Analysis of a jump-diffusion option pricing model with serially correlated jump sizes

This paper extends the classical jump-diffusion option pricing model to incorporate serially correlated jump sizes which have been documented in recent empirical studies. We model the series of jump sizes by an autoregressive process and provide an analysis on the underlying stock return process. Based on this analysis, the European option price and the hedging parameters under the extended model are derived analytically. Through numerical examples, we investigate how the autocorrelation of jump sizes influences stock returns, option prices and hedging parameters, and demonstrate its effects on hedging portfolios and implied volatility smiles. A calibration example based on real market data is provided to show the advantage of incorporating the autocorrelation of jump sizes.     

Estimating the Parameters of Stochastic Volatility Models Using Option Price Data

This article describes a maximum likelihood method for estimating the parameters of the standard square-root stochastic volatility model and a variant of the model that includes jumps in equity prices. The model is fitted to data on the S&P 500 Index and the prices of vanilla options written on the index, for the period 1990 to 2011. The method is able to estimate both the parameters of the physical measure (associated with the index) and the parameters of the risk-neutral measure (associated with the options), including the volatility and jump risk premia. The estimation is implemented using a particle filter whose efficacy is demonstrated under simulation. The computational load of this estimation method, which previously has been prohibitive, is managed by the effective use of parallel computing using graphics processing units (GPUs). The empirical results indicate that the parameters of the models are reliably estimated and consistent with values reported in previous work. In particular, both the volatility risk premium and the jump risk premium are found to be significant.     

ASSESSING AND TESTING FOR THRESHOLD NONLINEARITY IN STOCK RETURNS

This paper proposes a test for threshold nonlinearity in a time series with generalized autore‐gressive conditional heteroscedasticity (GARCH) volatility dynamics. This test is used to examine whether financial returns on market indices exhibit asymmetric mean and volatility around a threshold value, using a double‐threshold GARCH model. The test adopts the reversible‐jump Markov chain Monte Carlo idea of Green, proposed in 1995, to calculate the posterior probabilities for a conventional GARCH model and a double‐threshold GARCH model. Posterior evidence favouring the threshold GARCH model indicates threshold nonlinearity with asymmetric behaviour of the mean and volatility. Simulation experiments demonstrate that the test works very well in distinguishing between the conventional GARCH and the double‐threshold GARCH models. In an application to eight international financial market indices, including the G‐7 countries, clear evidence supporting the hypothesis of threshold nonlinearity is discovered, simultaneously indicating an uneven mean‐reverting pattern and volatility asymmetry around a threshold return value.     

How precise is the finite sample approximation of the asymptotic distribution of realised variation measures in the presence of jumps?

This paper studies the impact of jumps on volatility estimation and inference based on various realised variation measures such as realised variance, realised multipower variation and truncated realised multipower variation. We review the asymptotic theory of those realised variation measures and present a new estimator for the asymptotic ‘variance’ of the centered realised variance in the presence of jumps. Next, we compare the finite sample performance of the various estimators by means of detailed Monte Carlo studies. Here we study the impact of the jump activity, of the jump size of the jumps in the price and of the presence of additional independent or dependent jumps in the volatility. We find that the finite sample performance of realised variance and, in particular, of log-transformed realised variance is generally good, whereas the jump-robust statistics tend to struggle in the presence of a highly active jump process.     

Towards Uniformly Efficient Trend Estimation Under Weak/Strong Correlation and Non‐stationary Volatility

In this paper, we consider the deterministic trend model where the error process is allowed to be weakly or strongly correlated and subject to non‐stationary volatility. Extant estimators of the trend coefficient are analysed. We find that under heteroskedasticity, the Cochrane–Orcutt‐type estimator (with some initial condition) could be less efficient than Ordinary Least Squares (OLS) when the process is highly persistent, whereas it is asymptotically equivalent to OLS when the process is less persistent. An efficient non‐parametrically weighted Cochrane–Orcutt‐type estimator is then proposed. The efficiency is uniform over weak or strong serial correlation and non‐stationary volatility of unknown form. The feasible estimator relies on non‐parametric estimation of the volatility function, and the asymptotic theory is provided. We use the data‐dependent smoothing bandwidth that can automatically adjust for the strength of non‐stationarity in volatilities. The implementation does not require pretesting persistence of the process or specification of non‐stationary volatility. Finite‐sample evaluation via simulations and an empirical application demonstrates the good performance of proposed estimators.     

The Variance Risk Premium: Components,Term Structures,and Stock Return Predictability

This article examines the properties of the variance risk premium (VRP). We propose a flexible asset pricing model that captures co-jumps in prices and volatility, and self-exciting jump clustering. We estimate the model on equity returns and variance swap rates at different horizons. The total VRP is negative and has a downward-sloping term structure, while its jump component displays an upward-sloping term structure. The abrupt and persistent response of the short-term jump VRP to extreme events makes this specific premium a proxy for investors’ fear of a market crash. Furthermore, the use of the VRP level and slope, and of its components, helps improve the short-run predictability of equity excess returns.     

SVJD-LIBOR随机动态模型的市场校准估计与实证模拟

针对跳跃扩散LIBOR市场模型（JD-LIBOR）与随机波动率LIBOR市场模型（SVJD-LMM）各自应用局限,首先将正态跳跃扩散与Heston随机波动率同时引入标准化LIBOR市场模型中,建立一类新型双重驱动非标准化LIBOR市场模型（SVJD-LMM）。其次,运用Cap、Swaption等利率衍生产品市场数据和Black逆推校准方法,对模型的局部波动参数与瞬间相关性参数进行有效市场校准;并运用自适应马尔科夫链蒙特卡罗模拟方法（此后简称A-MCMC）对模型的随机波动率、跳跃扩散等其他主要参数进行有效理论估计与实证模拟。最后,针对六月期美元Libor远期利率实际数据,对上述三类市场模型进行了模拟比较分析。研究结论认为,若在单因子Libor利率市场模型基础上引入跳跃扩散过程,并且联立波动率的随机微分方程,则可极大地提高利率模型的解释力;加入随机波动率和跳跃扩散过程的模拟计算结果与实际利率的误差更小,从而更接近现实情况。     

Non‐parametric Threshold Estimation for Models with Stochastic Diffusion Coefficient and Jumps

Abstract. We consider a stochastic process driven by diffusions and jumps. Given a discrete record of observations, we devise a technique for identifying the times when jumps larger than a suitably defined threshold occurred. This allows us to determine a consistent non‐parametric estimator of the integrated volatility when the infinite activity jump component is Lévy. Jump size estimation and central limit results are proved in the case of finite activity jumps. Some simulations illustrate the applicability of the methodology in finite samples and its superiority on the multipower variations especially when it is not possible to use high frequency data.     

证券市场开放速度的影响和决定

加速开放中的中国证券市场,有必要衡量进程加速的效应和最优速度。通过股市波动率分解,将波动分解为普通市场反应带来的连续型和过度反应带来的跳跃型波动,细致分析了加速开放对股市的影响,并权衡总波动的大小和结构以估算最优开放增速。实证发现加速引入国际机构资本可以有效降低波动性,但股市信息融解效率仍较低,体现在跳跃型方差占比高;市场状况会影响最优增速,估算现阶段适当的外国资本审批额度月增速约为12%。最后对利用境外资本健全证券市场提出建议。     

HEAVY-TAILED-DISTRIBUTED THRESHOLD STOCHASTIC VOLATILITY MODELS IN FINANCIAL TIME SERIES

To capture mean and variance asymmetries and time‐varying volatility in financial time series, we generalize the threshold stochastic volatility (THSV) model and incorporate a heavy‐tailed error distribution. Unlike existing stochastic volatility models, this model simultaneously accounts for uncertainty in the unobserved threshold value and in the time‐delay parameter. Self‐exciting and exogenous threshold variables are considered to investigate the impact of a number of market news variables on volatility changes. Adopting a Bayesian approach, we use Markov chain Monte Carlo methods to estimate all unknown parameters and latent variables. A simulation experiment demonstrates good estimation performance for reasonable sample sizes. In a study of two international financial market indices, we consider two variants of the generalized THSV model, with US market news as the threshold variable. Finally, we compare models using Bayesian forecasting in a value‐at‐risk (VaR) study. The results show that our proposed model can generate more accurate VaR forecasts than can standard models.     

Bayesian inference for stable Lévy–driven stochastic differential equations with high‐frequency data

In this paper, we consider parametric Bayesian inference for stochastic differential equations driven by a pure‐jump stable Lévy process, which is observed at high frequency. In most cases of practical interest, the likelihood function is not available; hence, we use a quasi‐likelihood and place an associated prior on the unknown parameters. It is shown under regularity conditions that there is a Bernstein–von Mises theorem associated to the posterior. We then develop a Markov chain Monte Carlo algorithm for Bayesian inference, and assisted with theoretical results, we show how to scale Metropolis–Hastings proposals when the frequency of the data grows, in order to prevent the acceptance ratio from going to zero in the large data limit. Our algorithm is presented on numerical examples that help verify our theoretical findings.     

Testing for intercept‐scale switch in linear autoregression

Autoregressive models with switching regime are a frequently used class of nonlinear time series models, which are popular in finance, engineering, and other fields. We consider linear switching autoregressions in which the intercept and variance possibly switch simultaneously, while the autoregressive parameters are structural and hence the same in all states, and we propose quasi‐likelihood‐based tests for a regime switch in this class of models. Our motivation is from financial time series, where one expects states with high volatility and low mean together with states with low volatility and higher mean. We investigate the performance of our tests in a simulation study, and give an application to a series of IBM monthly stock returns. The Canadian Journal of Statistics 40: 427–446; 2012 © 2012 Statistical Society of Canada     

Value at risk estimation under stochastic volatility models using adaptive PMCMC methods

In this paper, we propose a value-at-risk (VaR) estimation technique based on a new stochastic volatility model with leverage effect, nonconstant conditional mean and jump. In order to estimate the model parameters and latent state variables, we integrate the particle filter and adaptive Markov Chain Monte Carlo (MCMC) algorithms to develop a novel adaptive particle MCMC (A-PMCMC) algorithm. Comprehensive simulation experiments based on three stock indices and two foreign exchange time series show effectiveness of the proposed A-PMCMC algorithm and the VaR estimation technique.     

A bayesian estimator for the dependence function of a bivariate extreme‐value distribution

Any continuous bivariate distribution can be expressed in terms of its margins and a unique copula. In the case of extreme‐value distributions, the copula is characterized by a dependence function while each margin depends on three parameters. The authors propose a Bayesian approach for the simultaneous estimation of the dependence function and the parameters defining the margins. They describe a nonparametric model for the dependence function and a reversible jump Markov chain Monte Carlo algorithm for the computation of the Bayesian estimator. They show through simulations that their estimator has a smaller mean integrated squared error than classical nonparametric estimators, especially in small samples. They illustrate their approach on a hydrological data set.     

Fully Bayesian Binary Markov Random Field Models: Prior Specification and Posterior Simulation

We propose a flexible prior model for the parameters of binary Markov random fields (MRF), defined on rectangular lattices and with maximal cliques defined from a template maximal clique. The prior model allows higher‐order interactions to be included. We also define a reversible jump Markov chain Monte Carlo algorithm to sample from the associated posterior distribution. The number of possible parameters for a higher‐order MRF becomes high, even for small template maximal cliques. We define a flexible parametric form where the parameters have interpretation as potentials for clique configurations, and limit the effective number of parameters by assigning apriori discrete probabilities for events where groups of parameter values are equal. To cope with the computationally intractable normalising constant of MRFs, we adopt a previously defined approximation of binary MRFs. We demonstrate the flexibility of our prior formulation with simulated and real data examples.     

Volatility estimation from short time series of stock prices

We consider estimation of the historical volatility of stock prices. It is assumed that the stock prices are represented as time series formed as samples of the solution of a stochastic differential equation with random and time-varying parameters; these parameters are not observable directly and have unknown evolution law. The price samples are available with limited frequency only. In this setting, the estimation has to be based on short time series, and the estimation error can be significant. We suggest some supplements to the existing nonparametric methods of volatility estimation. Two modifications of the standard summation formula for the volatility are derived. In addition, a linear transformation eliminating the appreciation rate and preserving the volatility is suggested.     

A New Approach to Volatility Modeling: The Factorial Hidden Markov Volatility Model

A new process—the factorial hidden Markov volatility (FHMV) model—is proposed to model financial returns or realized variances. Its dynamics are driven by a latent volatility process specified as a product of three components: a Markov chain controlling volatility persistence, an independent discrete process capable of generating jumps in the volatility, and a predictable (data-driven) process capturing the leverage effect. An economic interpretation is attached to each one of these components. Moreover, the Markov chain and jump components allow volatility to switch abruptly between thousands of states, and the transition matrix of the model is structured to generate a high degree of volatility persistence. An empirical study on six financial time series shows that the FHMV process compares favorably to state-of-the-art volatility models in terms of in-sample fit and out-of-sample forecasting performance over time horizons ranging from 1 to 100 days. Supplementary materials for this article are available online.     

Fourier-cosine method for pricing forward starting options with stochastic volatility and jumps

This article provides an efficient method for pricing forward starting options under stochastic volatility model with double exponential jumps. The forward characteristic function of the log asset price is derived and thereby forward starting options are well evaluated by Fourier-cosine technique. Based on adaptive simulated annealing algorithm, the model is calibrated to obtain the estimated parameters. Numerical results show that the pricing method is accurate and fast. Double exponential jumps have pronounced impacts on long-term forward starting options prices. Stochastic volatility model with double exponential jumps fits forward implied volatility smile pretty well in contrast to stochastic volatility model.     

A proportional hazards regression model with change-points in the baseline function

In this article, we consider a new regression model for counting processes under a proportional hazards assumption. This model is motivated by the need of understanding the evolution of the booking process of a railway company. The main novelty of the approach consists in assuming that the baseline hazard function is piecewise constant, with unknown times of jump (these times of jump are estimated from the data as model parameters). Hence, the parameters of the model can be separated into two different types: parameters that measure the influence of the covariates, and parameters from a multiple change-point model for the baseline. Cox??s semiparametric regression can be seen as a limit case of our model. We develop an iterative procedure to estimate the different parameters, and a test procedure that allows to perform change-point detection in the baseline. Our technique is supported by simulation studies and a real data analysis, which show that our model can be a reasonable alternative to Cox??s regression model, particularly in the presence of tied event times.     

Flexible Threshold Models for Modelling Interest Rate Volatility

This paper focuses on interest rate models with regime switching and extends previous nonlinear threshold models by relaxing the assumption of a fixed number of regimes. Instead we suggest automatic model determination through Bayesian inference via the reversible jump Markov Chain Monte Carlo (MCMC) algorithm. Moreover, we allow the thresholds in the volatility to be driven not only by the interest rate but also by other economic factors. We illustrate our methodology by applying it to interest rates and other economic factors of the American economy.