Realized volatility forecasting and option pricing

A growing literature advocates the use of microstructure noise-contaminated high-frequency data for the purpose of volatility estimation. This paper evaluates and compares the quality of several recently-proposed estimators in the context of a relevant economic metric, i.e., profits from option pricing and trading. Using forecasts obtained by virtue of alternative volatility estimates, agents price short-term options on the S&P 500 index before trading with each other at average prices. The agents’ average profits and the Sharpe ratios of the profits constitute the criteria used to evaluate alternative volatility estimates and the corresponding forecasts. For our data, we find that estimators with superior finite sample Mean-squared-error properties generate higher average profits and higher Sharpe ratios, in general. We confirm that, even from a forecasting standpoint, there is scope for optimizing the finite sample properties of alternative volatility estimators as advocated by Bandi and Russell [Bandi, F.M., Russell, J.R., 2005. Market microstructure noise, integrated variance estimators, and the accuracy of asymptotic approximations. Working Paper; Bandi, F.M., Russell, J.R., 2008b. Microstructure noise, realized variance, and optimal sampling. Review of Economic Studies 75, 339–369] in recent work.     

Predictive density estimators for daily volatility based on the use of realized measures

The main objective of this paper is to propose a feasible, model free estimator of the predictive density of integrated volatility. In this sense, we extend recent papers by Andersen et al. [Andersen, T.G., Bollerslev, T., Diebold, F.X., Labys, P., 2003. Modelling and forecasting realized volatility. Econometrica 71, 579–626], and by Andersen et al. [Andersen, T.G., Bollerslev, T., Meddahi, N., 2004. Analytic evaluation of volatility forecasts. International Economic Review 45, 1079–1110; Andersen, T.G., Bollerslev, T., Meddahi, N., 2005. Correcting the errors: Volatility forecast evaluation using high frequency data and realized volatilities. Econometrica 73, 279–296], who address the issue of pointwise prediction of volatility via ARMA models, based on the use of realized volatility. Our approach is to use a realized volatility measure to construct a non-parametric (kernel) estimator of the predictive density of daily volatility. We show that, by choosing an appropriate realized measure, one can achieve consistent estimation, even in the presence of jumps and microstructure noise in prices. More precisely, we establish that four well known realized measures, i.e. realized volatility, bipower variation, and two measures robust to microstructure noise, satisfy the conditions required for the uniform consistency of our estimator. Furthermore, we outline an alternative simulation based approach to predictive density construction. Finally, we carry out a simulation experiment in order to assess the accuracy of our estimators, and provide an empirical illustration that underscores the importance of using microstructure robust measures when using high frequency data.     

Out of sample forecasts of quadratic variation

We compare the forecasts of Quadratic Variation given by the Realized Volatility (RV) and the Two Scales Realized Volatility (TSRV) computed from high frequency data in the presence of market microstructure noise, under several different dynamics for the volatility process and assumptions on the noise. We show that TSRV largely outperforms RV, whether looking at bias, variance, RMSE or out-of-sample forecasting ability. An empirical application to all DJIA stocks confirms the simulation results.     

Volatility forecasting and microstructure noise

It is common practice to use the sum of frequently sampled squared returns to estimate volatility, yielding the so-called realized volatility. Unfortunately, returns are contaminated by market microstructure noise. Several noise-corrected realized volatility measures have been proposed. We assess to what extent correction for microstructure noise improves forecasting future volatility using a MIxed DAta Sampling (MIDAS) regression framework. We study the population prediction properties of various realized volatility measures, assuming i.i.d$i.i.d$. microstructure noise. Next we study optimal sampling issues theoretically, when the objective is forecasting and microstructure noise contaminates realized volatility. We distinguish between conditional and unconditional optimal sampling schemes, and find that conditional optimal sampling seems to work reasonably well in practice.     

Measuring volatility with the realized range

Realized variance, being the summation of squared intra-day returns, has quickly gained popularity as a measure of daily volatility. Following Parkinson [1980. The extreme value method for estimating the variance of the rate of return. Journal of Business 53, 61–65] we replace each squared intra-day return by the high–low range for that period to create a novel and more efficient estimator called the realized range. In addition, we suggest a bias-correction procedure to account for the effects of microstructure frictions based upon scaling the realized range with the average level of the daily range. Simulation experiments demonstrate that for plausible levels of non-trading and bid–ask bounce the realized range has a lower mean-squared error than the realized variance, including variants thereof that are robust to microstructure noise. Empirical analysis of the S&P500 index-futures and the S&P100 constituents confirms the potential of the realized range.     

Nonparametric transfer function models

In this paper a class of nonparametric transfer function models is proposed to model nonlinear relationships between ‘input’ and ‘output’ time series. The transfer function is smooth with unknown functional forms, and the noise is assumed to be a stationary autoregressive-moving average (ARMA) process. The nonparametric transfer function is estimated jointly with the ARMA parameters. By modeling the correlation in the noise, the transfer function can be estimated more efficiently. The parsimonious ARMA structure improves the estimation efficiency in finite samples. The asymptotic properties of the estimators are investigated. The finite-sample properties are illustrated through simulations and one empirical example.     

Subsampling high frequency data

The main contribution of this paper is to propose a novel way of conducting inference for an important general class of estimators that includes many estimators of integrated volatility. A subsampling scheme is introduced that consistently estimates the asymptotic variance for an estimator, thereby facilitating inference and the construction of valid confidence intervals. The new method does not rely on the exact form of the asymptotic variance, which is useful when the latter is of complicated form. The method is applied to the volatility estimator of Aït-Sahalia et al. (2011) in the presence of autocorrelated and heteroscedastic market microstructure noise.     

No-arbitrage semi-martingale restrictions for continuous-time volatility models subject to leverage effects,jumps and i.i.d. noise: Theory and testable distributional implications

We develop a sequential procedure to test the adequacy of jump-diffusion models for return distributions. We rely on intraday data and nonparametric volatility measures, along with a new jump detection technique and appropriate conditional moment tests, for assessing the import of jumps and leverage effects. A novel robust-to-jumps approach is utilized to alleviate microstructure frictions for realized volatility estimation. Size and power of the procedure are explored through Monte Carlo methods. Our empirical findings support the jump-diffusive representation for S&P500 futures returns but reveal it is critical to account for leverage effects and jumps to maintain the underlying semi-martingale assumption.     

Forecasting exchange rate volatility using high-frequency data: Is the euro different?

We assess the performances of alternative procedures for forecasting the daily volatility of the euro’s bilateral exchange rates using 15 min data. We use realized volatility and traditional time series volatility models. Our results indicate that using high-frequency data and considering their long memory dimension enhances the performance of volatility forecasts significantly. We find that the intraday FIGARCH model and the ARFIMA model outperform other traditional models for all exchange rate series.     

Quasi-maximum likelihood estimation of volatility with high frequency data

This paper investigates the properties of the well-known maximum likelihood estimator in the presence of stochastic volatility and market microstructure noise, by extending the classic asymptotic results of quasi-maximum likelihood estimation. When trying to estimate the integrated volatility and the variance of noise, this parametric approach remains consistent, efficient and robust as a quasi-estimator under misspecified assumptions. Moreover, it shares the model-free feature with nonparametric alternatives, for instance realized kernels, while being advantageous over them in terms of finite sample performance. In light of quadratic representation, this estimator behaves like an iterative exponential realized kernel asymptotically. Comparisons with a variety of implementations of the Tukey–Hanning₂ kernel are provided using Monte Carlo simulations, and an empirical study with the Euro/US Dollar future illustrates its application in practice.     

Asymptotically distribution-free tests for the volatility function of a diffusion

This paper develops two tests for parametric volatility function of a diffusion model based on Khmaladze (1981)’s martingale transformation. The tests impose no restrictions on the functional form of the drift function and are shown to be asymptotically distribution-free. The tests are consistent against a large class of fixed alternatives and have nontrivial power against a class of root-n$n$ local alternatives. The paper also extends the tests of volatility to testing for joint specifications of drift and volatility. Monte Carlo simulations show that the tests perform well in finite samples. The proposed tests are then applied to testing models of short-term interest, using data of Treasury bill rate and Eurodollar deposit rate. The empirical results show that the commonly used CKLS volatility function of Chan et al. (1992) fits volatility function poorly and none of the parametric interest rate models considered in the paper fit data well.     

Modeling spot rate using a realized stochastic volatility model with level effect and dynamic drift☆

This paper proposes a class of realized stochastic volatility model based on both various realized volatility measures and spot rate. It applies the realized stochastic volatility model (Takahashi, Omori, & Watanabe, 2009, and Koopman & Scharth, 2013) to the spot rate model with dynamic drift and level effect setups (RSVL). A jointly approximated maximum likelihood procedure is used to estimate this model. The simulation results show that the RSVL model can be consistently estimated and noise-and-jump-robust realized volatility measures improve the accuracy of the estimation. This study empirically investigates the Chinese interbank repo market with RSVL model, which manifested the advantage of taking the level effect and nonlinear drift into consideration. The noise-and-jump-robust realized volatility measures (e.g. subsample realized volatility and threshold pre-average realized volatility) decrease the volatility fitting error. The nonparametric testing suggests that the RSVL model with noise-and-jump-robust realized volatility measures has more power on forecasting excess kurtosis and fat tails and predicting dynamics of higher order autocorrelations.     

Forecasting stock market volatility: Can the risk aversion measure exert an important role?

In this paper, we predict realized volatility of stock return by utilizing time-varying risk aversion based on a simple linear autoregressive model. Our in-sample results suggest that time-varying risk aversion have significant impact for stock return volatility. In terms of out-of-sample forecasting performance, the empirical results indicate that the incorporation of time-varying risk aversion in the benchmark model can yield more accurate stock return volatility forecasts. Notably, the out-of-sample forecasting results confirm that our conclusions are robust when we apply alternative lag orders and alternative prediction evaluation periods. Finally, we study links between the prediction ability of time-varying risk aversion and the volatility of other stock indices and two kinds of crude oil, and find that the new predictor can effectively strengthen forecasting performance in most case. In view of the importance of volatility risk in the asset pricing process, our research is of great significance for financial asset participants.     

Realized jumps on financial markets and predicting credit spreads

This paper extends the jump detection method based on bipower variation to identify realized jumps on financial markets and to estimate parametrically the jump intensity, mean, and variance. Finite sample evidence suggests that the jump parameters can be accurately estimated and that the statistical inferences are reliable under the assumption that jumps are rare and large. Applications to equity market, treasury bond, and exchange rate data reveal important differences in jump frequencies and volatilities across asset classes over time. For investment grade bond spread indices, the estimated jump volatility has more forecasting power than interest rate factors and volatility factors including option-implied volatility, with control for systematic risk factors. The jump volatility risk factor seems to capture the low frequency movements in credit spreads and comoves countercyclically with the price–dividend ratio and corporate default rate.     

How much nominal rigidity is there in the US economy? Testing a new Keynesian DSGE model using indirect inference

We evaluate the Smets-Wouters New Keynesian model of the US postwar period, using indirect inference, the bootstrap and a VAR representation of the data. We find that the model is strongly rejected. While an alternative (New Classical) version of the model fares no better, adding limited nominal rigidity to it produces a ‘weighted’ model version closest to the data. But on data from 1984 onwards - the ‘great moderation’ - the best model version is one with a high degree of nominal rigidity, close to New Keynesian. Our results are robust to a variety of methodological and numerical issues.     

Ultra high frequency volatility estimation with dependent microstructure noise

We analyze the impact of time series dependence in market microstructure noise on the properties of estimators of the integrated volatility of an asset price based on data sampled at frequencies high enough for that noise to be a dominant consideration. We show that combining two time scales for that purpose will work even when the noise exhibits time series dependence, analyze in that context a refinement of this approach is based on multiple time scales, and compare empirically our different estimators to the standard realized volatility.     

Spurious regressions in technical trading

This paper investigates the spurious effect in forecasting asset returns when signals from technical trading rules are used as predictors. Against economic intuition, the simulation result shows that, even if past information has no predictive power, buy or sell signals based on the difference between the short-period and long-period moving averages of past asset prices can be statistically significant when the forecast horizon is relatively long. The theoretical analysis reveals that both ‘momentum’ and ‘contrarian’ strategies can be falsely supported, while the probability of obtaining each result depends on the type of the test statistics employed.     

Forecasting multivariate realized stock market volatility

We present a new matrix-logarithm model of the realized covariance matrix of stock returns. The model uses latent factors which are functions of lagged volatility, lagged returns and other forecasting variables. The model has several advantages: it is parsimonious; it does not require imposing parameter restrictions; and, it results in a positive-definite estimated covariance matrix. We apply the model to the covariance matrix of size-sorted stock returns and find that two factors are sufficient to capture most of the dynamics.     

Asymptotic normality of narrow-band least squares in the stationary fractional cointegration model and volatility forecasting

We consider semiparametric frequency domain analysis of cointegration between long memory processes, i.e. fractional cointegration, allowing derivation of useful long-run relations even among stationary processes. The approach is due to Robinson (1994b. Annals of Statistics 22, 515–539) and uses a degenerating part of the periodogram near the origin to form a narrow-band frequency domain least squares (FDLS) estimator of the cointegrating relation, which is consistent for arbitrary short-run dynamics. We derive the asymptotic distribution theory for the FDLS estimator of the cointegration vector in the stationary long memory case, thus complementing Robinson's consistency result. An application to the relation between the volatility realized in the stock market and the associated implicit volatility derived from option prices is offered.     

Estimation of partial differential equations with applications in finance

Linear parabolic partial differential equations (PDE’s) and diffusion models are closely linked through the celebrated Feynman–Kac representation of solutions to PDE’s. In asset pricing theory, this leads to the representation of derivative prices as solutions to PDE’s. Very often implied derivative prices are calculated given preliminary estimates of the diffusion model for the underlying variable. We demonstrate that the implied derivative prices are consistent and derive their asymptotic distribution under general conditions. We apply this result to three leading cases of preliminary estimators: Nonparametric, semiparametric and fully parametric ones. In all three cases, the asymptotic distribution of the solution is derived. We demonstrate the use of these results in obtaining confidence bands and standard errors for implied prices of bonds, options and other derivatives. Our general results also are of interest for the estimation of diffusion models using either historical data of the underlying process or option prices; these issues are also discussed.