yields reveals that both credit and liquidity factors were important sources of variation in swap spreads over the past decade. Copyright 1997 by American Finance Association.
Substantial progress has been made in developing more realistic option pricing models. Empirically, however, it is not known whether and by how much each generalization improves option pricing and hedging. The authors fill this gap by first deriving an option model that allows volatility, interest rates, and jumps to be stochastic. Using S&P 500 options, they examine several alternative models from three perspectives: (1) internal consistency of implied parameters/volatility with relevant time-series data, (2) out-of-sample pricing, and (3) hedging. Overall, incorporating stochastic volatility and jumps is important for pricing and internal consistency. But for hedging, modeling stochastic volatility alone yields the best performance. Copyright 1997 by American Finance Association.