A reduced form framework for modeling volatility of speculative prices based on realized variation measures

Building on realized variance and bipower variation measures constructed from high-frequency financial prices, we propose a simple reduced form framework for effectively incorporating intraday data into the modeling of daily return volatility. We decompose the total daily return variability into the continuous sample path variance, the variation arising from discontinuous jumps that occur during the trading day, as well as the overnight return variance. Our empirical results, based on long samples of high-frequency equity and bond futures returns, suggest that the dynamic dependencies in the daily continuous sample path variability are well described by an approximate long-memory HAR–GARCH model, while the overnight returns may be modeled by an augmented GARCH type structure. The dynamic dependencies in the non-parametrically identified significant jumps appear to be well described by the combination of an ACH model for the time-varying jump intensities coupled with a relatively simple log-linear structure for the jump sizes. Finally, we discuss how the resulting reduced form model structure for each of the three components may be used in the construction of out-of-sample forecasts for the total return volatility.     

Inference with non-Gaussian Ornstein–Uhlenbeck processes for stochastic volatility

Continuous-time stochastic volatility models are becoming an increasingly popular way to describe moderate and high-frequency financial data. Barndorff-Nielsen and Shephard (2001a) proposed a class of models where the volatility behaves according to an Ornstein–Uhlenbeck (OU) process, driven by a positive Lévy process without Gaussian component. These models introduce discontinuities, or jumps, into the volatility process. They also consider superpositions of such processes and we extend that to the inclusion of a jump component in the returns. In addition, we allow for leverage effects and we introduce separate risk pricing for the volatility components. We design and implement practically relevant inference methods for such models, within the Bayesian paradigm. The algorithm is based on Markov chain Monte Carlo (MCMC) methods and we use a series representation of Lévy processes. MCMC methods for such models are complicated by the fact that parameter changes will often induce a change in the distribution of the representation of the process and the associated problem of overconditioning. We avoid this problem by dependent thinning methods. An application to stock price data shows the models perform very well, even in the face of data with rapid changes, especially if a superposition of processes with different risk premiums and a leverage effect is used.     

Realized Laplace transforms for estimation of jump diffusive volatility models

We develop an efficient and analytically tractable method for estimation of parametric volatility models that is robust to price-level jumps. The method entails first integrating intra-day data into the Realized Laplace Transform of volatility, which is a model-free estimate of the daily integrated empirical Laplace transform of the unobservable volatility. The estimation is then done by matching moments of the integrated joint Laplace transform with those implied by the parametric volatility model. In the empirical application, the best fitting volatility model is a non-diffusive two-factor model where low activity jumps drive its persistent component and more active jumps drive the transient one.     

Statistical estimation of multivariate Ornstein–Uhlenbeck processes and applications to co-integration

Ornstein–Uhlenbeck models are continuous-time processes which have broad applications in finance as, e.g., volatility processes in stochastic volatility models or spread models in spread options and pairs trading. The paper presents a least squares estimator for the model parameter in a multivariate Ornstein–Uhlenbeck model driven by a multivariate regularly varying Lévy process with infinite variance. We show that the estimator is consistent. Moreover, we derive its asymptotic behavior and test statistics. The results are compared to the finite variance case. For the proof we require some new results on multivariate regular variation of products of random vectors and central limit theorems. Furthermore, we embed this model in the setup of a co-integrated model in continuous time.     

Option pricing with stochastic volatility models

A general class of models for derivative pricing with stochastic volatility is analyzed. We include the possibility of jumps for the paths of the asset's price and for those of its volatility. We also consider the case of correlation between the process of the asset's price and that of its volatility. In this way we are able to give a unifying view on most of the models studied in the literature. We will examine theoretical issues related to the market price of volatility risk, the equivalent martingale measures and the possibility of obtaining a numerically tractable formula for contingent claim pricing. Finally, we propose some methodologies to test the behavior of stochastic volatility models when applied to market data.     

Bayesian hypothesis testing in latent variable models

Hypothesis testing using Bayes factors (BFs) is known not to be well defined under the improper prior. In the context of latent variable models, an additional problem with BFs is that they are difficult to compute. In this paper, a new Bayesian method, based on the decision theory and the EM algorithm, is introduced to test a point hypothesis in latent variable models. The new statistic is a by-product of the Bayesian MCMC output and, hence, easy to compute. It is shown that the new statistic is appropriately defined under improper priors because the method employs a continuous loss function. In addition, it is easy to interpret. The method is illustrated using a one-factor asset pricing model and a stochastic volatility model with jumps.     

A unified approach to Bermudan and barrier options under stochastic volatility models with jumps

Many financial assets, such as currencies, commodities, and equity stocks, exhibit both jumps and stochastic volatility, which are especially prominent in the market after the financial crisis. Some strategic decision making problems also involve American-style options. In this paper, we develop a novel, fast and accurate method for pricing American and barrier options in regime switching jump diffusion models. By blending regime switching models and Markov chain approximation techniques in the Fourier domain, we provide a unified approach to price Bermudan, American options and barrier options under general stochastic volatility models with jumps. The models considered include Heston, Hull–White, Stein–Stein, Scott, the 3/2 model, and the recently proposed 4/2 model and the α-Hypergeometric model with general jump amplitude distributions in the return process. Applications include the valuation of discretely monitored contracts as well as continuously monitored contracts common in the foreign exchange markets. Numerical results are provided to demonstrate the accuracy and efficiency of the proposed method.     

A closed-form exact solution for pricing fixed-income variance swaps with affine-jump model

Fixed-income variance swaps became popular for investors to trade and hedge the fluctuation of interest rates after the recent global financial crisis over the past few decades, however, their valuations and risk management have not been studied sufficiently. This paper presents an analytic approach for pricing some discretely sampled fixed-income variance swaps under an affine-jump model with stochastic mean, stochastic volatility, and jumps. We employ a generalized characteristic function to derive the closed-form pricing formulas of these swaps, including two kinds of zero-coupon bond variance swap, Libor variance swap, and bond yield variance swap, to be precise. We also perform some numerical studies based on these models, which suggest that the fair strike values of these variance swaps are within a reasonable range regardless of estimation risk with data dependence and near-zero short rate regime. Our numerics show that the influences of varying sampling frequency and time-to-maturity on the values of these swaps are significant, and highlight the risks of specifying short rate model. Furthermore, the sensitivity analysis on the key parameters finds that the risks of stochastic volatility and jumps play prominent roles in pricing these variance swaps under the near-zero short rate regime.     

A discrete-time model for daily S & P500 returns and realized variations: Jumps and leverage effects

We develop an empirically highly accurate discrete-time daily stochastic volatility model that explicitly distinguishes between the jump and continuous-time components of price movements using nonparametric realized variation and Bipower variation measures constructed from high-frequency intraday data. The model setup allows us to directly assess the structural inter-dependencies among the shocks to returns and the two different volatility components. The model estimates suggest that the leverage effect, or asymmetry between returns and volatility, works primarily through the continuous volatility component. The excellent fit of the model makes it an ideal candidate for an easy-to-implement auxiliary model in the context of indirect estimation of empirically more realistic continuous-time jump diffusion and Lévy-driven stochastic volatility models, effectively incorporating the interdaily dependencies inherent in the high-frequency intraday data.     

The common and specific components of dynamic volatility

This paper develops a dynamic approximate factor model in which returns are time-series heteroskedastic. The heteroskedasticity has three components: a factor-related component, a common asset-specific component, and a purely asset-specific component. We develop a new multivariate GARCH model for the factor-related component. We develop a univariate stochastic volatility model linked to a cross-sectional series of individual GARCH models for the common asset-specific component and the purely asset-specific component. We apply the analysis to monthly US equity returns for the period January 1926 to December 2000. We find that all three components contribute to the heteroskedasticity of individual equity returns. Factor volatility and the common component in asset-specific volatility have long-term secular trends as well as short-term autocorrelation. Factor volatility has correlation with interest rates and the business cycle.     

Moving average stochastic volatility models with application to inflation forecast

We introduce a new class of models that has both stochastic volatility and moving average errors, where the conditional mean has a state space representation. Having a moving average component, however, means that the errors in the measurement equation are no longer serially independent, and estimation becomes more difficult. We develop a posterior simulator that builds upon recent advances in precision-based algorithms for estimating these new models. In an empirical application involving US inflation we find that these moving average stochastic volatility models provide better in-sample fitness and out-of-sample forecast performance than the standard variants with only stochastic volatility.     

Threshold bipower variation and the impact of jumps on volatility forecasting

This study reconsiders the role of jumps for volatility forecasting by showing that jumps have a positive and mostly significant impact on future volatility. This result becomes apparent once volatility is separated into its continuous and discontinuous components using estimators which are not only consistent, but also scarcely plagued by small sample bias. With the aim of achieving this, we introduce the concept of threshold bipower variation, which is based on the joint use of bipower variation and threshold estimation. We show that its generalization (threshold multipower variation) admits a feasible central limit theorem in the presence of jumps and provides less biased estimates, with respect to the standard multipower variation, of the continuous quadratic variation in finite samples. We further provide a new test for jump detection which has substantially more power than tests based on multipower variation. Empirical analysis (on the S&P500 index, individual stocks and US bond yields) shows that the proposed techniques improve significantly the accuracy of volatility forecasts especially in periods following the occurrence of a jump.     

Impact of volatility jumps in a mean-reverting model: Derivative pricing and empirical evidence

This paper investigates the critical role of volatility jumps under mean reversion models. Based on the empirical tests conducted on the historical prices of commodities, we demonstrate that allowing for the presence of jumps in volatility in addition to price jumps is a crucial factor when confronting non-Gaussian return distributions. By employing the particle filtering method, a comparison of results drawn among several mean-reverting models suggests that incorporating volatility jumps ensures an improved fit to the data. We infer further empirical evidence for the existence of volatility jumps from the possible paths of filtered state variables. Our numerical results indicate that volatility jumps significantly affect the level and shape of implied volatility smiles. Finally, we consider the pricing of options under the mean reversion model, where the underlying asset price and its volatility both have jump components.     

Multifrequency jump-diffusions: An equilibrium approach

This paper proposes that equilibrium valuation is a powerful method to generate endogenous jumps in asset prices. We specify an economy with continuous consumption and dividend paths, in which endogenous price jumps originate from the market impact of regime-switches in the drifts and volatilities of fundamentals. We parsimoniously incorporate regimes of heterogeneous durations and verify that the persistence of a shock endogenously increases the magnitude of the induced price jump. As the number of frequencies driving fundamentals goes to infinity, the price process converges to a novel stochastic process, which we call a multifractal jump-diffusion.     

Forecasting volatility with component conditional autoregressive range model

In this paper, we propose a component conditional autoregressive range (CCARR) model for forecasting volatility. The proposed CCARR model assumes that the price range comprises both a long-run (trend) component and a short-run (transitory) component, which has the capacity to capture the long memory property of volatility. The model is intuitive and convenient to implement by using the maximum likelihood estimation method. Empirical analysis using six stock market indices highlights the value of incorporating a second component into range (volatility) modelling and forecasting. In particular, we find that the proposed CCARR model fits the data better than the CARR model, and that it generates more accurate out-of-sample volatility forecasts and contains more information content about the true volatility than the popular GARCH, component GARCH and CARR models.     

A new class of multidimensional Wishart-based hybrid models

In this article, we present a new class of pricing models that extend the application of Wishart processes to the so-called stochastic local volatility (or hybrid) pricing paradigm. This approach combines the advantages of local and stochastic volatility models. Despite the growing interest on the topic, however, it seems that no particular attention has been paid to the use of multidimensional specifications for the stochastic volatility component. Our work tries to fill the gap: we introduce two hybrid models in which the stochastic volatility dynamics is described by means of a Wishart process. The proposed parametrizations not only preserve the desirable features of existing Wishart-based models but significantly enhance the ability of reproducing market prices of vanilla options.

     

随机波动模型下定价方差互换的一类控制变量

本文从风险中性定价的角度出发,给出了在随机波动模型下定价方差互换的一类控制变量,从而大大提高了使用蒙特卡罗方法计算方差互换价格时的效率,并在波动率的平方满足几何布朗运动(GBM)和波动率满足Ornstein-Uhlen-beck(OU)过程这两种随机波动情况下给出了控制变量的具体形式。特别对于GBM型随机波动模型,可以得到一系列控制变量,从而进行多元控制。     

Time-varying risk aversion and renminbi exchange rate volatility: Evidence from CARR-MIDAS model

In this paper, we investigate the relation between time-varying risk aversion and renminbi exchange rate volatility using the conditional autoregressive range-mixed-data sampling (CARR-MIDAS) model. The CARR-MIDAS model is a range-based volatility model, which exploits intraday information regarding the intraday trajectory of the price. Moreover, the model features a MIDAS structure allowing for time-varying risk aversion to drive the long-run volatility dynamics. Our empirical results show that time-varying risk aversion has a significantly negative effect on the long-run volatility of renminbi exchange rate. Moreover, we observe that both intraday ranges and time-varying risk aversion contain important information for forecasting renminbi exchange rate volatility. The range-based CARR-MIDAS model incorporating time-varying risk aversion provides more accurate out-of-sample forecasts of renminbi exchange rate volatility compared to a variety of competing models, including the return-based GARCH, GARCH-MIDAS and GARCH-MIDAS incorporating time-varying risk aversion as well as range-based CARR, CARR-MIDAS and heterogeneous autoregressive (HAR), for forecast horizons of 1 day up to 3 months. This result is robust to alternative risk aversion measure, alternative MIDAS lags as well as alternative out-of-sample periods. Overall, our findings highlight the value of incorporating intraday information and time-varying risk aversion for forecasting the renminbi exchange rate volatility.     

Time-varying leverage effects

Vast empirical evidence points to the existence of a negative correlation, named ”leverage effect”, between shocks to variance and shocks to returns. We provide a nonparametric theory of leverage estimation in the context of a continuous-time stochastic volatility model with jumps in returns, jumps in variance, or both. Leverage is defined as a flexible function of the state of the firm, as summarized by the spot variance level. We show that its point-wise functional estimates have asymptotic properties (in terms of rates of convergence, limiting biases, and limiting variances) which crucially depend on the likelihood of the individual jumps and co-jumps as well as on the features of the jump size distributions. Empirically, we find economically important time-variation in leverage with more negative values associated with higher variance levels.     

The role of implied volatility in forecasting future realized volatility and jumps in foreign exchange,stock, and bond markets

We study the forecasting of future realized volatility in the foreign exchange, stock, and bond markets from variables in our information set, including implied volatility backed out from option prices. Realized volatility is separated into its continuous and jump components, and the heterogeneous autoregressive (HAR) model is applied with implied volatility as an additional forecasting variable. A vector HAR (VecHAR) model for the resulting simultaneous system is introduced, controlling for possible endogeneity issues. We find that implied volatility contains incremental information about future volatility in all three markets, relative to past continuous and jump components, and it is an unbiased forecast in the foreign exchange and stock markets. Out-of-sample forecasting experiments confirm that implied volatility is important in forecasting future realized volatility components in all three markets. Perhaps surprisingly, the jump component is, to some extent, predictable, and options appear calibrated to incorporate information about future jumps in all three markets.