Pricing VIX options with stochastic skew and asymmetric jumps

This paper performs several empirical exercises to provide evidence that the stochas-tic skew behavior and asymmetric jumps exist in VIX markets.In order to adequately capture all of the features,we develop a general valuation model and obtain quasi-analytical solutions for pricing VIX options.In addition,we make comparative studies of alternative models to illustrate the e ects after taking into account these features on the valuation of VIX options and investigate the relative value of an additional volatility factor and jump components.The empirical results indicate that the multi-factor volatility structure is vital to VIX option pricing due to providing more exibility in the modeling of VIX dynamics,and the need for asymmetric jumps cannot be eliminated by an additional volatility factor.     

Pricing VXX option with default risk and positive volatility skew

This paper proposes and makes a comparative study of alternative models for VXX option pricing. Factors such as mean-reversion, jumps, default risk and positive volatility skew are taken into consideration. In particular, default risk is characterized by jump-to-default framework and the “positive volatility skew” issue is addressed by stochastic volatility of volatility and jumps. Daily calibration is conducted and comparative study of the models is performed to check whether they properly fit market prices and generate reasonable positive volatility skews and deltas. Overall, jump-to-default extended LRJ model with positive correlated stochastic volatility (called JDLRJSV in the paper) serves as the best model in all the required aspects.     

Pricing VIX options in a stochastic vol‐of‐vol model

This paper proposes and makes a study of a new model for volatility index option pricing. Factors such as mean‐reversion, jumps, and stochastic volatility are taken into consideration. In particular, the positive volatility skew is addressed by the jump and the stochastic volatility of volatility. Daily calibration is used to check whether the model fits market prices and generates positive volatility skews. Overall, the results show that the mean‐reverting logarithmic jump and stochastic volatility model (called MRLRJSV in the paper) serves as the best model in all the required aspects. Copyright © 2015 John Wiley & Sons, Ltd.     

Regime-switching stochastic volatility model: estimation and calibration to VIX options

We develop and implement a method for maximum likelihood estimation of a regime-switching stochastic volatility model. Our model uses a continuous time stochastic process for the stock dynamics with the instantaneous variance driven by a Cox–Ingersoll–Ross process and each parameter modulated by a hidden Markov chain. We propose an extension of the EM algorithm through the Baum–Welch implementation to estimate our model and filter the hidden state of the Markov chain while using the VIX index to invert the latent volatility state. Using Monte Carlo simulations, we test the convergence of our algorithm and compare it with an approximate likelihood procedure where the volatility state is replaced by the VIX index. We found that our method is more accurate than the approximate procedure. Then, we apply Fourier methods to derive a semi-analytical expression of S&P500 and VIX option prices, which we calibrate to market data. We show that the model is sufficiently rich to encapsulate important features of the joint dynamics of the stock and the volatility and to consistently fit option market prices.     

Analysis of VIX Markets with a Time-Spread Portfolio

This paper explores the relationship between option markets for the S&P500 (SPX) and Chicago Board Options Exchange’s CBOE’s Volatility Index (VIX). Results are obtained by using the so-called time-spread portfolio to replicate a future contract on the squared VIX. The time-spread portfolio is interesting because it provides a model-free link between derivative prices for SPX and VIX. Time spreads can be computed from SPX put options with different maturities, which results in a term structure for squared volatility. This term structure can be compared to the VIX-squared term structure that is backed-out from VIX call options. The time-spread portfolio is also used to measure volatility-of-volatility (vol-of-vol) and the volatility leverage effect. There may emerge small differences in these measurements, depending on whether time spreads are computed with options on SPX or options on VIX. A study of 2012 daily options data shows that vol-of-vol estimates utilizing SPX data will reflect the volatility leverage effect, whereas estimates that exclusively utilize VIX options will predominantly reflect the premia in the VIX-future term structure.     

Estimation of Tempered Stable Lévy Models of Infinite Variation

Truncated realized quadratic variations (TRQV) are among the most widely used high-frequency-based nonparametric methods to estimate the volatility of a process in the presence of jumps. Nevertheless, the truncation level is known to critically affect its performance, especially in the presence of infinite variation jumps. In this paper, we study the optimal truncation level, in the mean-square error sense, for a semiparametric tempered stable Lévy model. We obtain a novel closed-form 2nd-order approximation of the optimal threshold in a high-frequency setting. As an application, we propose a new estimation method, which combines iteratively an approximate semiparametric method of moment estimator and TRQVs with the newly found small-time approximation for the optimal threshold. The method is tested via simulations to estimate the volatility and the Blumenthal-Getoor index of a generalized CGMY model and, via a localization technique, to estimate the integrated volatility of a Heston type model with CGMY jumps. Our method is found to outperform other alternatives proposed in the literature when working with a Lévy process (i.e., the volatility is constant), or when the index of jump intensity Y is larger than 3/2 in the presence of stochastic volatility.

     

基于修正的已实现阈值幂变差的股市跳跃波动行为研究

基于非参数统计方法,利用考虑金融资产价格跳跃和杠杆效应的时点波动估计方法修正已实现阈值幂变差,构造甄别跳跃的检验统计量,对金融资产价格中的随机波动、有限活跃跳跃和无限活跃跳跃等问题进行综合研究。为同时吸收波动率的异方差集聚效应和收益率的非对称效应,对原有的已实现波动率异质自回归预测模型进行拓展,将非对称的异质性自回归模型的误差项设定为GARCH模型,以考察跳跃波动序列与连续波动序列之间的复杂关系。利用沪深股指高频数据进行实证研究,包括进行跳跃识别,跳跃活动程度检验和波动率预测效果对比。研究结果表明,沪深股市同时存在布朗运动成分、有限活跃跳跃和无限活跃跳跃成分,其中连续路径方差占主体。同时,收益和波动间的杠杆效应显著,无论短期还是长期,连续波动和跳跃波动对波动率的预测均具有显著影响,同时考虑股价的跳跃、波动和杠杆效应因素有助于更准确地刻画资产价格动态过程。     

A scaled version of the double-mean-reverting model for VIX derivatives

As the Heston model is not consistent with VIX data in real market well enough, alternative stochastic volatility models including the double-mean-reverting model of Gatheral (in: Bachelier Congress, 2008) have been developed to overcome its limitation. The double-mean-reverting model is a three factor model successfully reflecting the empirical dynamics of the variance but there is no closed form solution for VIX derivatives and SPX options and thus calibration using conventional techniques may be slow. In this paper, we propose a fast mean-reverting version of the double-mean-reverting model. We obtain a closed form approximation for VIX derivatives and show how it is effective by comparing it with the Heston model and the double-mean-reverting model.     

The Heston model with stochastic elasticity of variance

Empirical evidence suggests that single factor models would not capture the full dynamics of stochastic volatility such that a marked discrepancy between their predicted prices and market prices exists for certain ranges (deep in‐the‐money and out‐of‐the‐money) of time‐to‐maturities of options. On the other hand, there is an empirical reason to believe that volatility skew fluctuates randomly. Based upon the idea of combining stochastic volatility and stochastic skew, this paper incorporates stochastic elasticity of variance running on a fast timescale into the Heston stochastic volatility model. This multiscale and multifactor hybrid model keeps analytic tractability of the Heston model as much as possible, while it enhances capturing the complex nature of volatility and skew dynamics. Asymptotic analysis based on ergodic theory yields a closed form analytic formula for the approximate price of European vanilla options. Subsequently, the effect of adding the stochastic elasticity factor on top of the Heston model is demonstrated in terms of implied volatility surface. Copyright © 2016 John Wiley & Sons, Ltd.     

基于跳跃、好坏波动率和马尔科夫状态转换的高频波动率模型预测

基于跳跃、好坏波动率的视角,采用比ABD检测更稳健的ADS检测法进行甄别跳跃,提出HAR改进模型,进一步考虑到实际波动率的非线性和高持续性动态,文章引入马尔科夫状态转换机制以构建对应的MRS-HAR族模型,推导其参数估计方法,并运用滚动时间窗预测技术和MCS检验评估预测模型结果,并采取不同的窗口期进行稳健性检验.以上海期货交易所的黄金连续(AU0)期货合约为研究对象,实证研究表明:结合马尔科夫状态转换机制,跳跃波动在上涨行情时会抑制未来波动性;结合马尔科夫状态转换机制,好坏波动率在上涨行情时正负冲击相对平衡,而在下跌行情时好(坏)波动率抑制(加剧)未来波动性;MCS检验证实,结合马尔科夫状态转换的MRS-HAR族模型相比于HAR族模型具有更优的预测精度,进一步考虑由ADS检测修正的好坏波动率和符号跳跃能够改善波动率模型的预测能力,其中基于符号跳跃和马尔科夫状态转换的MRS-HAR-RV-SJ模型展现了最高的预测精度.     

Research on the Forecasting Performance of the HAR-Type Model Based on True and False Jumps

Since the jump of an asset price has a strong effect on the estimate and forecast volatility, it has received widespread attention. Following HAR-CJ model introduced by Andersen et al, lots of works focus on this problem. In this paper, through a threshold technique, we distinguish the true and false jumps. Then we introduce two models, HAR-CTFJ model and LHAR-CTFJ model. Our result shows that the effect from the true jumps is significant while that from the false jumps is not. Moreover, the SPA test shows that our models (i.e. HAR-CTJ and LHAR-CTJ)are better than the classical HAR-CJ model in the prediction of volatility.     

New approach and analysis of the generalized constant elasticity of variance model

Generally, it is well known that the constant elasticity of variance (CEV) model fails to capture the empirical results verifying that the implied volatility of equity options displays smile and skew curves at the same time. In this study, to overcome the limitation of the CEV model, we introduce a new model, which is a generalization of the CEV model, and show that it can capture the smile and skew effects of implied volatility. Using an asymptotic analysis for two small parameters that determine the volatility shape, we obtain approximated solutions for option prices in the extended model. In addition, we demonstrate the stability of the solution for the expansion of the option price. Furthermore, we show the convergence rate of the solutions in Monte-Carlo simulation and compare our model with the CEV, Heston, and other extended stochastic volatility models to verify its flexibility and efficiency compared with these other models when fitting option data from the S&P 500 index.     

Pricing Exotic Discrete Variance Swaps under the 3/2-Stochastic Volatility Models

Abstract

We consider pricing of various types of exotic discrete variance swaps, like the gamma swaps and corridor variance swaps, under the 3/2-stochastic volatility models (SVMs) with jumps in asset price. The class of SVMs that use a constant-elasticity-of-variance (CEV) process for the instantaneous variance exhibits good analytical tractability only when the CEV parameter takes just a few special values (namely 0, 1/2, 1 and 3/2). The popular Heston model corresponds to the choice of the CEV parameter to be 1/2. However, the stochastic volatility dynamics implied by the Heston model fails to capture some important empirical features of the market data. The choice of 3/2 for the CEV parameter in the SVM shows better agreement with empirical studies while it maintains a good level of analytical tractability. Using the partial integro-differential equation (PIDE) formulation, we manage to derive quasi-closed-form pricing formulas for the fair strike prices of various types of exotic discrete variance swaps with various weight processes and different return specifications under the 3/2-model. Pricing properties of these exotic discrete variance swaps with respect to various model parameters are explored.     

Bayesian modeling and forecasting of Value‐at‐Risk via threshold realized volatility

This study proposes a threshold realized generalized autoregressive conditional heteroscedastic (GARCH) model that jointly models daily returns and realized volatility, thereby taking into account the bias and asymmetry of realized volatility. We incorporate this threshold realized GARCH model with skew Student‐t innovations as the observation equation, view this model as a sharp transition model, and treat the realized volatility as a proxy for volatility under this nonlinear structure. Through the Bayesian Markov chain Monte Carlo method, the model can jointly estimate the parameters in the return equation, the volatility equation, and the measurement equation. As an illustration, we conduct a simulation study and apply the proposed method to the US and Japan stock markets. Based on quantile forecasting and volatility estimation, we find that the threshold heteroskedastic framework with realized volatility successfully models the asymmetric dynamic structure. We also investigate the predictive ability of volatility by comparing the proposed model with the traditional GARCH model as well as some popular asymmetric GARCH and realized GARCH models. This threshold realized GARCH model with skew Student‐t innovations outperforms the competing risk models in out‐of‐sample volatility and Value‐at‐Risk forecasting.     

Prices and Asymptotics for Discrete Variance Swaps

Abstract

We study the fair strike of a discrete variance swap for a general time-homogeneous stochastic volatility model. In the special cases of Heston, Hull–White and Schöbel–Zhu stochastic volatility models, we give simple explicit expressions (improving Broadie and Jain (2008a). The effect of jumps and discrete sampling on volatility and variance swaps. International Journal of Theoretical and Applied Finance, 11(8), 761–797) in the case of the Heston model). We give conditions on parameters under which the fair strike of a discrete variance swap is higher or lower than that of the continuous variance swap. The interest rate and the correlation between the underlying price and its volatility are key elements in this analysis. We derive asymptotics for the discrete variance swaps and compare our results with those of Broadie and Jain (2008a. The effect of jumps and discrete sampling on volatility and variance swaps. International Journal of Theoretical and Applied Finance, 11(8), 761–797), Jarrow et al. (2013. Discretely sampled variance and volatility swaps versus their continuous approximations. Finance and Stochastics, 17(2), 305–324) and Keller-Ressel and Griessler (2012. Convex order of discrete, continuous and predictable quadratic variation and applications to options on variance. Working paper. Retrieved from http://arxiv.org/abs/1103.2310).     

Stochastic differential equations with diffusion and jumps modeling currency markets

An efficient currency market with zero transaction costs is considered. The dynamics of the exchange rate in this market is described by stochastic differential equations (SDEs) with diffusion and jumps; the latter are assumed to be described by a Lévy process. Adjusting theoretical arbitrage-free option prices computed within these models to market option prices requires properly choosing the coefficients in the SDEs. For this purpose, an expression for local volatility in a diffusion model is found and a relation between local and implied volatilities is determined. For a market model with diffusion and jumps, expressions for the local volatility and the local rate function are given. Moreover, in Merton’s model, where the jump component is a compound Poisson process with normal jumps, a relation between the local and the implied volatilities is determined.     

股票价格序列中随机跳跃存在性的一种检验方法

提出股票价格序列跳跃的一种检验方法.假设价格具有连续样本路径,建立一个关于股票价格样本观察的统计量,利用中心极限定理求得该统计量的极限分布为正态分布,这样,当该统计量超出基于极限分布算出的临界水平时,可以拒绝原假设,认为样本中存在跳跃.用此方法来应用于中国股市沪深股票指数,得到了中国股市存在随机跳跃的直接证据.提出的跳跃检验方法无需对连续部分的波动率形式作过多的假设,克服了波动率模型对检验准确性的影响.结果对金融资产的定价、投资和风险管理都具有积极的意义.     

Implied Filtering Densities on the Hidden State of Stochastic Volatility

Abstract

We formulate and analyse an inverse problem using derivative prices to obtain an implied filtering density on volatility’s hidden state. Stochastic volatility is the unobserved state in a hidden Markov model (HMM) and can be tracked using Bayesian filtering. However, derivative data can be considered as conditional expectations that are already observed in the market, and which can be used as input to an inverse problem whose solution is an implied conditional density on volatility. Our analysis relies on a specification of the martingale change of measure, which we refer to as separability. This specification has a multiplicative component that behaves like a risk premium on volatility uncertainty in the market. When applied to SPX options data, the estimated model and implied densities produce variance-swap rates that are consistent with the VIX volatility index. The implied densities are relatively stable over time and pick up some of the monthly effects that occur due to the options’ expiration, indicating that the volatility-uncertainty premium could experience cyclic effects due to the maturity date of the options.     

On the Esscher transforms and other equivalent martingale measures for Barndorff-Nielsen and Shephard stochastic volatility models with jumps

We compute and then discuss the Esscher martingale transform for exponential processes, the Esscher martingale transform for linear processes, the minimal martingale measure, the class of structure preserving martingale measures, and the minimum entropy martingale measure for stochastic volatility models of the Ornstein–Uhlenbeck type as introduced by Barndorff-Nielsen and Shephard. We show that in the model with leverage, with jumps both in the volatility and in the returns, all those measures are different, whereas in the model without leverage, with jumps in the volatility only and a continuous return process, several measures coincide, some simplifications can be made and the results are more explicit. We illustrate our results with parametric examples used in the literature.     

国际油价波动跳跃性特征的实证分析

根据国际油价波动中存在异常跳跃的情况,本文运用EGARCH-Jump模型对国际油价波动的跳跃性特征进行了实证分析。结果表明,加入跳跃因素的模型减缓了国际油价波动的持续性,同时杠杆效应消失,表明跳跃性因素是国际油价波动的影响因素之一,也证实了国际油价波动的跳跃性特征是国际石油市场产生杠杆效应的原因。但从长期来看,跳跃性因素对国际油价波动的扰动影响并不大,国际油价的波动仍主要受正常信息的影响。总体上,EGARCH-Jump模型比普通GARCH族模型能更好地捕捉国际油价波动的动态性特征。