Mean Reversion Parameter Research Articles

System of Non-Linear Stochastic Differential Equations with Financial Market Quantities

In this paper, two systems of modified stochastic differential equations were considered. The variable coefficient problem was solved using Ito’s theorem to obtain an analytical solutions which was used to generate various behaviors of asset values which shows as follows: (i) increase in when are fixed increases the value of asset returns. (ii) a little increase on time when return rates and stock volatility are fixed increases the value of assets.(iii) an increase in the volatility parameter increases the value of asset pricing and parameter shows the various levels of long term investment plans, (iv) increase in rate of mean-reversion parameter reduces the value of asset. (v) An increase in the volatility parameter decreases the value of asset pricing (vi) The goodness of fit probability QQplots are not statistically significant and besides do come from a common distribution which has a vital meaning in the assessment of asset values for capital market investments. Nevertheless, the Tables 1,2 and 3 are best in comparisons with Tables 4,5 and 6 in terms of predictions for capital investments. The governing investment equations are unique and therefore are found to be satisfactory.

Improving the Estimation and Predictions of Small Time Series Models

Abstract A new approach is developed for improving the point estimation and predictions of parametric time-series models. The method targets performance criteria such as estimation bias, root mean squared error, variance, or prediction error, and produces closed-form estimators focused towards these targets via a computational approximation method. This is done for an autoregression coefficient, for the mean reversion parameter in Vasicek and CIR diffusion models, for the binomial thinning parameter in integer-valued autoregressive (INAR) models, and for predictions from a CIR model. The success of the prediction targeting approach is shown in Monte Carlo simulations and in out-of-sample forecasting of the US Federal Funds rate.

Standard and fractional reflected Ornstein–Uhlenbeck processes as the limits of square roots of Cox–Ingersoll–Ross processes

In this paper, we establish a new connection between Cox–Ingersoll–Ross (CIR) and reflected Ornstein–Uhlenbeck (ROU) models driven by either a standard Wiener process or a fractional Brownian motion with . We prove that, with probability 1, the square root of the CIR process converges uniformly on compacts to the ROU process as the mean reversion parameter tends to either (in the standard case) or to 0 (in the fractional case). This also allows to obtain a new representation of the reflection function of the ROU as the limit of integral functionals of the CIR processes. The results of the paper are illustrated by simulations.

Modeling and forecasting realized volatility with the fractional Ornstein–Uhlenbeck process

This paper proposes to model and forecast realized volatility (RV) using the fractional Ornstein–Uhlenbeck (fO–U) process with a general Hurst parameter, H. A two-stage method is introduced for estimating parameters in the fO–U process based on discrete-sampled observations. In the first stage, H is estimated based on the ratio of two second-order differences of observations from different frequencies. In the second stage, with the estimated H, the other parameters of the model are estimated by the method of moments. All estimators have closed-form expressions and are easy to implement. A large sample theory of the proposed estimators is derived. Extensive simulations show that the proposed estimators and the large-sample theory perform well in finite samples. We apply the model and the method to the logarithmic daily RV series of various financial assets. Our empirical findings suggest that H is much smaller than 1/2, indicating that the RV series have rough sample paths, and that the mean reversion parameter takes a small positive number, indicating that the RV series are stationary but have slow mean reversion. The proposed model is compared with many alternative models, including the fractional Brownian motion, ARFIMA, and HAR, in forecasting RV and logarithmic RV.

Quantifying uncertainty with a derivative tracking SDE model and application to wind power forecast data

We develop a data-driven methodology based on parametric Itô’s Stochastic Differential Equations (SDEs) to capture the real asymmetric dynamics of forecast errors, including the uncertainty of the forecast at time zero. Our SDE framework features time-derivative tracking of the forecast, time-varying mean-reversion parameter, and an improved state-dependent diffusion term. Proofs of the existence, strong uniqueness, and boundedness of the SDE solutions are shown by imposing conditions on the time-varying mean-reversion parameter. We develop the structure of the drift term based on sound mathematical theory. A truncation procedure regularizes the prediction function to ensure that the trajectories do not reach the boundaries almost surely in a finite time. Inference based on approximate likelihood, constructed through the moment-matching technique both in the original forecast error space and in the Lamperti space, is performed through numerical optimization procedures. We propose a fixed-point likelihood optimization approach in the Lamperti space. Another novel contribution is the characterization of the uncertainty of the forecast at time zero, which turns out to be crucial in practice. We extend the model specification by considering the length of the unknown time interval preceding the first time a forecast is provided through an additional parameter in the density of the initial transition. All the procedures are agnostic of the forecasting technology, and they enable comparisons between different forecast providers. We apply our SDE framework to model historical Uruguayan normalized wind power production and forecast data between April and December 2019. Sharp empirical confidence bands of wind power production forecast error are obtained for the best-selected model.

Bias Correction for Bond Option Greeks via Jackknife

The underlying models for bond options are often based on some linear drift functions such that the option Greeks depend crucially on the mean reversion parameters. Substantial estimation bias may arise when these parameters are estimated using standard methods such as maximum likelihood estimation, leading to a bias in estimating the Greeks. To address this issue, following Phillips and Yu (2005), a jackknife method is adopted in this article. In particular, we apply the method directly to the estimation of option Greeks, rather than the estimation of parameters. This approach is general and computationally inexpensive; hence, it is convenient in practice. The finite-sample performance is investigated in several Monte Carlo studies. At last, in dynamic Delta hedging, we show that the bias reduction in the estimation of option Greeks using the proposed method can achieve some economic value. <b>TOPICS:</b>Derivatives, options, fixed income and structured finance, quantitative methods, simulations <b>Key Findings</b> ▪ Like option pricing, the bias problem is still serious in estimating the Greeks of the options. ▪ The jackknife method is a good method to reduce the estimation bias in estimating the Greeks. ▪ The bias reduction in the estimation of option Greeks using the proposed method can achieve some economic values in option markets around the world.

One‐Factor Hull—White Model Calibration for CVA Part I: Instrument Selection with a Kink

This paper is the first of a multi-part series on the calibration of the one-factor Hull-White short rate model for the purpose of computing CVAs (and xVAs) with an xVA system. It introduces an atypical bootstrapping scheme for the calibration of the short rate volatility. The second part focuses on the selection of the mean reversion parameter. In both expositions we present long-term time series results for EUR, JPY, and USD, covering the period from the beginning of 2009 (at the earliest) to spring 2020.

One‐Factor Hull—White Model Calibration for CVA Part II: Optimizing the Mean Reversion Parameter

This paper is the second of a multi-part series on the calibration of the one-factor Hull—White short rate model for the purpose of computing CVAs (and XVas) with an XVa system. The first part introduces an atypical bootstrapping scheme for the calibration of the short rate volatility. The present second part focuses on the selection of the mean reversion parameter. In both expositions we present long-term time series results for EUR, JPY, and USD, covering the period from the beginning of 2009 (at the earliest) to spring 2020.

Out‐of‐sample performance of bias‐corrected estimators for diffusion processes

AbstractWe investigated the out‐of‐sample forecasting performance of six bias‐corrected estimators that have recently emerged in the literature for the Ornstein–Uhlenbeck process: the naïve estimator, the Tang and Chen estimator, the Bao et al. estimator, the bootstrap estimator, the Wang et al. estimator, and the bootstrap estimator based on the Euler method, along with the benchmark least squares (LS) estimator. Our Monte Carlo simulations illustrated that the bias‐corrected estimators, except for the Bao et al. estimator, produced much worse out‐of‐sample forecasting performance than the LS estimator because these other estimators have a tendency to generate negative estimations of the mean reversion parameter when the true value is close to zero. However, if we set a zero lower bound to all of these estimators, including the LS estimator, all the bias‐corrected estimators improved the out‐of‐sample forecasting performance of the LS estimator as long as the true mean reversion parameter was not very large. These main results also hold for the Cox–Ingersoll–Ross process. Our real data applications confirmed these overall findings.

One-factor Hull-White Model Calibration for CVA - Part II: Optimizing the Mean Reversion Parameter

This paper is the second of a multi-part series on the calibration of the one-factor Hull-White short rate model for the purpose of computing CVAs (and xVAs) with an xVA system. The first part introduces an atypical bootstrapping scheme for the calibration of the short rate volatility. The present second part focuses on the selection of the mean reversion parameter. In both expositions we present long-term time series results for EUR, JPY, and USD, covering the period from the beginning of 2009 (at the earliest) to spring 2020.

One-factor Hull-White Model Calibration for CVA - Part I: Instrument Selection With a Kink

This paper is the first of a multi-part series on the calibration of the one-factor Hull-White short rate model for the purpose of computing CVAs (and xVAs) with an xVA system. It introduces an atypical bootstrapping scheme for the calibration of the short rate volatility. The second part focuses on the selection of the mean reversion parameter. In both expositions we present long-term time series results for EUR, JPY, and USD, covering the period from the beginning of 2009 (at the earliest) to spring 2020.

Bias-corrected estimators for the Vasicek model: an application in risk measure estimation

We evaluate the usefulness of bias-correction methods in enhancing the Vasicek model for market risk and counterparty risk management practices. The naive bias-corrected estimator, the Tang and Chen bias-corrected estimator and the Bao et al bias-corrected estimator are selected to be compared against the benchmark least squares (LS) estimator. Our Monte Carlo experiment shows that the bias-corrected estimators substantially reduce the small sample bias of the LS estimator for the Vasicek model and project much more accurate value-at-risk and potential future exposure estimations. Even if the sample length is as long as 30 years, the improvements are still significant, especially for the cases where the mean-reversion parameter is close to zero. The applications to real data further demonstrate that the small sample bias of the LS estimator cannot be ignored and one should consider bias-corrected estimators for the Vasicek model.

Weak dependence and GMM estimation of supOU and mixed moving average processes

We consider a mixed moving average (MMA) process $X$ driven by a Lévy basis and prove that it is weakly dependent with rates computable in terms of the moving average kernel and the characteristic quadruple of the Lévy basis. Using this property, we show conditions ensuring that sample mean and autocovariances of $X$ have a limiting normal distribution. We extend these results to stochastic volatility models and then investigate a Generalized Method of Moments estimator for the supOU process and the supOU stochastic volatility model after choosing a suitable distribution for the mean reversion parameter. For these estimators, we analyze the asymptotic behavior in detail.

The use of bias correction versus the Jackknife when testing the mean reversion and long term mean parameters in continuous time models

Abstract In this paper we extend the results in [5] in two directions: First, we show that by bias correcting the estimated mean reversion parameter we can also have better finite sample properties of the testing procedure using a t-statistic in the near unit root situation when the mean reversion parameter is approaching its lower bound versus using the Jackknife estimator of Phillips and Yu [8]. Second, we show that although Tang and Chen [10] demonstrate that the variance of the maximum likelihood estimator of the long term mean parameter is of an order equal to the reciprocal of the sample size (the same order as that of the bias and variance of the mean reversion parameter estimator and so it does not converge very fast to its true value), the t-statistic related to that parameter does not exhibit large empirical size distortions and so does not need to be bias corrected in practice.

The Informational Content of Infinite Maturity Bonds

This paper investigates how the market implicitly prices very long-dated cash-flows. We do this empirically, using daily market data on the 3½% War Loan, a UK infinite maturity coupon bond. We price the War Loan as an American option with 100 years of maturity. To this end, we perform daily calibrations to observed swaption prices using the Hull-White (1990) one-factor term structure model. The mean-reversion and volatility parameters resulting from the calibrations are then used in a finite difference method, which is equivalent to the Hull-White (1994 and 1996) trinomial tree procedure. We calculate the constant long-term yield beyond the last observed yield, which equates the results of the finite difference procedure with the observed prices of the War Loan. We find that prior to the financial crisis the constant long-term yield was situated above the last observed yield, whereas after the crisis it was situated below the last observed yield.

Distribution of the mean reversion estimator in the Ornstein–Uhlenbeck process

ABSTRACTWe derive the exact distribution of the maximum likelihood estimator of the mean reversion parameter (κ) in the Ornstein–Uhlenbeck process using numerical integration through analytical evaluation of a joint characteristic function. Different scenarios are considered: known or unknown drift term, fixed or random start-up value, and zero or positive κ. Monte Carlo results demonstrate the remarkably reliable performance of our exact approach across all the scenarios. In comparison, misleading results may arise under the asymptotic distributions, including the advocated infill asymptotic distribution, which performs poorly in the tails when there is no intercept in the regression and the starting value of the process is nonzero.

Parameter estimation for the Langevin equation with stationary-increment Gaussian noise

We study the Langevin equation with stationary-increment Gaussian noise. We show the strong consistency and the asymptotic normality with Berry–Esseen bound of the so-called second moment estimator of the mean reversion parameter. The conditions and results are stated in terms of the variance function of the noise. We consider both the case of continuous and discrete observations. As examples we consider fractional and bifractional Ornstein–Uhlenbeck processes. Finally, we discuss the maximum likelihood and the least squares estimators.

Bias in the estimation of mean reversion in continuous-time Lévy processes

Abstract This paper develops the approximate bias of the ordinary least squares estimator of the mean reversion parameter in continuous-time Levy processes. Several cases are considered, depending on whether the long-run mean is known or unknown and whether the initial condition is fixed or random. The approximate bias is used to construct a bias corrected estimator. The performance of the approximate bias and the bias corrected estimator is examined using simulated data.

The Hull-White 1 Factor Convexity Adjustment and the Special Case when the Hull-White and Ho-Lee Models are Equivalent

In what follows we quote the Hull-White 1 factor and Ho-Lee model dynamics and their corresponding Eurodollar convexity adjustment formulas. We then show that, in the special case where the Hull-White mean reversion parameter is zero, the adjustment under the Hull-White and Ho-Lee models is identical.

Financial weather derivatives for corn production in Northern China: A comparison of pricing methods

Abstract The focus in this study is on the pricing of financial derivatives for hedging weather risks in crop production. Employing data from an earlier study, we compare different methods for pricing weather derivative options based on growing degree days (GDDs). We employ average daily temperatures to derive GDDs using three approaches: (1) An econometric approach with a sine function; (2) Monte Carlo simulation with a sine function and three methods to estimate the mean-reversion parameter; and (3) a historic approach (burn analysis) based on a 10-year moving average of GDDs. Results indicate that the historical average method provides the best fit, followed by the stochastic process with a high mean reversion speed, and, finally, the approach using the econometrically estimated sine function. Depending on the method used, premiums for weather derivative options vary from $21.27 to $24.39 per GDD index contract.