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Title:

Option Pricing in Multivariate Stochastic Volatility Models of OU Type

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We present a multivariate stochastic volatility model with leverage, which is flexible enough to recapture the individual dynamics as well as the interdependencies between several assets while still being highly analytically tractable. First we derive the characteristic function and give conditions that ensure its analyticity and absolute integr...

We present a multivariate stochastic volatility model with leverage, which is flexible enough to recapture the individual dynamics as well as the interdependencies between several assets while still being highly analytically tractable. First we derive the characteristic function and give conditions that ensure its analyticity and absolute integrability in some open complex strip around zero. Therefore we can use Fourier methods to compute the prices of multi-asset options efficiently. To show the applicability of our results, we propose a concrete specification, the OU-Wishart model, where the dynamics of each individual asset coincide with the popular Gamma-OU BNS model. This model can be well calibrated to market prices, which we illustrate with an example using options on the exchange rates of some major currencies. Finally, we show that covariance swaps can also be priced in closed form. Minimize

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preprint

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Title:

Absolute Moments of Generalized Hyperbolic Distributions and Approximate Scaling of Normal Inverse Gaussian Lévy Processes

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Expressions for (absolute) moments of generalized hyperbolic and normal inverse Gaussian (NIG) laws are given in terms of moments of the corresponding symmetric laws. For the (absolute) moments centred at the location parameter "μ" explicit expressions as series containing Bessel functions are provided. Furthermore, the derivatives of the logari...

Expressions for (absolute) moments of generalized hyperbolic and normal inverse Gaussian (NIG) laws are given in terms of moments of the corresponding symmetric laws. For the (absolute) moments centred at the location parameter "μ" explicit expressions as series containing Bessel functions are provided. Furthermore, the derivatives of the logarithms of absolute "μ"-centred moments with respect to the logarithm of time are calculated explicitly for NIG Lévy processes. Computer implementation of the formulae obtained is briefly discussed. Finally, some further insight into the apparent scaling behaviour of NIG Lévy processes is gained. Copyright 2005 Board of the Foundation of the Scandinavian Journal of Statistics. Minimize

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article

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Title:

MULTIVARIATE CARMA PROCESSES

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Abstract. A multivariate Lévy-driven continuous time autoregressive moving average (CARMA) model of order (p, q), q < p, is introduced. It extends the well-known univariate CARMA and multivariate discrete time ARMA models. We give an explicit construction using a state space representation and a spectral representation of the driving Lévy proces...

Abstract. A multivariate Lévy-driven continuous time autoregressive moving average (CARMA) model of order (p, q), q < p, is introduced. It extends the well-known univariate CARMA and multivariate discrete time ARMA models. We give an explicit construction using a state space representation and a spectral representation of the driving Lévy process. Furthermore, various probabilistic properties of the state space model and the multivariate CARMA process itself are discussed in detail. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2008-07-01

Source:

http://www-lit.ma.tum.de/veroeff/quel/050.60004.pdf

http://www-lit.ma.tum.de/veroeff/quel/050.60004.pdf Minimize

Document Type:

text

Language:

en

Subjects:

CARMA process ; Lévy process ; multivariate stochastic differential equation ; spectral representation

CARMA process ; Lévy process ; multivariate stochastic differential equation ; spectral representation Minimize

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Metadata may be used without restrictions as long as the oai identifier remains attached to it.

Metadata may be used without restrictions as long as the oai identifier remains attached to it. Minimize

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Title:

On Markov-switching ARMA processes – stationarity, existence of moments and geometric ergodicity, Submitted for publication

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The probabilistic properties of R d-valued Markov-Switching ARMA processes with a general state space parameter chain are analysed. Stationarity and ergodicity conditions are given and a feasible general stationarity condition based on a tailor-made norm is introduced. Moreover, it is shown that causality of all individual regimes is neither a n...

The probabilistic properties of R d-valued Markov-Switching ARMA processes with a general state space parameter chain are analysed. Stationarity and ergodicity conditions are given and a feasible general stationarity condition based on a tailor-made norm is introduced. Moreover, it is shown that causality of all individual regimes is neither a necessary nor a sufficient criterion for strict negativity of the associated Lyapunov exponent. We also consider finiteness of moments and prove geometric ergodicity and strong mixing. The feasible stationarity condition is extended to ensure these properties. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2008-07-17

Source:

http://www-lit.ma.tum.de/veroeff/quel/059.60002.pdf

http://www-lit.ma.tum.de/veroeff/quel/059.60002.pdf Minimize

Document Type:

text

Language:

en

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Metadata may be used without restrictions as long as the oai identifier remains attached to it.

Metadata may be used without restrictions as long as the oai identifier remains attached to it. Minimize

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Title:

Multivariate COGARCH(1, 1) processes

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Multivariate COGARCH(1, 1) processes are introduced as a continuous-time models for multidimensional heteroskedastic observations. Our model is driven by a single multivariate Lévy process and the latent timevarying covariance matrix is directly specified as a stochastic process in the positive semidefinite matrices. After defining the COGARCH(1...

Multivariate COGARCH(1, 1) processes are introduced as a continuous-time models for multidimensional heteroskedastic observations. Our model is driven by a single multivariate Lévy process and the latent timevarying covariance matrix is directly specified as a stochastic process in the positive semidefinite matrices. After defining the COGARCH(1, 1) process, we analyze its probabilistic properties. We show a sufficient condition for the existence of a stationary distribution for the stochastic covariance matrix process and present criteria ensuring the finiteness of moments. Under certain natural assumptions on the moments of the driving Lévy process, explicit expressions for the first and second-order moments and (asymptotic) secondorder stationarity of the covariance matrix process are obtained. Furthermore, we study the stationarity and second-order structure of the increments of the multivariate COGARCH(1, 1) process and their “squares”. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2011-02-10

Source:

http://www-m4.ma.tum.de/pers/stelzer/Stelzer2010Bernoulli.pdf

Document Type:

text

Language:

en

Subjects:

COGARCH ; Lévy process ; multivariate GARCH ; positive definite random matrix process ; second-order moment structure ; stationarity

COGARCH ; Lévy process ; multivariate GARCH ; positive definite random matrix process ; second-order moment structure ; stationarity Minimize

DDC:

519 Probabilities & applied mathematics *(computed)*

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Metadata may be used without restrictions as long as the oai identifier remains attached to it.

Metadata may be used without restrictions as long as the oai identifier remains attached to it. Minimize

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On the definition, stationary distribution and second order structure of positive semidefinite Ornstein–Uhlenbeck type processes

On the definition, stationary distribution and second order structure of positive semidefinite Ornstein–Uhlenbeck type processes Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-11-21

Source:

http://arxiv.org/pdf/0909.0851v1.pdf

http://arxiv.org/pdf/0909.0851v1.pdf Minimize

Document Type:

text

Language:

en

Subjects:

quadratic variation

quadratic variation Minimize

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Title:

On Markov-switching ARMA processes – stationarity, existence of moments and geometric ergodicity, Submitted for publication

Description:

The probabilistic properties of R d-valued Markov-Switching ARMA processes with a general state space parameter chain are analysed. Stationarity and ergodicity conditions are given and a feasible general stationarity condition based on a tailor-made norm is introduced. Moreover, it is shown that causality of all individual regimes is neither a n...

The probabilistic properties of R d-valued Markov-Switching ARMA processes with a general state space parameter chain are analysed. Stationarity and ergodicity conditions are given and a feasible general stationarity condition based on a tailor-made norm is introduced. Moreover, it is shown that causality of all individual regimes is neither a necessary nor a sufficient criterion for strict negativity of the associated Lyapunov exponent. We also consider finiteness of moments and prove geometric ergodicity and strong mixing. The feasible stationarity condition is extended to ensure these properties. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2008-07-17

Source:

http://www-lit.ma.tum.de/veroeff/quel/059.60002.ps.gz

http://www-lit.ma.tum.de/veroeff/quel/059.60002.ps.gz Minimize

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text

Language:

en

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Title:

On the definition, stationary distribution and second order structure of positive semidefinite Ornstein–Uhlenbeck type processes

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2011-02-10

Source:

http://www-m4.ma.tum.de/pers/stelzer/PigorschSteler2009Bernoulli.pdf

http://www-m4.ma.tum.de/pers/stelzer/PigorschSteler2009Bernoulli.pdf Minimize

Document Type:

text

Language:

en

Subjects:

completely positive matrix ; matrix subordinator

completely positive matrix ; matrix subordinator Minimize

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Title:

Multivariate Markov-switching ARMA processes with regularly varying noise

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The tail behaviour of stationary R d-valued Markov-Switching ARMA processes driven by a regularly varying noise is analysed. It is shown that under appropriate summability conditions the MS-ARMA process is again regularly varying as a sequence. Moreover, the feasible stationarity condition given in Stelzer (2006) is extended to a criterion for r...

The tail behaviour of stationary R d-valued Markov-Switching ARMA processes driven by a regularly varying noise is analysed. It is shown that under appropriate summability conditions the MS-ARMA process is again regularly varying as a sequence. Moreover, the feasible stationarity condition given in Stelzer (2006) is extended to a criterion for regular variation. Our results complement in particular those of Saporta (2005) where regularly varying tails of one-dimensional MS-AR(1) processes coming from consecutive large parameters were studied. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2008-07-01

Source:

http://www-lit.ma.tum.de/veroeff/quel/069.60012.ps.gz

http://www-lit.ma.tum.de/veroeff/quel/069.60012.ps.gz Minimize

Document Type:

text

Language:

en

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Title:

Multivariate Markov-switching ARMA processes with regularly varying noise

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The tail behaviour of stationary R d-valued Markov-Switching ARMA processes driven by a regularly varying noise is analysed. It is shown that under appropriate summability conditions the MS-ARMA process is again regularly varying as a sequence. Moreover, the feasible stationarity condition given in Stelzer (2006) is extended to a criterion for r...

The tail behaviour of stationary R d-valued Markov-Switching ARMA processes driven by a regularly varying noise is analysed. It is shown that under appropriate summability conditions the MS-ARMA process is again regularly varying as a sequence. Moreover, the feasible stationarity condition given in Stelzer (2006) is extended to a criterion for regular variation. Our results complement in particular those of Saporta (2005) where regularly varying tails of one-dimensional MS-AR(1) processes coming from consecutive large parameters were studied. Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2008-07-01

Source:

http://www-lit.ma.tum.de/veroeff/quel/069.60012.pdf

http://www-lit.ma.tum.de/veroeff/quel/069.60012.pdf Minimize

Document Type:

text

Language:

en

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