Unifying pricing formula for several stochastic volatility models with jumps

In this paper, we introduce a unifying approach to option pricing under continuous‐time stochastic volatility models with jumps. For European style options, a new semi‐closed pricing formula is derived using the generalized complex Fourier transform of the corresponding partial integro‐differential equation. This approach is successfully applied to models with different volatility diffusion and jump processes. We also discuss how to price options with different payoff functions in a similar way. In particular, we focus on a log‐normal and a log‐uniform jump diffusion stochastic volatility model, originally introduced by Bates and Yan and Hanson, respectively. The comparison of existing and newly proposed option pricing formulas with respect to time efficiency and precision is discussed. We also derive a representation of an option price under a new approximative fractional jump diffusion model that differs from the aforementioned models, especially for the out‐of‐the money contracts. Copyright © 2017 John Wiley & Sons, Ltd.     

基于高频波动率模型与R-vine copula的行业资产组合风险测度研究

相较于低频波动率模型,高频波动率模型在单资产的波动和风险预测中均取得了更好效果,因此如何将高频波动率模型引入组合风险分析具有重要的理论和现实意义。本文以沪深300指数中的6种行业高频数据为例,运用滚动时间窗技术建立9类已实现波动率异质自回归(HAR-RV-type)模型刻画行业指数波动,同时使用R-vine copula模型描述行业资产间相依结构,进一步结合均值-CVaR模型优化行业资产组合投资比例,构建组合风险的预期损失模型,并通过返回测试比较不同风险模型的精度差异。研究结果表明:将HAR族高频波动率模型引入组合风险分析框架,能够有效预测行业资产组合风险状况;高频波动率预测的准确性将进而影响组合风险测度效果,跳跃、符号跳跃变差以及符号正向、负向跳跃变差均有助于提高行业组合风险的预测精度。     

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.     

Rare Shock,Two-Factor Stochastic Volatility and Currency Option Pricing

Abstract

In this paper, we develop an option valuation model where the dynamics of the spot foreign exchange rate is governed by a two-factor Markov-modulated jump-diffusion process. The short-term fluctuation of stochastic volatility is driven by a Cox–Ingersoll–Ross (CIR) process and the long-term variation of stochastic volatility is driven by a continuous-time Markov chain which can be interpreted as economy states. Rare events are governed by a compound Poisson process with log-normal jump amplitude and stochastic jump intensity is modulated by a common continuous-time Markov chain. Since the market is incomplete under regime-switching assumptions, we determine a risk-neutral martingale measure via the Esscher transform and then give a pricing formula of currency options. Numerical results are presented for investigating the impact of the long-term volatility and the annual jump intensity on option prices.     

时间序列中方差的结构变点的小波识别(英文)

本文给出了时间序列中方差的小波系数的两种估计:连续估计和离散估计.这两种估计可以用来检测时间序列中方差的结构变点.利用这两种估计我们给出了方差变点的位置和跳跃幅度的估计,并且显示出这些估计可达到最佳收敛速度.同时,我们还给出了这些估计的收敛速度以及检验统计量的渐进分布!     

Robust static hedging of barrier options in stochastic volatility models

Static hedge portfolios for barrier options are extremely sensitive with respect to changes of the volatility surface. In this paper we develop a semi-infinite programming formulation of the static super-replication problem in stochastic volatility models which allows to robustify the hedge against model parameter uncertainty in the sense of a worst case design. From a financial point of view this robustness guarantees the hedge performance for an infinite number of future volatility surface scenarios including volatility shocks and changes of the skew. After proving existence of such robust hedge portfolios and presenting an algorithm to numerically solve the underlying optimization problem, we apply the approach to a detailed example. Surprisingly, the optimal robust portfolios are only marginally more expensive than the barrier option itself.     

考虑微观结构噪声与跳跃影响的波动建模

现有的金融高频数据研究,并未充分考虑微观结构噪声对波动建模和预测的影响.以非参数化方法为理论框架,基于高频数据,采用适当方法分离出波动中的微观结构噪声成份,构建了新的跳跃方差和连续样本路径方差,将已实现波动分解为连续样本路径方差、跳跃方差和微观结构噪声方差.同时考虑微观结构噪声和跳跃对波动的影响,对HAR-RV-CJ模型进行改进,提出了HAR-RV-N-CJ模型和LHAR-RV-N-CJ模型.通过上证综指高频数据进行实证,结果表明新模型在模型拟合和预测方面均优于HAR-RV-CJ模型.     

Pricing Forward Starting Call Options under a Markov-Modulated Jump Diffusion Process

The pricing problem of forward starting call options under a Markov-modulated jump diffusion process is studied. Under the assumption that the dynamics of risky asset follows a Markov-modulated jump diffusion process, the explicit analytical formula of forward starting call options is obtained by the change of measure and no arbitrage pricing theory. Moreover, the numerical results of option value are provided by the Monte Carlo method, and the value of forward starting call options is compared when the risky asset satisfies different financial models.     

The Gumbel test and jumps in the volatility process

In the framework of jump detection in stochastic volatility models the Gumbel test based on extreme value theory has recently been introduced. Compared to other jump tests it possesses the advantages that the direction and location of jumps may also be detected. Furthermore, compared to the Barndorff–Nielsen and Shephard test based on bipower variation the Gumbel test possesses a larger power. However, so far one assumption was that the volatility process is Hölder continuous, though there is empirical evidence for jumps in the volatility as well. In this paper we derive that the Gumbel test still works under the setting of finitely many jumps not exceeding a certain size. This maximal jump size depends on the relative sampling frequencies involved in the definition of the test statistics. Furthermore, we show that the given bound on the jump size is sharp and investigate the details of the phase transition at this critical bound.     

沪深300仿真股指期货价格不对称跳跃波动的实证分析

本研究利用2006年10月30日至2009年3月13日期间的仿真的沪深300指数期货每日结算价,探讨了期货价格的不对称跳跃波动行为。在实证研究方法上,本文以Chan和Maheu的GARCH(1,1)-ARJI模型为基础并进行了扩展,以EGARCH(1,1)-CJI和EGARCH(1,1)-ARJI两种模型来刻画股指期货价格的不对称和跳跃波动行为。实证结果显示:(1)沪深300仿真股指期货价格存在不对称跳跃波动,而且跳跃强度不为一固定常数,异常信息所产生的跳跃强度是随着时间变动的。(2)经过似然比检验,结果显示EGARCH(1,1)-ARJI模型比EGARCH(1,1)-CJI模型具有更好的拟合能力。     

Variational Solutions of the Pricing PIDEs for European Options in Lévy Models

Abstract

One of the fundamental problems in financial mathematics is to develop efficient algorithms for pricing options in advanced models such as those driven by Lévy processes. Essentially there are three approaches in use. These are Monte Carlo, Fourier transform and partial integro-differential equation (PIDE)-based methods. We focus our attention here on the latter. There is a large arsenal of numerical methods for efficiently solving parabolic equations that arise in this context. Especially Galerkin and Galerkin-inspired methods have an impressive potential. In order to apply these methods, what is required is a formulation of the equation in the weak sense.

The contribution of this paper is therefore to analyse weak solutions of the Kolmogorov backward equations which are related to prices of European options in (time-inhomogeneous) Lévy models and to establish a precise link between the prices and the weak solutions of these equations. The resulting relation is a Feynman–Kac representation of the solution as a conditional expectation. Our special concern is to provide a framework that is able to cover both, the common types of European options and a wide range of advanced models in which these derivatives are priced.

An application to financial models requires in particular to admit pure jump processes such as generalized hyperbolic processes as well as unbounded domains of the equation. In order to deal at the same time with the typical pay-offs that can arise, the weak formulation of the equation is based on exponentially weighted Sobolev–Slobodeckii spaces. We provide a number of examples of models that are covered by this general framework. Examples of options for which such an analysis is required are calls, puts, digital and power options as well as basket options.     

Stochastic volatility Gaussian Heath-Jarrow-Morton models

This paper extends the class of deterministic volatility Heath-Jarrow-Morton models to a Markov chain stochastic volatility framework allowing for jump discontinuities and a variety of deformations of the term structure of forward rate volatilities. Analytical solutions for the dynamics of the volatility term structure are obtained. Semimartingale decompositions of the interest rates under a spot and forward martingale measures are identified. Stochastic volatility versions of the continuous time Ho-Lee and Hull-White extended Vasicek models are obtained. Introducing a regime shift in volatility that is an exponential function of time to maturity leads to a Vasicek dynamics with regime switching coefficients of the short rate.     

机制转换条件下股市收益率跳跃行为研究:基于RS-ARJI模型的以中国股市为例的实证分析

为检验股市收益率机制转换特性,考察机制转换条件下股市收益率的跳跃特征,以及在不同机制下跳跃行为对股市收益率的冲击效应,将Markov机制转换思想引入自回归跳跃(ARJI)模型,构建一个机制转换自回归跳跃(RS-ARM)模型.基于该模型对中国股市进行实证研究,结果表明:股市存在高、低波动两种机制,高波动时期的跳跃幅度和强度及其对股市收益率的冲击均大于低波动时期.同时,波动率估计和预测评价指标显示,RS-ARJI模型优于目前被广泛使用的GARCH模型和ARJI模型.     

Markov-modulated jump-diffusions for currency option pricing

This paper introduces dynamic models for the spot foreign exchange rate with capturing both the rare events and the time-inhomogeneity in the fluctuating currency market. For the rare events, we use a compound Poisson process with log-normal jump amplitude to describe the jumps. As for the time-inhomogeneity in the market dynamics, we particularly stress the strong dependence of the domestic/foreign interest rates, the appreciation rate and the volatility of the foreign currency on the time-varying sovereign ratings in the currency market. The time-varying ratings are formulated by a continuous-time finite-state Markov chain. Based on such a spot foreign exchange rate dynamics, we then study the pricing of some currency options. Here we will adopt a so-called regime-switching Esscher transform to identify a risk-neutral martingale measure. By determining the regime-switching Esscher parameters we then get an integral expression on the prices of European-style currency options. Finally, numerical illustrations are given.     

Convexity preserving jump-diffusion models for option pricing

We investigate which jump-diffusion models are convexity preserving. The study of convexity preserving models is motivated by monotonicity results for such models in the volatility and in the jump parameters. We give a necessary condition for convexity to be preserved in several-dimensional jump-diffusion models. This necessary condition is then used to show that, within a large class of possible models, the only convexity preserving models are the ones with linear coefficients.     

基于傅立叶变换的SVJ模型的欧式期权定价

为了更加精确的计算期权价格,将结合随机波动和跳扩散模型(以下简称SVJ模型)以更好的描述期权标的资产价格过程,然而这样的价格过程无法得到概率密度函数的封闭形式,而只能得到包含特殊函数和无限求和的复杂的表达式.不过它们的特征函数都是封闭且是唯一的,因而可以通过它们的特征函数,并运用两种傅立叶变换的方法来求出期权价格.其中FFT算法计算的结果将与Monte Carlo模拟得出的结果进行比较,然后再将SVJ模型的计算结果和Black-Scholes模型进行比较.     

双跳跃仿射扩散模型的几何平均型水平重置期权定价

在资产收益率及其波动率均满足随机跳跃且具有跳跃相关性的仿射扩散模型下,用广义双指数分布和伽玛分布分别刻画非对称性收益率及其波动率的跳跃波动变化,研究了具有几何平均特征的水平重置期权定价问题.通过Girsanov测度变换和多维Fourier逆变换方法,给出了此类重置期权定价的解析公式.最后,通过数值实例着重分析了联合跳跃...     

非参数回归中方差变点的小波检测(英文)

本文主要研究了非参数回归模型中方差函数的变点, 利用小波方法构造的检验量来检测方差中的变点,建立了这些检验量的渐近分布, 并且运用这些检验量构造了方差变点的位置和跳跃幅度的估计, 给出了这些估计的渐近性质, 并进一步通过随机模拟验证了本文方法在有限样本下的性质.     

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.     

Modelling stochastic skew of FX options using SLV models with stochastic spot/vol correlation and correlated jumps

It is known that the implied volatility skew of Forex (FX) options demonstrates a stochastic behaviour which is called stochastic skew. In this paper, we create stochastic skew by assuming the spot/instantaneous variance (InV) correlation to be stochastic. Accordingly, we consider a class of Stochastic Local Volatility (SLV) models with stochastic correlation where all drivers – the spot, InV and their correlation – are modelled by processes. We assume all diffusion components to be fully correlated, as well as all jump components. A new fully implicit splitting finite-difference scheme is proposed for solving forward PIDE which is used when calibrating the model to market prices of the FX options with different strikes and maturities. The scheme is unconditionally stable, of second order of approximation in time and space, and achieves a linear complexity in each spatial direction. The results of simulation obtained by using this model demonstrate the capacity of the presented approach in modelling stochastic skew.