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Neil Shephard - One of the best experts on this subject based on the ideXlab platform.
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Limit theorems for bipower Variation in financial econometrics
2005Co-Authors: Ole Barndorff-nielsen, Jean Jacod, Svend Graversen, Neil ShephardAbstract:In this paper we provide an asymptotic analysis of generalised bipower measures of the Variation of price processes in financial economics. These measures encompass the usual Quadratic Variation, power Variation and bipower Variations which have been highlighted in recent years in financial econometrics. The analysis is carried out under some rather general Brownian semimartingale assumptions, which allow for standard leverage effects.
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power Variation and stochastic volatility a review and some new results
Journal of Applied Probability, 2004Co-Authors: Ole E Barndorffnielsen, S. E. Graversen, Neil ShephardAbstract:In this paper we review some recent work on limit results on realised power Variation, that is sums of powers of absolute increments of various semimartingales. A special case of this analysis is realised variance and its probability limit, Quadratic Variation. Such quantities often appear in financial econometrics in the analysis of volatility. The paper also provides some new results and discusses open issues.
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power and bipower Variation with stochastic volatility and jumps
Journal of Financial Econometrics, 2004Co-Authors: Ole E Barndorffnielsen, Neil ShephardAbstract:This article shows that realized power Variation and its extension, realized bipower Variation, which we introduce here, are somewhat robust to rare jumps. We demonstrate that in special cases, realized bipower Variation estimates integrated variance in stochastic volatility models, thus providing a model-free and consistent alternative to realized variance. Its robustness property means that if we have a stochastic volatility plus infrequent jumps process, then the difference between realized variance and realized bipower Variation estimates the Quadratic Variation of the jump component. This seems to be the first method that can separate Quadratic Variation into its continuous and jump components. Various extensions are given, together with proofs of special cases of these results. Detailed mathematical results are reported in Barndorff-Nielsen and Shephard (2003a).
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econometric analysis of realized volatility and its use in estimating stochastic volatility models
Journal of The Royal Statistical Society Series B-statistical Methodology, 2002Co-Authors: Ole E Barndorffnielsen, Neil ShephardAbstract:Summary. The availability of intraday data on the prices of speculative assets means that we can use Quadratic Variation-like measures of activity in financial markets, called realized volatility, to study the stochastic properties of returns. Here, under the assumption of a rather general stochastic volatility model, we derive the moments and the asymptotic distribution of the realized volatility error—the difference between realized volatility and the discretized integrated volatility (which we call actual volatility). These properties can be used to allow us to estimate the parameters of stochastic volatility models without recourse to the use of simulation-intensive methods.
Francesco Russo - One of the best experts on this subject based on the ideXlab platform.
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Generalized coVariation for Banach space valued processes, Itô formula and applications
2015Co-Authors: Cristina Di Girolami, Francesco RussoAbstract:This paper discusses a new notion of Quadratic Variation and coVariation for Banach space valued processes (not necessarily semimartingales) and related Itô formula. If X and Y take respectively values in Banach spaces B1 and B2 and χ is a suitable subspace of the dual of the projective tensor product of B1 and B2 (denoted by (B1⊗̂πB2) ∗), we define the so-called χ-coVariation of X and Y. If X = Y, the χ-coVariation is called χ-Quadratic Variation. The notion of χ-Quadratic Variation is a natural generalization of the one introduced by Métivier-Pellaumail and Dinculeanu which is too restrictive for many applications. In particular, if χ is the whole space (B1⊗̂πB1) ∗ then the χ-Quadratic Variation coincides with the Quadratic Variation of a B1-valued semimartingale. We evaluate the χ-coVariation of various processes for several examples of χ with a particular attention to the case B1 = B2 = C([−τ, 0]) for some τ> 0 and X and Y being window processes. If X is a real valued process, we call window process associated with X the C([−τ, 0])-valued process X: = X(·) defined by Xt(y) = Xt+y, where y ∈ [−τ, 0]. The Itô formula introduced here is an important instrument to establish a representation result of Clark-Ocone type for a class of path dependent random variables of type h = H(XT (·)), H: C([−T, 0]) − → R for not-necessarily semimartingales X with finite Quadratic Variation. This representation will be linked to a function u: [0, T]×C([−T, 0]) − → R solving an infinite dimensional partial differential equation
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generalized coVariation for banach space valued processes ito formula and applications
Osaka Journal of Mathematics, 2014Co-Authors: Cristina Di Girolami, Francesco RussoAbstract:This paper discusses a new notion of Quadratic Variation and coVariation for Banach space valued processes (not necessarily semimartingales) and related Ito formula. If $\X$ and $\Y$ take respectively values in Banach spaces $B_{1}$ and $B_{2}$ and $\chi$ is a suitable subspace of the dual of the projective tensor product of $B_{1}$ and $B_{2}$ (denoted by $(B_{1}\hat{\otimes}_{\pi}B_{2})^{\ast}$), we define the so-called $\chi$-coVariation of $\X$ and $\Y$. If $\X=\Y$, the $\chi$-coVariation is called $\chi$-Quadratic Variation. The notion of $\chi$-Quadratic Variation is a natural generalization of the one introduced by Metivier-Pellaumail and Dinculeanu which is too restrictive for many applications. In particular, if $\chi$ is the whole space $(B_{1}\hat{\otimes}_{\pi}B_{1})^{\ast}$ then the $\chi$-Quadratic Variation coincides with the Quadratic Variation of a $B_{1}$-valued semimartingale. We evaluate the $\chi$-coVariation of various processes for several examples of $\chi$ with a particular attention to the case $B_{1}=B_{2}=C([-\tau,0])$ for some $\tau>0$ and $\X$ and $\Y$ being \textit{window processes}. If $X$ is a real valued process, we call window process associated with $X$ the $C([-\tau,0])$-valued process $\X:=X(\cdot)$ defined by $X_t(y) = X_{t+y}$, where $y \in [-\tau,0]$. The Ito formula introduced here is an important instrument to establish a representation result of Clark-Ocone type for a class of path dependent random variables of type $h=H(X_{T}(\cdot))$, $H:C([-T,0])\longrightarrow\R$ for not-necessarily semimartingales $X$ with finite Quadratic Variation. This representation will be linked to a function $u:[0,T]\times C([-T,0])\longrightarrow \mathbb{R}$ solving an infinite dimensional partial differential equation.
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Generalized coVariation for Banach space valued processes, Itô formula and applications
2013Co-Authors: Cristina Di Girolami, Francesco RussoAbstract:This paper discusses a new notion of Quadratic Variation and coVariation for Banach space valued processes (not necessarily semimartingales) and related Itô formula. If $\X$ and $\Y$ take respectively values in Banach spaces $B_{1}$ and $B_{2}$ and $\chi$ is a suitable subspace of the dual of the projective tensor product of $B_{1}$ and $B_{2}$ (denoted by $(B_{1}\hat{\otimes}_{\pi}B_{2})^{\ast}$), we define the so-called $\chi$-coVariation of $\X$ and $\Y$. If $\X=\Y$, the $\chi$-coVariation is called $\chi$-Quadratic Variation. The notion of $\chi$-Quadratic Variation is a natural generalization of the one introduced by Métivier-Pellaumail and Dinculeanu which is too restrictive for many applications. In particular, if $\chi$ is the whole space $(B_{1}\hat{\otimes}_{\pi}B_{1})^{\ast}$ then the $\chi$-Quadratic Variation coincides with the Quadratic Variation of a $B_{1}$-valued semimartingale. We evaluate the $\chi$-coVariation of various processes for several examples of $\chi$ with a particular attention to the case $B_{1}=B_{2}=C([-\tau,0])$ for some $\tau>0$ and $\X$ and $\Y$ being \textit{window processes}. If $X$ is a real valued process, we call window process associated with $X$ the $C([-\tau,0])$-valued process $\X:=X(\cdot)$ defined by $X_t(y) = X_{t+y}$, where $y \in [-\tau,0]$. The Itô formula introduced here is an important instrument to establish a representation result of Clark-Ocone type for a class of path dependent random variables of type $h=H(X_{T}(\cdot))$, $H:C([-T,0])\longrightarrow\R$ for not-necessarily semimartingales $X$ with finite Quadratic Variation. This representation will be linked to a function $u:[0,T]\times C([-T,0])\longrightarrow \mathbb{R}$ solving an infinite dimensional partial differential equation.
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the coVariation for banach space valued processes and applications
arXiv: Probability, 2013Co-Authors: Cristina Di Girolami, Giorgio Fabbri, Francesco RussoAbstract:This article focuses on a new concept of Quadratic Variation for processes taking values in a Banach space $B$ and a corresponding coVariation. This is more general than the classical one of M\'etivier and Pellaumail. Those notions are associated with some subspace $\chi$ of the dual of the projective tensor product of $B$ with itself. We also introduce the notion of a convolution type process, which is a natural generalization of the It\^o process and the concept of $\bar \nu_0$-semimartingale, which is a natural extension of the classical notion of semimartingale. The framework is the stochastic calculus via regularization in Banach spaces. Two main applications are mentioned: one related to Clark-Ocone formula for finite Quadratic Variation processes; the second one concerns the probabilistic representation of a Hilbert valued partial differential equation of Kolmogorov type.
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the coVariation for banach space valued processes and applications
Documents de recherche, 2013Co-Authors: Cristina Di Girolami, Giorgio Fabbri, Francesco RussoAbstract:This article focuses on a new concept of Quadratic Variation for processes taking values in a Banach space B and a corresponding coVariation. This is more general than the classical one of Metivier and Pellaumail. Those notions are associated with some subspace ? of the dual of the projective tensor product of B with itself. We also introduce the notion of a convolution type process, which is a natural generalization of the Ito process and the concept of ¯V0-semimartingale, which is a natural extension of the classical notion of semimartingale. The framework is the stochastic calculus via regularization in Banach spaces. Two main applications are mentioned: one related to Clark-Ocone formula for finite Quadratic Variation processes; the second one concerns the probabilistic representation of a Hilbert valued partial differential equation of Kolmogorov type. 2010 Math Subject Classification: 60G22, 60H05, 60H07, 60H15, 60H30, 26E20, 35K90 46G05
Ole E Barndorffnielsen - One of the best experts on this subject based on the ideXlab platform.
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power Variation and stochastic volatility a review and some new results
Journal of Applied Probability, 2004Co-Authors: Ole E Barndorffnielsen, S. E. Graversen, Neil ShephardAbstract:In this paper we review some recent work on limit results on realised power Variation, that is sums of powers of absolute increments of various semimartingales. A special case of this analysis is realised variance and its probability limit, Quadratic Variation. Such quantities often appear in financial econometrics in the analysis of volatility. The paper also provides some new results and discusses open issues.
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power and bipower Variation with stochastic volatility and jumps
Journal of Financial Econometrics, 2004Co-Authors: Ole E Barndorffnielsen, Neil ShephardAbstract:This article shows that realized power Variation and its extension, realized bipower Variation, which we introduce here, are somewhat robust to rare jumps. We demonstrate that in special cases, realized bipower Variation estimates integrated variance in stochastic volatility models, thus providing a model-free and consistent alternative to realized variance. Its robustness property means that if we have a stochastic volatility plus infrequent jumps process, then the difference between realized variance and realized bipower Variation estimates the Quadratic Variation of the jump component. This seems to be the first method that can separate Quadratic Variation into its continuous and jump components. Various extensions are given, together with proofs of special cases of these results. Detailed mathematical results are reported in Barndorff-Nielsen and Shephard (2003a).
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econometric analysis of realized volatility and its use in estimating stochastic volatility models
Journal of The Royal Statistical Society Series B-statistical Methodology, 2002Co-Authors: Ole E Barndorffnielsen, Neil ShephardAbstract:Summary. The availability of intraday data on the prices of speculative assets means that we can use Quadratic Variation-like measures of activity in financial markets, called realized volatility, to study the stochastic properties of returns. Here, under the assumption of a rather general stochastic volatility model, we derive the moments and the asymptotic distribution of the realized volatility error—the difference between realized volatility and the discretized integrated volatility (which we call actual volatility). These properties can be used to allow us to estimate the parameters of stochastic volatility models without recourse to the use of simulation-intensive methods.
Nizar Touzi - One of the best experts on this subject based on the ideXlab platform.
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moral hazard in dynamic risk management
Management Science, 2017Co-Authors: Jaksa Cvitanic, Dylan Possamai, Nizar TouziAbstract:We consider a contracting problem in which a principal hires an agent to manage a risky project. When the agent chooses volatility components of the output process and the principal observes the output continuously, the principal can compute the Quadratic Variation of the output, but not the individual components. This leads to moral hazard with respect to the risk choices of the agent. To find the optimal contract, we develop a novel approach to solving principal–agent problems: first, we identify a family of admissible contracts for which the optimal agent’s action is explicitly characterized; then, we show that we do not lose on generality when finding the optimal contract inside this family, up to integrability conditions. To do this, we use the recent theory of singular changes of measures for Ito processes. We solve the problem in the case of CARA preferences and show that the optimal contract is linear in these factors: the contractible sources of risk, including the output, the Quadratic Variation of the output and the cross-Variations between the output and the contractible risk sources. Thus, like sample Sharpe ratios used in practice, path-dependent contracts naturally arise when there is moral hazard with respect to risk management. In a numerical example, we show that the loss of efficiency can be significant if the principal does not use the Quadratic Variation component of the optimal contract.
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moral hazard in dynamic risk management
arXiv: Portfolio Management, 2014Co-Authors: Jaksa Cvitanic, Dylan Possamai, Nizar TouziAbstract:We consider a contracting problem in which a principal hires an agent to manage a risky project. When the agent chooses volatility components of the output process and the principal observes the output continuously, the principal can compute the Quadratic Variation of the output, but not the individual components. This leads to moral hazard with respect to the risk choices of the agent. We identify a family of admissible contracts for which the optimal agent's action is explicitly characterized, and, using the recent theory of singular changes of measures for It\^o processes, we study how restrictive this family is. In particular, in the special case of the standard Homlstr\"om-Milgrom model with fixed volatility, the family includes all possible contracts. We solve the principal-agent problem in the case of CARA preferences, and show that the optimal contract is linear in these factors: the contractible sources of risk, including the output, the Quadratic Variation of the output and the cross-Variations between the output and the contractible risk sources. Thus, like sample Sharpe ratios used in practice, path-dependent contracts naturally arise when there is moral hazard with respect to risk management. In a numerical example, we show that the loss of efficiency can be significant if the principal does not use the Quadratic Variation component of the optimal contract.
Huu-tai Thai - One of the best experts on this subject based on the ideXlab platform.
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a nonlocal beam theory for bending buckling and vibration of nanobeams
International Journal of Engineering Science, 2012Co-Authors: Huu-tai ThaiAbstract:Abstract A nonlocal shear deformation beam theory is proposed for bending, buckling, and vibration of nanobeams using the nonlocal differential constitutive relations of Eringen. The theory, which does not require shear correction factor, accounts for both small scale effects and Quadratic Variation of shear strains and consequently shear stresses through the thickness of the beam. In addition, it has strong similarities with nonlocal Euler–Bernoulli beam theory in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The equations of motion are derived from Hamilton’s principle. Analytical solutions of deflection, buckling load, and natural frequency are presented for a simply supported beam, and the obtained results compare well with those predicted by the nonlocal Timoshenko and Reddy beam theories.
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a refined plate theory for functionally graded plates resting on elastic foundation
Composites Science and Technology, 2011Co-Authors: Huu-tai Thai, Dongho ChoiAbstract:Abstract A refined plate theory for functionally graded plates resting on elastic foundation is developed in this paper. The theory accounts for a Quadratic Variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. The number of independent unknowns of present theory is four, as against five in other shear deformation theories. The material properties of plate are assumed to vary according to power law distribution of the volume fraction of the constituents. The elastic foundation is modeled as two-parameter Pasternak foundation. Equations of motion are derived using Hamilton’s principle. The closed-form solutions of rectangular plates are obtained. Numerical results are presented to verify the accuracy of present theory.
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levy type solution for buckling analysis of orthotropic plates based on two variable refined plate theory
Composite Structures, 2011Co-Authors: Huu-tai Thai, Seungeock KimAbstract:Abstract This paper presents closed-form solution for buckling analysis of orthotropic plates using two variable refined plate theory. The theory accounts for a Quadratic Variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. Governing equations are derived from the principle of minimum total potential energy. The closed-form solutions of rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions are obtained by applying the state space approach to the Levy-type solution. Comparison studies are performed to verify the validity of the present results. The effects of boundary condition, loading condition, and Variations of modulus ratio, aspect ratio, and thickness ratio on the critical buckling load of orthotropic plates are investigated and discussed in detail.