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.     

GARCH(1,1)模型及其在汇率条件波动预测中的应用

检验人民币/日元汇率与波动的时间序列特征,证实存在简单单位根过程及条件异方差性。计算表明,其汇率变化率的ARMA及ARMA/GARCH组合模型的建模不成立,GARCH、EGARCH、IGARCH模型的建模效果接近,且GARCH(1,1)拟合效果最好。GARCH(1,1)模型的跨度为一年的样本外条件异方差预测,显示出该年末汇率的震荡,与实际情况一致。GARCH(1,1)是汇率数据建娱的首选模型。     

American Options Exercise Boundary When the Volatility Changes Randomly

The American put option exercise boundary has been studied extensively as a function of time and the underlying asset price. In this paper we analyze its dependence on the volatility, since the Black and Scholes model is used in practice via the (varying) implied volatility parameter. We consider a stochastic volatility model for the underlying asset price. We provide an extension of the regularity results of the American put option price function and we prove that the optimal exercise boundary is a decreasing function of the current volatility process realization. Accepted 13 January 1998     

Local Volatility Pricing Models for Long-Dated FX Derivatives

Abstract

We study the local volatility function in the foreign exchange (FX) market, where both domestic and foreign interest rates are stochastic. This model is suitable to price long-dated FX derivatives. We derive the local volatility function and obtain several results that can be used for the calibration of this local volatility on the FX option's market. Then, we study an extension to obtain a more general volatility model and propose a calibration method for the local volatility associated with this model.     

随机波动率模型下技术分析指标的一些性质(英文)

一些流行的技术指标(例如布林带,RSI,ROC等)被股市交易者广为使用.交易者将每日(小时,周,……)的实际股价作为计算某个技术指标的样本,通过观察相关频率来指导投资.技术指标的有效性已在广泛的应用中得到了验证.我们已经证明在Black-Scholes模型下,某些技术指标有许多有用的统计性质.作为更一般的情况,随机波动率模型在金融数学中得到了广泛的讨论.本文基于随机波动率模型对技术指标的统计性质进行了研究.研究结果表明,如果股票价格服从随机波动率模型,则技术指标的合理性可以得到有力的证明,从这个角度我们为技术分析奠定理论基础.     

Investment timing under hybrid stochastic and local volatility

We consider an investment timing problem under a real option model where the instantaneous volatility of the project value is given by a combination of a hidden stochastic process and the project value itself. The stochastic volatility part is given by a function of a fast mean-reverting process as well as a slowly varying process and the local volatility part is a power (the elasticity parameter) of the project value itself. The elasticity parameter controls directly the correlation between the project value and the volatility. Knowing that the project value represents the market price of a real asset in many applications and the value of the elasticity parameter depends on the asset, the elasticity parameter should be treated with caution for investment decision problems. Based on the hybrid structure of volatility, we investigate the simultaneous impact of the elasticity and the stochastic volatility on the real option value as well as the investment threshold.     

A Matched Asymptotic Expansions Approach to Continuity Corrections for Discretely Sampled Options. Part 1: Barrier Options

We propose a general framework to model equity volatility for a firm financed by equity and additional non-equity sources of funds. The stochastic nature of equity volatility is endogenous, and comes from the impact of a change in the value of the firm's assets on the financial leverage. We first present the basic model, which is an extension of the Black-Scholes model, to value corporate securities. Second, we show for the first time in the option literature, that instantaneous equity volatility is a solution of a partial differential equation similar to Black-Scholes', although it is non-linear and in general does not have any analytical solution. However, analytical approximations for equity volatility are proposed for different capital structures: (1) equity and debt, (2) equity and warrants, and (3) equity, debt and warrants. They are shown to be very accurate.     

Multifactor Heston's stochastic volatility model for European option pricing

In this study, we extend the multiscale stochastic volatility model of [Fouque J‐P, Lorig MJ, SIAM J Financial Math. 2011;2(1):221‐254] by incorporating a slow varying factor of volatility. The resulting model can be viewed as a multifactor extension of the Heston model with two additional factors driving the volatility levels. An asymptotic analysis consisting of singular and regular perturbation expansions is developed to obtain an approximation to European option prices. We also find explicit expressions for some essential functions that are available only in integral formulas in the work of [Fouque J‐P, Lorig MJ, SIAM J Financial Math. 2011;2(1):221‐254]. This finding basically leads to considerable reduction in computational time for numerical calculation as well as calibration problems. An accuracy result of the asymptotic approximation is also provided. For numerical illustration, the multifactor Heston model is calibrated to index options on the market, and we find that the resulting implied volatility surfaces fit the market data better than those produced by the multiscale stochastic volatility model of [Fouque J‐P, Lorig MJ, SIAM J Financial Math. 2011;2(1):221‐254], particularly for long‐maturity call options.     

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.     

Ornstein–Uhlenbeck processes in Hilbert space with non-Gaussian stochastic volatility

We propose a non-Gaussian operator-valued extension of the Barndorff-Nielsen and Shephard stochastic volatility dynamics, defined as the square-root of an operator-valued Ornstein–Uhlenbeck process with Lévy noise and bounded drift. We derive conditions for the positive definiteness of the Ornstein–Uhlenbeck process, where in particular we must restrict to operator-valued Lévy processes with “non-decreasing paths”. It turns out that the volatility model allows for an explicit calculation of its characteristic function, showing an affine structure. We introduce another Hilbert space-valued Ornstein–Uhlenbeck process with Wiener noise perturbed by this class of stochastic volatility dynamics. Under a strong commutativity condition between the covariance operator of the Wiener process and the stochastic volatility, we can derive an analytical expression for the characteristic functional of the Ornstein–Uhlenbeck process perturbed by stochastic volatility if the noises are independent. The case of operator-valued compound Poisson processes as driving noise in the volatility is discussed as a particular example of interest. We apply our results to futures prices in commodity markets, where we discuss our proposed stochastic volatility model in light of ambit fields.     

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.     

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.     

Robust investment–reinsurance optimization with multiscale stochastic volatility

This paper investigates the investment and reinsurance problem in the presence of stochastic volatility for an ambiguity-averse insurer (AAI) with a general concave utility function. The AAI concerns about model uncertainty and seeks for an optimal robust decision. We consider a Brownian motion with drift for the surplus of the AAI who invests in a risky asset following a multiscale stochastic volatility (SV) model. We formulate the robust optimal investment and reinsurance problem for a general class of utility functions under a general SV model. Applying perturbation techniques to the Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation associated with our problem, we derive an investment–reinsurance strategy that well approximates the optimal strategy of the robust optimization problem under a multiscale SV model. We also provide a practical strategy that requires no tracking of volatility factors. Numerical study is conducted to demonstrate the practical use of theoretical results and to draw economic interpretations from the robust decision rules.     

Upper bounds for ruin probabilities under stochastic interest rate and optimal investment strategies

In this paper, we study the upper bounds for ruin probabilities of an insurance company which invests its wealth in a stock and a bond. We assume that the interest rate of the bond is stochastic and it is described by a Cox-Ingersoll-Ross (CIR) model. For the stock price process, we consider both the case of constant volatility (driven by an O-U process) and the case of stochastic volatility (driven by a CIR model). In each case, under certain conditions, we obtain the minimal upper bound for ruin probability as well as the corresponding optimal investment strategy by a pure probabilistic method.     

模糊集理论在随机波动率期权定价中的应用

目的是对基于随机波动率模型的期权定价问题应用模糊集理论.主要思想是把波动率的概率表示转换为可能性表示,从而把关于股票价格的带随机波动率的随机过程简化为带模糊参数的随机过程.然后建立非线性偏微分方程对欧式期权进行定价.     

Pricing currency options under two-factor Markov-modulated stochastic volatility models

This article investigates the valuation of currency options when the dynamic of the spot Foreign Exchange (FX) rate is governed by a two-factor Markov-modulated stochastic volatility model, with the first stochastic volatility component driven by a lognormal diffusion process and the second independent stochastic volatility component driven by a continuous-time finite-state Markov chain model. The states of the Markov chain can be interpreted as the states of an economy. We employ the regime-switching Esscher transform to determine a martingale pricing measure for valuing currency options under the incomplete market setting. We consider the valuation of the European-style and American-style currency options. In the case of American options, we provide a decomposition result for the American option price into the sum of its European counterpart and the early exercise premium. Numerical results are included.     

基于ARMA-GARCH-SN模型的沪深300股指期货日内波动率研究与预测

运用五个交易日的股指期货高频数据(每秒两笔),本文主要研究了沪深300股指期货日内波动率特征并对日内波动率预测。研究发现高频股指期货日内收益率有明显的波动率聚集和条件异方差现象,但无尖峰厚尾现象,收益率序列分布符合有偏正态分布。因此,我们对时间序列建立了最优的ARMA-GARCH-SN模型,并对模型拟合充分性做了验证,拟合结果发现ARMA(1,2)-GARCH(1,1)-SN模型基本能够刻画股指期货高频日内波动特征,条件方差所受的冲击具有很强的持续性、日内波动也具有长记忆性,最后我们还利用自助法对高频股指期货日内波动率两步预测、利用滚动回归预测方法对样本做了样本内预测。预测结果表明,波动率预测结果能够较好地反映股指期货日内波动特征。     

An integral representation of elasticity and sensitivity for stochastic volatility models

This paper presents a generic probabilistic approach to study elasticities and sensitivities of financial quantities under stochastic volatility models. We describe the shock elasticity, the quantile sensitivity and the vega value of cash flows with respect to perturbation of the volatility function of the model. The main contribution is to establish explicit formulae for these elasticities and sensitivities based on a novel application of the exponential measure change technique in Palmowski and Rolski (Bernoulli 8(6):767–785 2002). We carry out explicit calculations for the Heston model and the 3/2 stochastic volatility model, and derive explicit expressions in terms of model parameters.     

On filtering and estimation of a threshold stochastic volatility model

We derive a nonlinear filter and the corresponding filter-based estimates for a threshold autoregressive stochastic volatility (TARSV) model. Using the technique of a reference probability measure, we derive a nonlinear filter for the hidden volatility and related quantities. The filter-based estimates for the unknown parameters are then obtained from the EM algorithm.     

上交所国债市场波动性研究

本文基于2003～2007五年间上证国债指数数据,选择建立了AR(2)-GARCH(1,1)、AR(2) -EGARCH(1,1)、AR(2)-CARCH(1,1)。(γ1,γ2)和GARCH(1,1).(m,n)-M四个模型,从不同的视角分析了我国上交所国债市场的波动性状况。实证结果表明,模型中引入利率调整因子γ1、准备金率调整因子γ2及流动性因子m和n作为外生解释变量,能够较好地刻画我国货币政策对国债市场波动的冲击、市场流动性对国债收益率波动的影响及国债收益率波动与收益率的关系。本文的研究结果对完善我国国债市场管理,促进市场发展有一定的指导意义。