The Fractional OU Process: Term Structure Theory and Application

The paper revisits dynamic term structure models (DTSMs) and proposes a new way in dealing with the limitation of the classical affine models. In particular, this paper expands the flexibility of the DTSMs by applying a fractional Brownian motion as the governing force of the state variable instead of the standard Brownian motion. This is a new direction in pricing non defaultable bonds with offspring in the arbitrage free pricing of weather derivatives based on fractional Brownian motions. By applying fractional Ito calculus and a fractional version of the Girsanov transform, a no arbitrage price of the bond is recovered by solving a fractional version of the fundamental bond pricing equation. Besides this theoretical contribution, the paper proposes an estimation methodology based on the Kalman filter approach, which is applied to the US term structure of interest ratesFractional bond pricing equation, fractional Brownian motion, fractional Ornstein-Uhlenbeck process, long memory, Kalman Filter

Fully Modified Narrow-Band Least Squares Estimation of Weak Fractional Cointegration

We consider estimation of the cointegrating relation in the weak fractional cointegration model, where the strength of the cointegrating relation (difference in memory parameters) is less than one-half. A special case is the stationary fractional cointegration model, which has found important application recently, especially in financial economics. Previous research on this model has considered a semiparametric narrow-band least squares (NBLS) estimator in the frequency domain, but in the stationary case its asymptotic distribution has been derived only under a condition of non-coherence between regressors and errors at the zero frequency. We show that in the absence of this condition, the NBLS estimator is asymptotically biased, and also that the bias can be consistently estimated. Consequently, we introduce a fully modified NBLS estimator which eliminates the bias, and indeed enjoys a faster rate of convergence than NBLS in general. We also show that local Whittle estimation of the integration order of the errors can be conducted consistently based on NBLS residuals, but the estimator has the same asymptotic distribution as if the errors were observed only under the condition of non-coherence. Furthermore, compared to much previous research, the development of the asymptotic distribution theory is based on a different spectral density representation, which is relevant for multivariate fractionally integrated processes, and the use of this representation is shown to result in lower asymptotic bias and variance of the narrow-band estimators. We present simulation evidence and a series of empirical illustrations to demonstrate the feasibility and empirical relevance of our methodology.Fractional cointegration, frequency domain, fully modified estimation, long memory, semiparametric

Fully Modified Narrow-Band Least Squares Estimation of Stationary Fractional Cointegration

We consider estimation of the cointegrating relation in the stationary fractional cointegration model which has found important application recently, especially in financial economics. Previous research on this model has considered a semiparametric narrow-band least squares (NBLS) estimator in the frequency domain, often under a condition of non-coherence between regressors and errors at the zero frequency. We show that in the absence of this condition, the NBLS estimator is asymptotically biased, and also that the bias can be consistently estimated. Consequently, we introduce a fully modified NBLS estimator which eliminates the bias, and indeed enjoys a faster rate of convergence than NBLS in general. We also show that local Whittle estimation of the integration order of the errors can be conducted consistently on the residuals from NBLS regression, whereas the estimator has the same asymptotic distribution as if the errors were observed only under the condition of non-coherence. Furthermore, compared to much previous research, the development of the asymptotic distribution theory is based on a different spectral density representation, which is relevant for multivariate fractionally integrated processes, and the use of this representation is shown to result in lower asymptotic bias and variance of the narrow-band estimators. We also present simulation evidence and a series of empirical illustrations to demonstrate the feasibility and empirical relevance of our methodology.Fractional cointegration, frequency domain, fully modified estimation, long memory, semiparametric

Finite Sample Comparison of Parametric, Semiparametric, and Wavelet Estimators of Fractional Integration

In this paper we compare through Monte Carlo simulations the finite sample properties of estimators of the fractional differencing parameter, d. This involves frequency domain, time domain, and wavelet based approaches and we consider both parametric and semiparametric estimation methods. The estimators are briefly introduced and compared, and the criteria adopted for measuring finite sample performance are bias and root mean squared error. Most importantly, the simulations reveal that 1) the frequency domain maximum likelihood procedure is superior to the time domain parametric methods, 2) all the estimators are fairly robust to conditionally heteroscedastic errors, 3) the local polynomial Whittle and bias reduced log-periodogram regression estimators are shown to be more robust to short-run dynamics than other semiparametric (frequency domain and wavelet) estimators and in some cases even outperform the time domain parametric methods, and 4) without sufficient trimming of scales the wavelet based estimators are heavily biased.bias, finite sample distribution, fractional integration, maximum likelihood, Monte Carlo simulation, parametric estimation, semiparametric estimation, wavelet

The Fractional Ornstein-Uhlenbeck Process: Term Structure Theory and Application

The paper revisits dynamic term structure models (DTSMs) and proposes a new way in dealing with the limitation of the classical affine models. In particular, this paper expands the flexibility of the DTSMs by applying a fractional Brownian motion as the governing force of the state variable instead of the standard Brownian motion. This is a new direction in pricing non defaultable bonds with offspring in the arbitrage free pricing of weather derivatives based on fractional Brownian motions. By applying fractional Ito calculus and a fractional version of the Girsanov transform, a no arbitrage price of the bond is recovered by solving a fractional version of the fundamental bond pricing equation. Besides this theoretical contribution, the paper proposes an estimation methodology based on the Kalman filter approach, which is applied to the US term structure of interest rates.Fractional bond pricing equation; fractional Brownian motion; fractional Ornstein-Uhlenbeck process; long memory; Kalman filter

Finite Sample Accuracy of Integrated Volatility Estimators

We consider the properties of three estimation methods for integrated volatility, i.e. realized volatility, the Fourier estimator, and the wavelet estimator, when a typical sample of high-frequency data is observed. We employ several different generating mechanisms for the instantaneous volatility process, e.g. Ornstein-Uhlenbeck, long memory, and jump processes. The possibility of market microstructure contamination is also entertained using a model with bid-ask bounce in which case alternative estimators with theoretical justification under market microstructure noise are also examined. The estimation methods are compared in a simulation study which reveals a general robustness towards persistence or jumps in the latent stochastic volatility process. However, bid-ask bounce effects render realized volatility and especially the wavelet estimator less useful in practice, whereas the Fourier method remains useful and is superior to the other two estimators in that case. More strikingly, even compared to bias correction methods for microstructure noise, the Fourier method is superior with respect to RMSE while having only slightly higher bias.Bid-ask bounce, finite sample bias, integrated volatility, long memory, market microstructure, Monte Carlo simulation, realized volatility, wavelet

Local polynomial Whittle estimation of perturbed fractional processes

We propose a semiparametric local polynomial Whittle with noise estimator of the memory parameter in long memory time series perturbed by a noise term which may be serially correlated. The estimator approximates the log-spectrum of the short-memory component of the signal as well as that of the perturbation by two separate polynomials. Including these polynomials we obtain a reduction in the order of magnitude of the bias, but also inflate the asymptotic variance of the long memory estimator by a multiplicative constant. We show that the estimator is consistent for d in (0,1), asymptotically normal for d in (0,3/4), and if the spectral density is sufficiently smooth near frequency zero, the rate of convergence can become arbitrarily close to the parametric rate, sqrt(n). A Monte Carlo study reveals that the proposed estimator performs well in the presence of a serially correlated perturbation term. Furthermore, an empirical investigation of the 30 DJIA stocks shows that this estimator indicates stronger persistence in volatility than the standard local Whittle (with noise) estimator.Bias reduction, local Whittle, long memory, perturbed fractional process, semiparametric estimation, stochastic volatility

Continuous-Time Models, Realized Volatilities, and Testable Distributional Implications for Daily Stock Returns

We provide an empirical framework for assessing the distributional properties of daily speculative returns within the context of the continuous-time jump diffusion models traditionally used in asset pricing finance. Our approach builds directly on recently developed realized variation measures and non-parametric jump detection statistics constructed from high-frequency intraday data. A sequence of simple-to-implement moment-based tests involving various transformations of the daily returns speak directly to the importance of different distributional features, and may serve as useful diagnostic tools in the specification of empirically more realistic continuous-time asset pricing models. On applying the tests to the thirty individual stocks in the Dow Jones Industrial Average index, we find that it is important to allow for both time-varying diffusive volatility, jumps, and leverage effects to satisfactorily describe the daily stock price dynamics.return distributions, continuous-time models, mixture-of-distributions hypothesis, financial-time sampling, high-frequency data, volatility signature plots, realized volatilities, jumps, leverage and volatility feedback effects

Project Half Double: Addendum: Current Results for Phase 1, January 2017

The Half Double mission: Project Half Double has a clear mission. We want to succeed in finding a project methodology that can increase the success rate of our projects while increasing the development speed of new products and services. We are convinced that by doing so we can strengthen the competitiveness of Denmark and play an important role in the battle for jobs and future welfare. The overall goal is to deliver “Projects in half the time with double the impact” where projects in half the time should be understood as half the time to impact (benefit realization, effect is achieved) and not as half the time for project execution. The Half Double project journey: It all began in May 2013 when we asked ourselves: How do we create a new and radical project paradigm that can create successful projects? Today we are a movement of hundreds of passionate project people, and it grows larger by the day. The formal part of Project Half Double was initiated in June 2015; it is divided into two phases where phase 1 took place from June 2015 to June 2016 with seven pilot projects, and phase 2 is in progress from July 2016 to July 2017 with 10 pilot projects.The Half Double consortium: Implement Consulting Group is leading the project as well as establishing and managing the collaboration with the pilot project companies in terms of methodology. Aarhus University and the Technical University of Denmark will evaluate the impact of the pilot projects and legitimize the methodology in academia.The Danish Industry Foundation, an independent philanthropic foundation, is contributing to the project financially with DKK 13.8 million.About the addendum: We published the report “Preliminary results for phase 1” in June 2016 (Svejvig, Ehlers et al. 2016). It is time to follow up on the Phase 1 pilot projects and to document their development.The purpose of this addendum is thus to document the development in the pilot projects from June 2016 to January 2017 with particular focus on the impact they have created.This Addendum is a supplement and should be read in conjunction with the Phase 1 report, which will give the reader relevant further information.The target group for this report is practitioners in Danish industry and society in general.The report was prepared by a responsible editorial team from Aarhus University.The addendum was prepared from December 2016 to January 2017, which means that late data about pilot projects from January 2017 is not included in this report