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.     

我国燃料油期货价格与现货价格动态关系:基于TVECM的实证研究

利用门限向量误差修正模型(TVECM)研究了我国燃料油期货价格与现货价格之间的动态关系,实证分析发现,两者之间存在显著的非线性关系,门限值0.100将系统分为两个状态,在极端状态中期货价格和现货价格的调整速度均比标准状态快.然而,在两个状态中期货市场均具有较高的价格发现功能.     

中美燃料油期货价格引导研究

本文利用ADF单整检验、EG协整检验、误差修正模型、引导关系检验等方法研究中美燃料油期货价格之间的互动关系,结果表明:我国燃料油期货在价格和价格收益方面均与国际燃料油市场存在明显的即时引导关系;国际燃料油期货市场上的价格变动对国内燃料油期货价格以及价格收益的变动具有单向的滞后引导作用;我国燃料油期货市场的价格和收益方面的信息对于国际市场几乎不产生影响,我国尚不具备制定价格的话语权。     

关联规则及其在石油期货价格预测中的应用研究

针对Apriori算法及其变种的不足,提出了一种基于支持度计数矩阵和事务数据库布尔矩阵的新算法,利用此算法挖掘石油期货价格历史数据中的频繁项集,根据给定的最小支持度和最小置信度从挖掘结果产生关联规则,对关联规则在石油期货价格预测中的应用进行了探索,并提出了进一步的研究方向.     

Energy futures prices: term structure models with Kalman filter estimation

We present a class of multi-factor stochastic models for energy futures prices, similar to the interest rate futures models recently formulated by Heath. We do not postulate directly the risk-neutral processes followed by futures prices, but define energy futures prices in terms of a spot price, not directly observable, driven by several stochastic factors. Our formulation leads to an expression for futures prices which is well suited to the application of Kalman filtering techniques together with maximum likelihood estimation methods. Based on these techniques, we perform an empirical study of a one- and a two-factor model for futures prices for natural gas.     

Calibration and Filtering for Multi Factor Commodity Models with Seasonality: Incorporating Panel Data from Futures Contracts

We construct a general multi-factor model for estimation and calibration of commodity spot prices and futures valuation. This extends the multi-factor long-short model in Schwartz and Smith (Manag Sci 893–911, 2000) and Yan (Review of Derivatives Research 5(3):251–271, 2002) in two important aspects: firstly we allow for both the long and short term dynamic factors to be mean reverting incorporating stochastic volatility factors and secondly we develop an additive structural seasonality model. In developing this non-linear continuous time stochastic model we maintain desirable model properties such as being arbitrage free and exponentially affine, thereby allowing us to derive closed form futures prices. In addition the models provide an improved capability to capture dynamics of the futures curve calibration in different commodities market conditions such as backwardation and contango. A Milstein scheme is used to provide an accurate discretized representation of the s.d.e. model. This results in a challenging non-linear non-Gaussian state-space model. To carry out inference, we develop an adaptive particle Markov chain Monte Carlo method. This methodology allows us to jointly calibrate and filter the latent processes for the long-short and volatility dynamics. This methodology is general and can be applied to the estimation and calibration of many of the other multi-factor stochastic commodity models proposed in the literature. We demonstrate the performance of our model and algorithm on both synthetic data and real data for futures contracts on crude oil.     

Structural Electricity Models and Asymptotically Normal Estimators to Quantify Parameter Risk

ABSTRACT

We estimate a structural electricity (multi-commodity) model based on historical spot and futures data (fuels and power prices, respectively) and quantify the inherent parameter risk using an average value at risk approach (‘expected shortfall’). The mathematical proofs use the theory of asymptotic statistics to derive a parameter risk measure. We use far in-the-money options to derive a confidence level and use it as a prudent present value adjustment when pricing a virtual power plant. Finally, we conduct a present value benchmarking to compare the approach of temperature-driven demand (based on load data) to an ‘implied demand approach’ (demand implied from observable power futures prices). We observe that the implied demand approach can easily capture observed electricity price volatility whereas the estimation against observable load data will lead to a gap, because – amongst others – the interplay of demand and supply is not captured in the data (i.e., unexpected mismatches).     

Ornstein–Uhlenbeck processes time changed with additive subordinators and their applications in commodity derivative models

We characterize Ornstein–Uhlenbeck processes time changed with additive subordinators as time-inhomogeneous Markov semimartingales, based on which a new class of commodity derivative models is developed. Our models are tractable for pricing European, Bermudan and American futures options. Calibration examples show that they can be better alternatives than those developed in Li and Linetsky (2012)  [6]. Our method can be applied to many other processes popular in various areas besides finance to develop time-inhomogeneous Markov processes with desirable features and tractability.     

碳市场EUA与CER 期货价格变动关系的实证研究

基于VAR模型,对碳市场中的EUA期货价格和CER期货价格的变动关系进行了实证研究．选取欧洲气候交易所（ECX）的EUA期货价格和CER期货价格作为研究对象,运用Johansen协整检验、Granger因果关系检验、向量误差修正模型、广义脉冲响应函数和方差分解方法形成递进式的计量分析框架．研究结果表明：第一,EUA期货价格与CER期货价格之间存在着相互影响关系;第二,CER期货价格对市场信息的反映比EUA期货价格更为敏感,反映速度更快;第三,两种价格之间,CER期货价格变动的影响起主导作用,更好地发挥了期货的定价功能,两市场间存在杠杆效应．     

基于二次优化BP神经网络的期货价格预测

针对BP算法存在的不足,结合神经网络、遗传算法和主成分分析的优点,提出基于二次优化BP神经网络的期货价格预测算法.初次优化采用主成分分析法对网络结构进行优化,第二次优化采用自适应遗传算法对网络参数进行优化,将经过二次优化后建立的BP神经网络模型用于期货价格预测.经仿真检验,用新方法建立的模型对期货价格进行预测,在预测的精度和速度方面都优于单纯BP神经网络模型.     

A mean reverting process for pricing treasury bills and futures contracts

An analytically tractable, discrete-time single-factor model is developed for valuing treasury bills and futures contracts. It uses a multiplicative binomial foward process that creates neither negative nor implausibly large positive interest factors, and which can incorporate different possible degrees of mean reversion. The paper derives explicit formulae for bill prices, futures prices, their conditional variances and risk premia in a setting that relates the evolution of the term structure more closely to both model and data than do other similar works. In contrast to other term-structure constrained models, this paper emphasizes that in a one-factor model the martingale probabilities cannot be treated independently of the perturbation functions. The paper's empirical methods also differ from the customary approaches. Instead of comparing differences between model-predicted and observed prices, the paper applies ARCH methodology to test model-predicted ratios of conditional variances to risk premia. Our tests find influences exogenous to the model, but these factors do not seem capable of being explained with two-factor models using only interest rates.     

On martingale diffusions describing the ‘smile‐effect’ for implied volatilities

This paper discusses diffusion models describing the ‘smile‐effect’ of implied volatilities for option prices partly following the new approach of Bruno Dupire. If one restricts to the time homogeneous case, a careful study of this approach shows that the call option prices considered as a function of the price x of the underlying security, remaining time to maturity T–t and strike price K have necessarily to satisfy a certain functional equation, in order to fit into a coherent model. It is shown that for certain examples of empirically observed option prices which are reported in the literature, this functional equation does not hold. © 2000 John Wiley & Sons, Ltd.     

A forward–backward SDE approach to affine models

We consider factor models for interest rates and asset prices where the risk- neutral dynamics of the factors process is modelled by an affine diffusion. We characterize the factors process and bond price in terms of forward–backward stochastic differential equations (FBSDEs), prove an existence and uniqueness theorem which gives the solution explicitly, and characterize the bond price as an exponential affine function of the factors in a new way. Our approach unifies the results, based on stochastic flows, of Elliott and van der Hoek (Finance Stoch 5:511–525, 2001) with the approach, based on the Feynman-Kac formula, of Duffie and Kan (Math Finance 6(4):379–406, 1996), and addresses a mistake in the approach of Elliott and van der Hoek (Finance Stoch 5:511–525, 2001). We extend our results on the bond price to consider the futures and forward price of a risky asset or commodity.      

Small-noise limit of the quasi-Gaussian log-normal HJM model

Quasi-Gaussian HJM models are a popular approach for modeling the dynamics of the yield curve. This is due to their low dimensional Markovian representation, which greatly simplifies their numerical implementation. We present a qualitative study of the solutions of the quasi-Gaussian log-normal HJM model. Using a small-noise deterministic limit we show that the short rate may explode to infinity in finite time. This implies the explosion of the Eurodollar futures prices in this model. We derive explicit explosion criteria under mild assumptions on the shape of the yield curve.     

Valuation of Two-Factor Interest Rate Contingent Claims Using Green's Theorem

Abstract

Over the years a number of two-factor interest rate models have been proposed that have formed the basis for the valuation of interest rate contingent claims. This valuation equation often takes the form of a partial differential equation that is solved using the finite difference approach. In the case of two-factor models this has resulted in solving two second-order partial derivatives leading to boundary errors, as well as numerous first-order derivatives. In this article we demonstrate that using Green's theorem, second-order derivatives can be reduced to first-order derivatives that can be easily discretized; consequently, two-factor partial differential equations are easier to discretize than one-factor partial differential equations. We illustrate our approach by applying it to value contingent claims based on the two-factor CIR model. We provide numerical examples that illustrate that our approach shows excellent agreement with analytical prices and the popular Crank–Nicolson method.     

基于高频数据的中国国债期货价格发现能力研究

期货的价格发现能力是近几年国际学术界关注的热点问题,但目前理论界相关研究主要集中于商品期货和股指期货,尚缺乏专门针对中国国债期货价格发现方面的研究。随着中国5年期国债期货于2013年9月上市交易,深入研究中国市场结构下的国债期货价格发现能力有助于从微观视角掌握与其它期货品种内在运行规律的差异性。本文运用中国5年期国债期货上市交易后的5分钟高频数据,采取向量误差修正(VECM)模型和Granger因果关系检验等计量分析方法检验中国国债期货与现货价格之间的关系,并创新性地使用共同因子贡献法和信息份额法分析我国国债期货市场与现货市场对价格发现功能的贡献程度。结果表明,中国国债期货价格与现货价格之间存在长期协整关系。中国国债期货价格是现货价格的Granger成因,且两者之间存在单向的价格引导关系。同时通过实证得出中国国债期货市场在对价格发现的贡献程度上占主导地位的结论。     

Simulating Financial Prices

A method is described for simulating daily financial prices using appropriate statistical models. Simulated open, high, low and close prices can be obtained. These simulated prices are required for a rigorous evaluation of trading systems. Appropriate software based on the models has been used by a futures trader to improve his trading decisions.     

对我国期货市场波动性的分阶段实证研究

波动性是经济和金融研究的热点问题。本文首先采用无条件波动度量方法对我国三大期货市场1997年—2004年的波动性进行了估计,发现第一阶段97年—02年期货价格总体呈下跌趋势,三大市场整体波动性不大,较高的波动性都出现在期货价格下跌时期,较低的波动性都出现在期货价格上涨时期;第二阶段03年—04年三大市场波动性显著提高,总体价格呈上升趋势,较高的波动性都出现在期货价格上涨时期,而较低的波动性都出现在期货价格下跌时期;本文进一步采用条件波动模型对我国三大期货市场两个阶段收益率与波动性的相关关系及波动性的杠杆效应进行了研究,结果表明铜期货收益率与波动性显著相关,大豆期货收益率与波动性不显著相关;我国三大期货市场均存在杠杆效应,并且两个阶段波动性的杠杆效应相反,其中铜期货市场的杠杆效应更显著。