关于股票价格模型的假设检验

股票模型是人们用来描述股票价格波动特征的重要工具,包括Black-scholes'模型,随机波动率模型和更一般化的指数Lévy过程模型等.同时,一些流行的技术指标(例如布林带,RSI,ROC)被股市交易者广泛使用.交易者将每日(小时,周)实际股价作为计算某个技术指标的样本,通过观察相关频率指导投资.技术指标的有效性已在实际应用中得到充分验证.已有研究成果证明股价模型成立时,一些常用的技术指标有良好的统计性质,从而解释了技术分析的合理性.本文将从另一个角度出发,利用技术指标ROC的历史数据,拒绝股票价格波动服从指数Lévy过程模型的假设.     

Ornstein-Uhlenbeck随机波动率模型的参数估计

本文研究了波动率过程为Ornstein-Uhlenbeck过程,且波动率过程与股票价格过程的相关系数ρ可为[0,1]中的任一数的随机波动率模型参数估计.给出了OU过程的统计性质,并利用鞅极限理论给出模型中参数的估计式,证明估计量是具有渐进正态性从而是相合的.     

氏调制的几何布朗运动与布林带

在证券市场, 布林带作为流行的技术分析工具被广泛的运用\bd 到目前为止有许多模型被建立用来预测证券的价格, 因此研究这些模型是否具有布林带性质是一个重要的问题\bd Liu, Huang and Zheng (2006)和Liu and Zheng (2006)分别讨论了Black-Scholes模型和随机波动率模型作为真实的股票市场的布林带, 并且证明了相应的统计量的平稳性和大数定律成立\bd 本文我们将上述结果推广到马氏调制的几何布朗运动模型.     

基于真实波动率和高斯估计的随机波动率模型研究

在随机波动率模型中,由于波动率是不可观测,因此相应的参数估计和统计推断比较困难.将应用真实波动率近似估计积分波动率,进一步基于高斯估计方法给出非线性扩散模型的线性估计,而后再给出随机波动率模型精确的极大似然估计方法.最后,采用上证综合指数和深证成份指数对一系列随机波动率模型进行实证的研究.实证结果表明,均方根模型(Heston模型)较好地描述上证综合指数动态行为,而对于深证成份指数的描述在统计意义上没有显著地解释力.     

在4/2随机波动率下正态调和稳态过程模型对期权的定价和对冲

本文把4/2随机波动率添加到用于模拟资产价格的正态调和稳态过程中,得到了一种全新的Lévy随机波动率模型.更进一步地,本文在此基础上还构造了考虑杠杆效应的Lévy随机波动率模型.本文从数学上证明了模型可行性,并且推导了在这种模型下可被半显式表达出来的期权定价公式和对冲策略公式.作为应用,本文还推导了在上面模型下VIX(波动率指数)波动率衍生品的定价公式,最后数值检验证实了模型的实用性.     

动态随机弹性在期权定价中的应用

给出动态随机弹性的概念及运算性质,讨论了动态随机弹性在期权定价模型中的应用.主要结果有:(1)在波动率为常数时,期权价格对的弹性,得到了动态随机弹性服从运动,并给出了相应的经济解释;(2)由于波动率一般不是常数,也是随机过程,因此本文进一步研究了期权价格对波动率的弹性,就股票价格的波动情况给出了数学描述和金融意义上的解释.     

信息不对称下带有“随机压力”定价规则的连续时间模型

本文是对Back(1992)和Cho(2003)关于内部交易模型的拓展.在金融市场中一共有3类人:内部交易者,不知情交易者和做市商.考虑一类比Cho研究的模型更广的定价规则.主要用动态规划的方法,证明了当内部交易者是风险中性时,定价规则中"随机压力"消失,均衡价格还是仅依赖市场上累计交易量.相应地,本文的结论推广了Back和Cho在经典模型中的结论.     

股指期货对现货市场微观结构影响研究

着眼于股指期货对现货市场微观结构的影响,基于2015-2017年三次股指期货交易机制重大调整前后两个月的1分钟高频数据,利用ACD-EGARCH模型对不同市场波动率背景下的股指期货是否改善现货市场微观质量进行了实证研究.文章的主要结论为:股指期货在不同市场波动率背景下均能降低现货市场波动率,且新进入的投机者比信息交易者贡献更高的波动率;而只在平稳市背景下,股指期货能增强现货市场流动性,在波动市背景下,股指期货吸引的信息交易者超过现货市场增加的非信息交易者,现货市场流动性减弱.建议在平稳市背景下恢复股指期货的常态化交易,但需要防范利好消息和投机者入市对市场波动率的冲击风险.     

基于贝叶斯ARFIMA-WRV模型高频数据长记忆性研究

考虑到高频时间序列波动率的长记忆性问题,构建了赋权已实现波动分数整合自回归移动平均(ARFIMA-WRV)模型对其进行了研究.利用贝叶斯统计方法对模型做了相应的贝叶斯分析,并对我国中小板股市收益波动率的长记忆性特征进行了实证分析.实证结果表明我国中小板股市收益波动率存在长记忆性特征;采用消除日历效应影响的赋权已实现波动作为波动度量和贝叶斯参数估计方法,很大程度上提高了模型的参数精度.     

随机波动率与双指数跳扩散组合模型的美式期权定价

在股价满足Cox-Ingersoll-Ross(CIR)随机波动率与Kou的双指数跳扩散组合模型下,利用随机分析方法讨论了美式看跌期权函数及最佳实施边界的性质.应用一阶线性近似实施边界获得了期权价格的拟解析式和实施边界满足的非线性方程.进一步,应用梯形法离散处理方程式内积分表达式,建立了期权最佳实施边界和价格的数值算法.最后分别给出了常数波动率或CIR随机波动率的数值实例.     

Stochastic volatility model and technical analysis of stock price

In the stock market, some popular technical analysis indicators (e.g. Bollinger Bands, RSI, ROC, ...) are widely used by traders. They use the daily (hourly, weekly, ...) stock prices as samples of certain statistics and use the observed relative frequency to show the validity of those well-known indicators. However, those samples are not independent, so the classical sample survey theory does not apply. In earlier research, we discussed the law of large numbers related to those observations when one assumes Black-Scholes’ stock price model. In this paper, we extend the above results to the more popular stochastic volatility model.     

Non-linear filtering and optimal investment under partial information for stochastic volatility models

This paper studies the question of filtering and maximizing terminal wealth from expected utility in partial information stochastic volatility models. The special feature is that the only information available to the investor is the one generated by the asset prices, and the unobservable processes will be modeled by stochastic differential equations. Using the change of measure techniques, the partial observation context can be transformed into a full information context such that coefficients depend only on past history of observed prices (filter processes). Adapting the stochastic non-linear filtering, we show that under some assumptions on the model coefficients, the estimation of the filters depend on a priori models for the trend and the stochastic volatility. Moreover, these filters satisfy a stochastic partial differential equations named “Kushner–Stratonovich equations”. Using the martingale duality approach in this partially observed incomplete model, we can characterize the value function and the optimal portfolio. The main result here is that, for power and logarithmic utility, the dual value function associated to the martingale approach can be expressed, via the dynamic programming approach, in terms of the solution to a semilinear partial differential equation which depends on the filters estimate and the volatility. We illustrate our results with some examples of stochastic volatility models popular in the financial literature.     

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.     

Volatility Targeting Using Delayed Diffusions

ABSTRACT

A target volatility strategy (TVS) is a risky asset-riskless bond dynamic portfolio allocation which makes use of the risky asset historical volatility as an allocation rule with the aim of maintaining the instantaneous volatility of the investment constant at a target level. In a market with stochastic volatility, we consider a diffusion model for the value of a target volatility fund (TVF) which employs a system of stochastic delayed differential equations (SDDEs) involving the asset realized variance. First we prove that under some technical assumptions, contingent claim valuation on a TVF is approximately of Black-Scholes type, which is consistent with and supports the standing market practice. In second place, we develop a computational framework using recent results on Markovian approximations of SDDEs systems, which we then implement in the Heston variance model using an ad hoc Euler scheme. Our framework allows for efficient numerical valuation of derivatives on TVFs, whose typical purpose is the assessment of the guarantee costs of such funds for insurers.     

随机波动模型的马尔可夫链—蒙特卡洛模拟方法——在沪市收益率序列上的应用

针对具有Markov区制转移的、波动均值状态相依的随机波动模型,基于贝叶斯分析,我们推导并给出了对区制转移随机波动模型的MCMC估计方法,其中对参数估计采用Gibbs抽样方法,对潜在对数波动和区制的状态变量估计采用"向前滤波、向后抽样"的多步移动方法;利用该模型,对我国上证综指周收益率进行了实证分析,发现对沪市波动性有较好的描述,捕捉了波动的时变性、聚类性和非线性特征,同时刻画了沪市的高低波动状态转换过程。     

Market calibration under a long memory stochastic volatility model

In this article, we study a long memory stochastic volatility model (LSV), under which stock prices follow a jump-diffusion stochastic process and its stochastic volatility is driven by a continuous-time fractional process that attains a long memory. LSV model should take into account most of the observed market aspects and unlike many other approaches, the volatility clustering phenomenon is captured explicitly by the long memory parameter. Moreover, this property has been reported in realized volatility time-series across different asset classes and time periods. In the first part of the article, we derive an alternative formula for pricing European securities. The formula enables us to effectively price European options and to calibrate the model to a given option market. In the second part of the article, we provide an empirical review of the model calibration. For this purpose, a set of traded FTSE 100 index call options is used and the long memory volatility model is compared to a popular pricing approach – the Heston model. To test stability of calibrated parameters and to verify calibration results from previous data set, we utilize multiple data sets from NYSE option market on Apple Inc. stock.     

Smooth Solutions to Optimal Investment Models with Stochastic Volatilities and Portfolio Constraints

Abstract. This paper deals with an extension of Merton's optimal investment problem to a multidimensional model with stochastic volatility and portfolio constraints. The classical dynamic programming approach leads to a characterization of the value function as a viscosity solution of the highly nonlinear associated Bellman equation. A logarithmic transformation expresses the value function in terms of the solution to a semilinear parabolic equation with quadratic growth on the derivative term. Using a stochastic control representation and some approximations, we prove the existence of a smooth solution to this semilinear equation. An optimal portfolio is shown to exist, and is expressed in terms of the classical solution to this semilinear equation. This reduction is useful for studying numerical schemes for both the value function and the optimal portfolio. We illustrate our results with several examples of stochastic volatility models popular in the financial literature.     

Smooth Solutions to Optimal Investment Models with Stochastic Volatilities and Portfolio Constraints

   Abstract. This paper deals with an extension of Merton's optimal investment problem to a multidimensional model with stochastic volatility and portfolio constraints. The classical dynamic programming approach leads to a characterization of the value function as a viscosity solution of the highly nonlinear associated Bellman equation. A logarithmic transformation expresses the value function in terms of the solution to a semilinear parabolic equation with quadratic growth on the derivative term. Using a stochastic control representation and some approximations, we prove the existence of a smooth solution to this semilinear equation. An optimal portfolio is shown to exist, and is expressed in terms of the classical solution to this semilinear equation. This reduction is useful for studying numerical schemes for both the value function and the optimal portfolio. We illustrate our results with several examples of stochastic volatility models popular in the financial literature.     

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.