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.     

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.     

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.     

On the simulation of portfolios of interest rate and credit risk sensitive securities

We discuss extensions of reduced-form and structural models for pricing credit risky securities to portfolio simulation and valuation. Stochasticity in interest rates and credit spreads is captured via reduced-form models and is incorporated with a default and migration model based on the structural credit risk modelling approach. Calculated prices are consistent with observed prices and the term structure of default-free and defaultable interest rates. Three applications are discussed: (i) study of the inter-temporal price sensitivity of credit bonds and the sensitivity of future portfolio valuation with respect to changes in interest rates, default probabilities, recovery rates and rating migration, (ii) study of the structure of credit risk by investigating the impact of disparate risk factors on portfolio risk, and (iii) tracking of corporate bond indices via simulation and optimisation models. In particular, we study the effect of uncertainty in credit spreads and interest rates on the overall risk of a credit portfolio, a topic that has been recently discussed by Kiesel et al. [The structure of credit risk: spread volatility and ratings transitions. Technical report, Bank of England, ISSN 1268-5562, 2001], but has been otherwise mostly neglected. We find that spread risk and interest rate risk are important factors that do not diversify away in a large portfolio context, especially when high-quality instruments are considered.     

基于SV模型的国际现货贵金属市场收益波动的杠杆效应分析

针对国际现货贵金属市场收益波动中是否存在杠杆效应的问题,选取2008年至今的黄金、白银市场数据进行分析,运用具有杠杆效应的SV模型对其收益波动建模,并采取MCMC法—Gibbs法进行参数估计.结果表明:与股票市场的研究结论不同,国际现货黄金、白银市场在整个观察期内几乎不存在杠杆效应;但其震荡期内存在较弱的杠杆效应.     

Optimal selection of a portfolio of options under Value-at-Risk constraints: a scenario approach

This paper introduces a multiperiod model for the optimal selection of a financial portfolio of options linked to a single index. The objective of the model is to maximize the expected return of the portfolio under constraints limiting its Value-at-Risk. We rely on scenarios to represent future security prices. The model contains several interesting features, like the consideration of transaction costs, bid-ask spreads, arbitrage-free option pricing, and the possibility to rebalance the portfolio with options introduced at the start of each period. The resulting mixed integer programming model is applied to realistic test instances involving options on the S&P500 index. In spite of the large size and of the numerical difficulty of this model, near-optimal solutions can be computed by a standard branch-and-cut solver or by a specialized heuristic. The structure and the financial features of the selected portfolios are also investigated.     

Implied Volatility of Leveraged ETF Options

Abstract

This paper studies the problem of understanding implied volatilities from options written on leveraged exchanged-traded funds (LETFs), with an emphasis on the relations between LETF options with different leverage ratios. We first examine from empirical data the implied volatility skews for LETF options based on the S&P 500. In order to enhance their comparison with non-leveraged ETFs, we introduce the concept of moneyness scaling and provide a new formula that links option implied volatilities between leveraged and unleveraged ETFs. Under a multiscale stochastic volatility framework, we apply asymptotic techniques to derive an approximation for both the LETF option price and implied volatility. The approximation formula reflects the role of the leverage ratio, and thus allows us to link implied volatilities of options on an ETF and its leveraged counterparts. We apply our result to quantify matches and mismatches in the level and slope of the implied volatility skews for various LETF options using data from the underlying ETF option prices. This reveals some apparent biases in the leverage implied by the market prices of different products, long and short with leverage ratios two times and three times.     

Financial Risk Modelling and Econometric Inference

In finance, the explicit modelling of uncertainty takes on a particularly important role. The values of financial derivatives increase in the return volatility of the underlying security. This notion requires a concept of volatility and hence uncertainty. In addition, the choice between modelling in discrete and continuous time is not arbitrary, since it corresponds to a distinction between incomplete and complete markets, respectively, and this distinction matters for asset pricing, financial risk modelling, and inference. Risk and volatility are closely connected, and implied volatility, volatility forecasting, volatility in term structure models, stochastic volatility, and portfolio analysis are considered and related to a more general interplay between cross-sectional and dynamic aspects in finance. Stocks, bonds, and options are considered and placed in the context of efficiency and separation in inference.     

基于TGARCH-t的混合Copula投资组合风险测度研究

在分析了现有Copula函数在测度投资组合风险不足的情况下,首先充分考虑资产波动的时变性、杠杆效应等特征,选择了TGARCH-t模型进行边缘分布建模.接着引入混合Copula模型来描述投资组合的复杂相关结构,同时利用构造的主对角线距离统计量等方法验证了混合Copula模型的优势.最后通过VaR的蒙特卡洛模拟结果看到,这种方法能更为精确的测度投资组合风险值.     

Estimating and forecasting portfolio’s Value-at-Risk with wavelet-based extreme value theory: Evidence from crude oil prices and US exchange rates

This article proposes a wavelet-based extreme value theory (W-EVT) approach to estimate and forecast portfolio’s Value-at-Risk (VaR) given the stylized facts and complex structure of financial data. Our empirical application to portfolios of crude oil prices and US dollar exchange rates shows that the W-EVT models provide an effective and powerful tool for gauging extreme moments and improving the accuracy of portfolio’s VaR estimates and forecasts after noise is removed from the original data.     

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。     

American options under uncertain volatility

The uncertain volatility approach to financial derivatives is extended to American options (which allow early exercise before expiry). The requirement to model at the portfolio level made necessary by the non-linearity of the approach is found to lead to a recursive structure to the exercise possibilities across options. Other novel features include: the optimality sometimes of partial exercise; an interesting resolution to the issues surrounding short options whose exercise is controlled by a buyer counterparty; and the occurrence of a simple game structure for portfolios containing both long and short options. It is demonstrated that the exercise strategies resulting can significantly alter measured uncertain volatility risk. Contrary to the set of attributes for sensible risk measures put forward by Artzner, Delbaen, Eber and Heath, this risk need not be homogenous in portfolio size- forming a convincing argument for weakening this particular requirement.     

Forward Variance Dynamics: Bergomi’s Model Revisited

Abstract

In this article, we propose an arbitrage-free modelling framework for the joint dynamics of forward variance along with the underlying index, which can be seen as a combination of the two approaches proposed by Bergomi. The difference between our modelling framework and the Bergomi (2008. Smile dynamics III. Risk, October, 90–96) models is mainly the ability to compute the prices of VIX futures and options by using semi-analytic formulas. Also, we can express the sensitivities of the prices of VIX futures and options with respect to the model parameters, which enables us to propose an efficient and easy calibration to the VIX futures and options. The calibrated model allows to Delta-hedge VIX options by trading in VIX futures, the corresponding hedge ratios can be computed analytically.     

An E-ARCH model for the term structure of implied volatility of FX options

We construct a statistical model for the term-structure of implied volatilities of currency options based on daily historical data for 13 currency pairs over a 19-month period. We examine the joint evolution of 1 month, 2 month, 3 month, 6 month and 1 year at-the-money (50 δ) options in all the currency pairs. We show that there exist three uncorrelated state variables (principal components) which account for the parallel movement, slope oscillation, and curvature of the term structure and which explain, on average, the movements of the termstructure of volatility to more than 95% in all cases. We test and construct an exponential ARCH, or E-ARCH, model for each state variable. One of the applications of this model is to produce confidence bands for the term structure of volatility.     

基于联合估计法的债券信用利差期限结构拟合及应用

基于2016~2018年月度数据,通过独立估计的单曲线样条模型和SV 模型、联合估计的多曲线样条模型和SV模型拟合公司债信用利差期限结构,进而对模型拟合效果进行比较,讨论模型在宏观经济预测中的应用,得到以下结论:(1)拟合模型的函数形式是导致理论信用利差期限结构曲线翻折的原因。样条模型和SV模型拟合的信用利差曲线形状完全不同,且模型函数变动引起的误差变动大于曲线变动引起的误差变动。(2)联合估计模型可以修正独立估计模型的人为扭曲形式。多曲线模型的结果更接近实际信用利差,误差波动性明显减小,曲线更为平滑,且联合估计的多曲线样条模型优于独立估计的单曲线样条模型、独立估计的SV 模型和联合估计的SV模型。(3)公司债信用利差期限结构在一定程度上蕴含了市场对未来宏观经济的预期信息,且在短期内预测结果随先行期限延长而改善。因此,宏观经济政策制定者需关注信用利差和期限结构模型拟合研究,重视对信用利差期限结构的深度信息挖掘,从而提高中国宏观政策制定者调控手段的前瞻性和有效性。     

Liquidity and credit risk

The paper uses fuzzy measure theory to represent liquidity risk, i.e. the case in which the probability measure used to price contingent claims is not known precisely. This theory enables one to account for different values of long and short positions. Liquidity risk is introduced by representing the upper and lower bound of the price of the contingent claim computed as the upper and lower Choquet integral with respect to a subadditive function. The use of a specific class of fuzzy measures, known as g λ measures enables one to easily extend the available asset pricing models to the case of illiquid markets. As the technique is particularly useful in corporate claims evaluation, a fuzzified version of Merton's model of credit risk is presented. Sensitivity analysis shows that both the level and the range (the difference between upper and lower bounds) of credit spreads are positively related to the ‘quasi debt to firm value ratio’ and to the volatility of the firm value. This finding may be read as correlation between credit risk and liquidity risk, a result which is particularly useful in concrete risk-management applications. The model is calibrated on investment grade credit spreads, and it is shown that this approach is able to reconcile the observed credit spreads with risk premia consistent with observed default rate. Default probability ranges, rather than point estimates, seem to play a major role in the determination of credit spreads.     

Heston随机波动率市场中带VaR约束的最优投资策略

本文研究了Heston随机波动率市场下, 基于VaR约束下的动态最优投资组合问题。
假设Heston随机波动率市场由一个无风险资产和一个风险资产构成,投资者的目标为最大化其终端的期望效用。与此同时, 投资者将动态地评估其待选的投资组合的VaR风险,并将其控制在一个可接受的范围之内。本文在合理的假设下,使用动态规划的方法,来求解该问题的最优投资策略。在特定的参数范围内,利用数值方法计算出近似的最优投资策略和相应值函数, 并对结果进行了分析。     

The long‐term extreme price risk measure of portfolio in inventory financing: An application to dynamic impawn rate interval

Different from the short‐term risk measure for traditional financial assets (stocks, bonds, etc.), the key to illiquid inventory portfolio traded in the over‐the‐counter markets is to estimate the long‐term extreme price risk with time varying volatility. In this article, a new long‐term extreme price risk (value at risk and conditional value at risk) measure method for inventory portfolio and an application to dynamic impawn rate interval are proposed. To realize this, we first establish AutoRegressive Moving Average‐Exponential Generalized Autoregressive Conditional Heteroskedasticity‐Extreme Value Theory model and multivariatet‐Copula to depict the autocorrelation, fat tails, and volatility clustering of returns of inventories and the nonlinear dependence structure of inventories. Furthermore, we obtain the long‐term extreme price risk with time varying volatility via Monte Carlo simulation instead of square‐root‐of time rule. The results show that, first, benefits from risk diversification is significant; second, long‐term extreme price risk measure of inventory portfolio via Monte Carlo method outperforms the square‐root‐of time rule; the last is that the dynamic rate interval based on the long‐term price risk is superior to the crude rules of thumb in terms of reducing efficiency loss and improving risk coverage. In summary, this article provides a new quantitative framework for managing the risk of portfolio in inventory financing practice for banks constrained by risk limitation. © 2014 Wiley Periodicals, Inc. Complexity 20: 17–34, 2015     

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.     

金融危机下中美两国利率互换市场的特征及互动性分析

以2008～2009年中美两国利率互换市场的日交易数据为样本,分析比较了影响两国利率互换利差的主要因素,进而实证研究了危机期间中美两国利率互换市场的动态互动效应。结果表明:两国利率的水平和利率期限结构斜率是影响互换利差的主要因素,另外,中国的流动性溢价和美国的违约溢价对互换利差的影响也较为显著;研究发现:中美两国互换利差均受对方市场因素的影响,特别地,在金融危机期间,中美两国利率互换市场间存在着明显的互动效应,一方面,美国利率互换市场信息能够对中国利率互换市场产生较强的冲击,虽然冲击的程度受制于美国的经济状况;另一方面,中国市场对美国市场也形成了一定的反向冲击,且程度受制于中国的货币政策。