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21
Affine processes and applications in finance
 Annals of Applied Probability
, 2003
"... Abstract. We provide the definition and a complete characterization of regular affine processes. This type of process unifies the concepts of continuousstate branching processes with immigration and OrnsteinUhlenbeck type processes. We show, and provide foundations for, a wide range of financial ap ..."
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Cited by 38 (5 self)
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Abstract. We provide the definition and a complete characterization of regular affine processes. This type of process unifies the concepts of continuousstate branching processes with immigration and OrnsteinUhlenbeck type processes. We show, and provide foundations for, a wide range of financial applications for regular affine processes.
On the geometry of the term structure of interest rates
 Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences
"... Abstract. We present recently developed geometric methods for the analysis of finite dimensional term structure models of the interest rates. This includes an extension of the Frobenius theorem for Fréchet spaces in particular. This approach puts new light on many of the classical models, such as th ..."
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Cited by 10 (5 self)
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Abstract. We present recently developed geometric methods for the analysis of finite dimensional term structure models of the interest rates. This includes an extension of the Frobenius theorem for Fréchet spaces in particular. This approach puts new light on many of the classical models, such as the HullWhite extended Vasicek and CoxIngersollRoss short rate models. The notion of a finite dimensional realization (FDR) is central for this analysis: we motivate it, classify all generic FDRs and provide some new results for the corresponding factor processes, such as hypoellipticity of its generators and the existence of smooth densities. Furthermore we include finite dimensional external factors, thus admitting a stochastic volatility structure. 1.
A characterization of hedging portfolios for interest rate contingent claims
, 2004
"... We consider the problem of hedging a European interest rate contingent claim with a portfolio of zerocoupon bonds and show that an HJM type Markovian model driven by an infinite number of sources of randomness does not have some of the shortcomings found in the classical finite factor models. Indee ..."
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Cited by 10 (1 self)
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We consider the problem of hedging a European interest rate contingent claim with a portfolio of zerocoupon bonds and show that an HJM type Markovian model driven by an infinite number of sources of randomness does not have some of the shortcomings found in the classical finite factor models. Indeed, under natural conditions on the model, we find that there exists a unique hedging strategy, and that this strategy has the desirable property that at all times it consists of bonds with maturities that are less than or equal to the longest maturity of the bonds underlying the claim.
A Filtered No Arbitrage Model for the Term Structures from Noisy Data
, 2002
"... We consider the problem of pricing in financial markets when agents do not have access to full information. The particular problem concerns the pricing of non traded or illiquid bonds on the basis of the observations of the yields of traded zerocoupon bonds. ..."
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Cited by 9 (7 self)
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We consider the problem of pricing in financial markets when agents do not have access to full information. The particular problem concerns the pricing of non traded or illiquid bonds on the basis of the observations of the yields of traded zerocoupon bonds.
Existence of invariant manifolds for stochastic equations in infinite dimension
 J. Funct. Anal
, 2003
"... Abstract. We provide a Frobenius type existence result for finitedimensional invariant submanifolds for stochastic equations in infinite dimension, in the spirit of Da Prato and Zabczyk [5]. We recapture and make use of the convenient calculus on Fréchet spaces, as developed by Kriegl and Michor [1 ..."
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Cited by 4 (1 self)
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Abstract. We provide a Frobenius type existence result for finitedimensional invariant submanifolds for stochastic equations in infinite dimension, in the spirit of Da Prato and Zabczyk [5]. We recapture and make use of the convenient calculus on Fréchet spaces, as developed by Kriegl and Michor [16]. Our main result is a weak version of the Frobenius theorem on Fréchet spaces. As an application we characterize all finitedimensional realizations for a stochastic equation which describes the evolution of the term structure of interest rates. 1.
A theory of stochastic integration for bond markets. Working paper
, 2004
"... We introduce a theory of stochastic integration with respect to a family of semimartingales depending on a continuous parameter, as a mathematical background to the theory of bond markets. We apply our results to the problem of superreplication and utility maximization from terminal wealth in a bon ..."
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Cited by 3 (0 self)
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We introduce a theory of stochastic integration with respect to a family of semimartingales depending on a continuous parameter, as a mathematical background to the theory of bond markets. We apply our results to the problem of superreplication and utility maximization from terminal wealth in a bond market. Finally, we compare our approach to those already existing in literature. 1. Introduction. In
Finite dimensional Markovian realizations for stochastic volatility forward rate models
, 2001
"... We consider forward rate rate models of HeathJarrowMorton type, as well as more general infinite dimensional SDEs, where the volatility/diffusion term is stochastic in the sense of being driven by a separate hidden Markov process. Within this framework we use the previously developed Hilbert space ..."
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Cited by 3 (0 self)
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We consider forward rate rate models of HeathJarrowMorton type, as well as more general infinite dimensional SDEs, where the volatility/diffusion term is stochastic in the sense of being driven by a separate hidden Markov process. Within this framework we use the previously developed Hilbert space realization theory in order provide general necessary and sufficent conditions for the existence of a finite dimensional Markovian realizations for the stochastic volatility models. We illustrate the theory by analyzing a number of concrete examples.
Term structure models driven by Wiener process and Poisson measures: Existence and positivity
, 2009
"... Abstract. In the spirit of [4], we investigate term structure models driven by Wiener process and Poisson measures with forward curve dependent volatilities. This includes a full existence and uniqueness proof for the corresponding Heath–Jarrow–Morton type term structure equation. Furthermore, we ch ..."
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Cited by 1 (1 self)
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Abstract. In the spirit of [4], we investigate term structure models driven by Wiener process and Poisson measures with forward curve dependent volatilities. This includes a full existence and uniqueness proof for the corresponding Heath–Jarrow–Morton type term structure equation. Furthermore, we characterize positivity preserving models by means of the characteristic coefficients, which was open for jumpdiffusions. Additionally we treat existence, uniqueness and positivity of the BrodyHughston equation [7, 8] of interest rate theory with jumps, an equation which we believe to be very useful for applications. A key role in our investigation is played by the method of the moving frame, which allows to transform the Heath–Jarrow–Morton–Musiela equation to a timedependent SDE. Key Words: term structure models driven by Wiener process and Poisson measures, HeathJarrowMortonMusiela equation, positivity preserving models, BrodyHughston equation. 91B28, 60H15 1.
The Role of NoArbitrage on Forecasting: Lessons from a Parametric Term Structure Model ∗
, 2008
"... Fontaine, René Garcia, and Lotfi Karoui for important comments. We also thank comments and suggestions from seminar participants at the 26th Brazilian Colloquium of Mathematics, the Sofie 2008 Conference, Getulio Vargas Foundation, HEC Montreal, and Catholic University in Rio de Janeiro. The views e ..."
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Cited by 1 (0 self)
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Fontaine, René Garcia, and Lotfi Karoui for important comments. We also thank comments and suggestions from seminar participants at the 26th Brazilian Colloquium of Mathematics, the Sofie 2008 Conference, Getulio Vargas Foundation, HEC Montreal, and Catholic University in Rio de Janeiro. The views expressed are those of the authors and do not necessarily reflect those of the Central Bank of Brazil. The first author gratefully acknowledges financial support from CNPqBrazil. Parametric term structure models have been successfully applied to innumerous problems in fixed income markets, including pricing, hedging, managing risk, as well as studying monetary policy implications. On their turn, dynamic term structure models, equipped with stronger economic structure, have been mainly adopted to price derivatives and explain empirical stylized facts. In this paper, we combine flavors of those two classes of models to test if noarbitrage affects forecasting. We construct cross section (allowing arbitrages) and arbitragefree versions of a parametric polynomial model to analyze how well they predict outofsample interest rates. Based on U.S. Treasury yield data, we find that noarbitrage restrictions significantly improve forecasts. Arbitragefree versions achieve overall smaller biases and Root Mean Square Errors for most maturities and forecasting horizons. Furthermore, a decomposition of forecasts into forwardrates and holding return premia indicates that the superior performance of noarbitrage versions is due to a better identification of bond risk premium.
Local wellposedness of Musiela’s SPDE with Lévy noise
, 2007
"... We determine sufficient conditions on the volatility coefficient of Musiela’s stochastic partial differential equation driven by an infinite dimensional Lévy process so that it admits a unique local mild solution in spaces of functions whose first derivative is square integrable with respect to a we ..."
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Cited by 1 (1 self)
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We determine sufficient conditions on the volatility coefficient of Musiela’s stochastic partial differential equation driven by an infinite dimensional Lévy process so that it admits a unique local mild solution in spaces of functions whose first derivative is square integrable with respect to a weight.