美式期权隐含波动率校准问题的研究

研究的是美式期权的隐含波动率校准问题.首先提出一个正则化的最小二乘方法,在对其惩罚问题研究后找到最小二乘问题的最优条件,并给出美式期权波动率校准问题的算法.最后,通过数值算例说明了方法的有效性.     

认股权证隐含波动率迭代算法的改进

为了有效估计认股权证的隐含波动率,以及有利于观测权证的市场风险,对常用的迭代算法进行改进可加快计算速度.利用认股权证的Delta、Vega和Theta值以及多元泰勒公式导出认股权证隐含波动率变化与权证价格变化关系的不等式.根据导出的这个不等式给出了具体算法.实证结果说明这一算法改进了原有的算法,从而这个算法解决了原来的随意给定一个价格后不断修正的算法.     

快速均值回归随机波动率模型参数估计及应用

基于快速均值回归随机波动率模型, 研究双限期权的定价问题, 同时推导了考虑均值回归随机波动率的双限期权的定价公式。 根据金融市场中SPDR S&P 500 ETF期权的隐含波动率数据和标的资产的历史收益数据, 对快速均值回归随机波动率模型中的两个重要参数进行估计。 利用估计得到的参数以及定价公式, 对双限期权价格做了数值模拟。 数值模拟结果发现, 考虑了随机波动率之后双限期权的价格在标的资产价格偏高的时候会小于基于常数波动率模型的期权价格。     

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 Approximations for Pricing Asian Options and Volatility Swaps Under Stochastic Volatility

Abstract

We show that if the discounted Stock price process is a continuous martingale, then there is a simple relationship linking the variance of the terminal Stock price and the variance of its arithmetic average. We use this to establish a model-independent upper bound for the price of a continuously sampled fixed-strike arithmetic Asian call option, in the presence of non-zero time-dependent interest rates (Theorem 1.2). We also propose a model-independent lognormal moment-matching procedure for approximating the price of an Asian call, and we show how to apply these approximations under the Black–Scholes and Heston models (subsection 1.3). We then apply a similar analysis to a time-dependent Heston stochastic volatility model, and we show how to construct a time-dependent mean reversion and volatility-of-variance function, so as to be consistent with the observed variance swap curve and a pre-specified term structure for the variance of the integrated variance (Theorem 2.1). We characterize the small-time asymptotics of the first and second moments of the integrated variance (Proposition 2.2) and derive an approximation for the price of a volatility swap under the time-dependent Heston model ( Equation (52)), using the Brockhaus–Long approximation (Brockhaus, and Long, 2000 Brockhaus, O. and Long, D. 2000. Volatility Swaps made simple. Risk, 13(1) January: 92–96.  [Google Scholar]). We also outline a bootstrapping procedure for calibrating a piecewise-linear mean reversion level and volatility-of-volatility function (Subsection 2.3.2).     

Asymptotic Solutions for Australian Options with Low Volatility

Abstract

In this paper we derive asymptotic expansions for Australian options in the case of low volatility using the method of matched asymptotics. The expansion is performed on a volatility scaled parameter. We obtain a solution that is of up to the third order. In case that there is no drift in the underlying, the solution provided is in closed form, for a non-zero drift, all except one of the components of the solutions are in closed form. Additionally, we show that in some non-zero drift cases, the solution can be further simplified and in fact written in closed form as well. Numerical experiments show that the asymptotic solutions derived here are quite accurate for low volatility.     

欧式幂期权定价模型中参数及隐含波动率的统计特性

研究了欧式幂期权定价公式中价格的渐近无偏估计和隐含波动率估计的统计特性。利用Chaudhury M.M(1989)提出的研究欧式期极定价公式中渐近无偏估计的方法以及隐含波动率求解方法,研究了两种欧式幂型看涨期权定价公式(欧式看涨期权的价值定义分别为m ax(STα-X,0)和m ax(STα-Xa,0)中的隐含波动率的估计的统计特征、幂函数的幂指数选取以及两种幂函数期权定价公式的优劣。Monte-Carlo统计计算的模拟结果说明。幂期权定价公式中幂指数α取值应为α>0,而且欧式看涨期权的价值定义为m ax(STα-Xα,0)更为合理。     

基于ARMA-GARCH模型的恒指隐含波动率指数预测及其在期权交易中的应用

基于ARMA-GARCH模型,并结合均值回归效应,溢出效应和周内效应,本文研究了恒指隐含波动率指数(VHSI)能否被预测及预测是否有助于期权投资实践的问题.研究结果验证了香港股市具有均值回归的特性,标准普尔500指数对恒指隐含波动率指数有明显的溢出效应.此外,恒指隐含波动率指数呈现出周一上涨,周五下跌的特征,具有明显的周内效应.最后,本文运用ARMA-GARCH模型对恒指隐含波动率指数进行预测,并结合实际的市场数据做了期权交易模拟.结果显示,ARMA-GARCH模型比ARMA模型更适合对恒指隐含波动率进行建模;考虑了均值回归效应,溢出效应和周内效应之后,ARMAGARCH模型对恒指隐含波动率指数的预测能力显著提高,并且预测结果有助于期权交易获得较好的收益.     

随机波动率跳-扩散模型下外汇期权本外币对称公式

外汇期权本外币对称公式表示本币看涨/看跌期权与外币看跌/看涨期权用同类定价函数表示的等价关系.通过测度变换法指出本币测度下的Bates模型和Heston模型在外币测度下保持模型类型不变,并且由此证明这两个模型下的本外币对称公式,其中的定价函数由Attari公式给出.数值分析给出了本外币对称公式的应用示范,并且详细分析了Attari公式的计算速度优势.     

Pricing Options on Defaultable Stocks*

? ?. This work was inspired by the SAMSI workshops on Financial Mathematics, Statistics and Econometrics (Fall 2005, Spring 2006 North Carolina). The author wishes to thank the organizers for the travel grant to participate in this stimulating event. I also would like to thank Bo Yang for his research assistance and the two anonymous referees and an anonymous associate editor for their valuable suggestions. Stock option price approximations are developed for a model which takes both the risk of default and the stochastic volatility into account. The intensity of defaults is assumed to be influenced by the volatility. It is shown that it might be possible to infer the risk neutral default intensity from the stock option prices. The proposed option price approximation has a rich implied volatility surface structure and fits the data implied volatility well. A calibration exercise shows that an effective hazard rate from bonds issued by a company can be used to explain the impliedvolatility skew of the option prices issued by the same company. It is also observed that the implied yield spread obtained from calibrating all the model parameters to the option prices matches the observed yield spread.     

干散货运价波动的杠杆效应特征研究——基于非对称随机波动模型

由信息冲击引起的干散货运价的剧烈波动给航运实体市场带来巨大风险,同等强度的利空消息通常要比利好消息引起更大的市场波动,本文对干散货航运市场运价波动存在的杠杆效应特征进行研究,为航运企业和租船人等把握市场态势、规避风险提供重要依据。考虑运价收益分布的厚尾特征,改变传统的非对称随机波动模型中随机误差项的正态分布假定,建立基于student-t分布的改进的非对称随机波动模型,在贝叶斯分析的基础上通过MCMC方法进行参数估计。通过实证研究发现,在考虑了极端风险情况后,改进的厚尾分布的非对称随机波动模型对干散货运价波动的杠杆效应特征刻画更加准确和优越。     

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。     

仿射期限结构模型的非包含随机波动局限

一般的,含随机波动率成分的仿射期限结构模型认为,即时收益率瞬时方差是收益率水平的线性组合.本文利用我国银行间固定利率国债数据,构建了不依赖于特定仿射模型的检验方法,并对该推论进行了检验.实证结果表明,无论是事前估计还是事后估计的收益率方差,都不能表示成为横截面收益率的仿射函数.即尽管先前许多研究说明仿射模型能非常好地描...     

基于时变波动率的50ETF参数欧式期权定价

以上海证券交易所的首个股票期权品种50ETF为例,首先以时变波动率对经典BlackScholes期权定价公式常数波动率假设修正,然后基于正态分布、广义学生t分布和融入高阶矩的Edgeworth expansion渐近分布构建三种参数期权定价模型,最后采用参数显著性检验(Significance testing)、定价误差(Mispricing)、预测偏差(Forecastability)、对冲误差(Hedging errors)和波动率偏离修正(Volatility skew correction)5种严谨系统的评价标准,实证对比了在3种参数期权定价模型下的定价精度。研究结果表明:在时变波动率下,基于广义学生t分布和Edgeworth expansion渐近分布相比于正态分布显著提高了参数期权定价模型的定价精度。论文的研究成果为投资商和监管者提供了相对更为精确的期权定价模型。同时在相对更为准确的定价方法下,进一步利于50ETF期权在我国金融市场发挥价格发现和风险管理的作用。     

基于随机波动率模型的上证50ETF期权定价研究

传统上,期权定价主要基于Black-Scholes (B-S)模型。但B-S模型不能描述时变波动率以及解释"波动率微笑"现象,导致期权定价存在较大的误差。随机波动率模型克服了B-S模型的这些缺陷,能够合理地刻画波动率动态性和波动率微笑。基于此,本文考虑随机波动率模型下的期权定价问题,并针对我国上证50ETF期权进行实证分析。为了解决定价模型的参数估计问题,采用上证50ETF及其期权价格数据,建立两步法对定价模型的参数进行估计。该估计方法保证了定价模型在客观与风险中性测度下的一致性。采用2016年1月到2017年10月的上证50ETF期权价格数据为研究样本,对随机波动率模型进行了实证检验。结果表明,无论是在样本内还是样本外,随机波动率模型相比传统的常数波动率B-S模型都能够获得明显更为精确和稳定的定价结果,B-S模型的定价误差总体偏大且呈现较高波动,凸显了随机波动率对于期权定价的重要性。另外,随机波动率模型对于短期实值期权的定价相比对于其它期权的定价要更精确。     

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.     

A study of leveraged ETF based on geometric return

Leveraged (inverse) exchange-traded funds (LETFs) seek to deliver multiples (opposite) of the performance of the index or benchmark they track. LETFs typically are designed to achieve their stated performance objectives on a daily basis. Many real-life and hypothetical examples have been given to show that the performance of these ETFs over a period longer than one day can differ from their stated daily performance objectives. Formulae have been found using both continuous method and discrete method. A discrete method was used to find a formula linking the return of a leveraged fund with the corresponding multiple of the return of the unleveraged fund and its realized variance but the method needs to use some assumptions and statistical properties to create the volatility term. A CME report finds a very simple way to include volatility in their formula but fails to link to the return of the corresponding unleveraged product. In this paper, we find a natural way to link a leveraged fund with its corresponding unleveraged product and its realized variance in a discrete manner. Our derivation process is similar to that in the CME report, so we do not need to use assumptions and statistical properties to create the volatility term. Unlike the CME method, we use geometric return as opposed to arithmetic return. So, we are able to connect with the return of the corresponding unleveraged product.     

基于结构突变的波动率分解模型

在Heston-Nandi模型的基础上提出了一种波动率分解模型,分解模型同时考虑了金融波动的长记忆性和杠杆效应.从资产收益率的无条件方差发生结构突变出发,认为收益率的无条件方差随时间变化,将波动率分解为长期影响和短期冲击两部分,其中长期影响用来刻画波动率的持续性,短期冲击刻画金融波动的短期扰动.上证综指数据实证表明上海证券综合指数收益率序列的波动性同时具有长记忆性和杠杆效应,利用模型能很好的刻画这两种性质.     

随机波动率与投资中的q理论

本文将随机波动率引入托宾q模型,讨论生产率冲击的波动率大小对公司价值与投资决策的影响.研究发现,公司托宾q值会受到生产率冲击波动率的显著影响,波动率的增大会降低托宾q值,且该影响会随着波动率的增大而加剧.此外,波动率对托宾q值的影响也会传导到公司的投资决策上来,托宾q降低会使得投资额减小,该影响同样会随波动率的增大而加剧.本文还考虑了波动率对公司在用资产与成长机会价值的影响,更为合理地刻画了公司生产过程中生产率冲击的波动率为随机的假设现实情况,所得结论对公司估值及投资决策均有一定的参考价值.     

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.