The Experts below are selected from a list of 207 Experts worldwide ranked by ideXlab platform

Simone Farinelli - One of the best experts on this subject based on the ideXlab platform.

  • when risks and uncertainties collide mathematical finance for arbitrage markets in a quantum mechanical view
    arXiv: Risk Management, 2019
    Co-Authors: Simone Farinelli, Hideyuki Takada
    Abstract:

    Geometric Arbitrage Theory reformulates a generic asset model possibly allowing for arbitrage by packaging all assets and their forwards dynamics into a stochastic Principal Fibre Bundle, with a connection whose parallel transport encodes discounting and portfolio rebalancing, and whose curvature measures, in this geometric language, the ''instantaneous arbitrage capability'' generated by the market itself. The asset and market portfolio dynamics have a quantum mechanical description, which is constructed by quantizing the deterministic version of the stochastic Lagrangian system describing a market allowing for arbitrage. Results, obtained by solving explicitly the Schr\"odinger equations by means of spectral decomposition of the Hamilton operator, coincides with those obtained by solving the stochastic Euler Lagrange equations derived by a variational principle and providing therefore consistency. Arbitrage bubbles are computed.

  • Credit Risk in a Geometric Arbitrage Perspective
    arXiv: Pricing of Securities, 2015
    Co-Authors: Simone Farinelli
    Abstract:

    Geometric Arbitrage Theory, where a generic market is modelled with a Principal Fibre Bundle and arbitrage corresponds to its curvature, is applied to credit markets to model default risk and recovery, leading to closed form no arbitrage characterizations for corporate bonds.

  • Can You hear the Shape of a Market? Geometric Arbitrage and Spectral Theory
    SSRN Electronic Journal, 2015
    Co-Authors: Simone Farinelli, Hideyuki Takada
    Abstract:

    Geometric Arbitrage Theory reformulates a generic asset model possibly allowing for arbitrage by packaging all assets and their forwards dynamics into a stochastic Principal Fibre Bundle, with a connection whose parallel transport encodes discounting and portfolio rebalancing, and whose curvature measures, in this geometric language, the ''instantaneous arbitrage capability'' generated by the market itself. The cashflow Bundle is the vector Bundle associated to this stochastic Principal Fibre Bundle for the natural choice of the vector space Fibre. The cashflow Bundle carries a stochastic covariant differentiation induced by the connection on the Principal Fibre Bundle. The link between arbitrage theory and spectral theory of the connection Laplacian on the vector Bundle is given by the zero eigenspace resulting in a parametrization of all risk neutral measures equivalent to the statistical one. This indicates that a market satisfies the (NFLVR) condition if and only if $0$ is in the discrete spectrum of the connection Laplacian on the cash flow Bundle or of the Dirac Laplacian of the twisted cash flow Bundle with the exterior algebra Bundle. We apply this result by extending Jarrow-Protter-Shimbo theory of asset bubbles for complete arbitrage free markets to markets not satisfying the (NFLVR). Moreover, by means of the Atiyah-Singer index theorem, we prove that the Euler characteristic of the asset nominal space is a topological obstruction to the the (NFLVR) condition, and, by means of the Bochner-Weitzenb\ock formula, the non vanishing of the homology group of the cash flow Bundle is revealed to be a topological obstruction to (NFLVR), too. Asset bubbles are defined, classified and decomposed for markets allowing arbitrage.

  • Geometric Arbitrage Theory and Calibration of a Generator of Consistent Economic Scenarios
    SSRN Electronic Journal, 2008
    Co-Authors: Simone Farinelli
    Abstract:

    On the theoretical side: we introduced the differential geometric framework to translate any market model into a Principal Fibre Bundle allowing to interpret arbitrage as curvature, parameterizing arbitrage opportunities with the Lie Algebra of the holonomy group. The no arbitrage condition is equivalent to a continuity equation. On the practical side: we propose a methodology to simulate the future evolution of asset values such that: - the dimension of risk factors can be reduced. - the no arbitrage condition is satisfied. - the simulated moments of asset returns match the empirical ones.

Sylvie Paycha - One of the best experts on this subject based on the ideXlab platform.

  • Elliptic operators in the functional quantisation for gauge field theories
    Communications in Mathematical Physics, 1995
    Co-Authors: Sylvie Paycha
    Abstract:

    Given a gauge theory with gauge group G acting on a path space X , G and X being both infinite dimensional manifolds modelled on spaces of sections of vector Bundles on a compact riemannian manifold without boundary, it is shown that when the action of G on X is smooth, free and proper, the same ellipticity condition on an operator naturally given by the geometry of the problem yields both the existence of a Principal Fibre Bundle structure induced by the canonical projection π: X → X/G and the existence of the Faddeev-Popov determinant arising in the functional quantisation of the gauge theory. This holds for certain gauge theories with anomalies like bosonic closed string theory in non-critical dimension and also holds for a class of gauge theories which includes Yang-Mills theory.

Hideyuki Takada - One of the best experts on this subject based on the ideXlab platform.

  • when risks and uncertainties collide mathematical finance for arbitrage markets in a quantum mechanical view
    arXiv: Risk Management, 2019
    Co-Authors: Simone Farinelli, Hideyuki Takada
    Abstract:

    Geometric Arbitrage Theory reformulates a generic asset model possibly allowing for arbitrage by packaging all assets and their forwards dynamics into a stochastic Principal Fibre Bundle, with a connection whose parallel transport encodes discounting and portfolio rebalancing, and whose curvature measures, in this geometric language, the ''instantaneous arbitrage capability'' generated by the market itself. The asset and market portfolio dynamics have a quantum mechanical description, which is constructed by quantizing the deterministic version of the stochastic Lagrangian system describing a market allowing for arbitrage. Results, obtained by solving explicitly the Schr\"odinger equations by means of spectral decomposition of the Hamilton operator, coincides with those obtained by solving the stochastic Euler Lagrange equations derived by a variational principle and providing therefore consistency. Arbitrage bubbles are computed.

  • Can You hear the Shape of a Market? Geometric Arbitrage and Spectral Theory
    SSRN Electronic Journal, 2015
    Co-Authors: Simone Farinelli, Hideyuki Takada
    Abstract:

    Geometric Arbitrage Theory reformulates a generic asset model possibly allowing for arbitrage by packaging all assets and their forwards dynamics into a stochastic Principal Fibre Bundle, with a connection whose parallel transport encodes discounting and portfolio rebalancing, and whose curvature measures, in this geometric language, the ''instantaneous arbitrage capability'' generated by the market itself. The cashflow Bundle is the vector Bundle associated to this stochastic Principal Fibre Bundle for the natural choice of the vector space Fibre. The cashflow Bundle carries a stochastic covariant differentiation induced by the connection on the Principal Fibre Bundle. The link between arbitrage theory and spectral theory of the connection Laplacian on the vector Bundle is given by the zero eigenspace resulting in a parametrization of all risk neutral measures equivalent to the statistical one. This indicates that a market satisfies the (NFLVR) condition if and only if $0$ is in the discrete spectrum of the connection Laplacian on the cash flow Bundle or of the Dirac Laplacian of the twisted cash flow Bundle with the exterior algebra Bundle. We apply this result by extending Jarrow-Protter-Shimbo theory of asset bubbles for complete arbitrage free markets to markets not satisfying the (NFLVR). Moreover, by means of the Atiyah-Singer index theorem, we prove that the Euler characteristic of the asset nominal space is a topological obstruction to the the (NFLVR) condition, and, by means of the Bochner-Weitzenb\ock formula, the non vanishing of the homology group of the cash flow Bundle is revealed to be a topological obstruction to (NFLVR), too. Asset bubbles are defined, classified and decomposed for markets allowing arbitrage.

Farinelli Simone - One of the best experts on this subject based on the ideXlab platform.

  • Geometric Arbitrage Theory and Market Dynamics Reloaded
    2020
    Co-Authors: Farinelli Simone
    Abstract:

    We have embedded the classical theory of stochastic finance into a differential geometric framework called Geometric Arbitrage Theory and show that it is possible to: --Write arbitrage as curvature of a Principal Fibre Bundle. --Parameterize arbitrage strategies by its holonomy. --Give the Fundamental Theorem of Asset Pricing a differential homotopic characterization. --Characterize Geometric Arbitrage Theory by five principles and show they they are consistent with the classical theory of stochastic finance. --Derive for a closed market the equilibrium solution for market portfolio and dynamics in the cases where: -->Arbitrage is allowed but minimized. -->Arbitrage is not allowed. --Prove that the no-free-lunch-with-vanishing-risk condition implies the zero curvature condition. The converse is in general not true and additionally requires the Novikov condition for the instantaneous Sharpe Ratio Dynamics to be satisfied.Comment: This paper is essentially a new version of [Fa15], where some flaws have been amende

  • Can You hear the Shape of a Market? Geometric Arbitrage and Spectral Theory
    2020
    Co-Authors: Farinelli Simone, Takada Hideyuki
    Abstract:

    Geometric Arbitrage Theory reformulates a generic asset model possibly allowing for arbitrage by packaging all assets and their forwards dynamics into a stochastic Principal Fibre Bundle, with a connection whose parallel transport encodes discounting and portfolio rebalancing, and whose curvature measures, in this geometric language, the 'instantaneous arbitrage capability' generated by the market itself. The cashflow Bundle is the vector Bundle associated to this stochastic Principal Fibre Bundle for the natural choice of the vector space Fibre. The cashflow Bundle carries a stochastic covariant differentiation induced by the connection on the Principal Fibre Bundle. The link between arbitrage theory and spectral theory of the connection Laplacian on the vector Bundle is given by the zero eigenspace resulting in a parametrization of all risk neutral measures equivalent to the statistical one. This indicates that a market satisfies the (NFLVR) condition if and only if $0$ is in the discrete spectrum of the connection Laplacian on the cash flow Bundle or of the Dirac Laplacian of the twisted cash flow Bundle with the exterior algebra Bundle. We apply this result by extending Jarrow-Protter-Shimbo theory of asset bubbles for complete arbitrage free markets to markets not satisfying the (NFLVR). Moreover, by means of the Atiyah-Singer index theorem, we prove that the Euler characteristic of the asset nominal space is a topological obstruction to the the (NFLVR) condition, and, by means of the Bochner-Weitzenb\"ock formula, the non vanishing of the homology group of the cash flow Bundle is revealed to be a topological obstruction to (NFLVR), too. Asset bubbles are defined, classified and decomposed for markets allowing arbitrage.Comment: arXiv admin note: substantial text overlap with arXiv:1406.6805, arXiv:0910.167

Stefan Waldmann - One of the best experts on this subject based on the ideXlab platform.

  • Deformation quantization of surjective submersions and Principal Fibre Bundles
    Journal für die reine und angewandte Mathematik (Crelles Journal), 2010
    Co-Authors: Martin Bordemann, Nikolai Neumaier, Stefan Waldmann, Stefan Weiss
    Abstract:

    In this paper we establish a notion of deformation quantization of a surjective submersion which is specialized further to the case of a Principal Fibre Bundle: the functions on the total space are deformed into a right module for the star product algebra of the functions on the base manifold. In case of a Principal Fibre Bundle we require in addition invariance under the Principal action. We prove existence and uniqueness of such deformations. The commutant within all differential operators on the total space is computed and gives a deformation of the algebra of vertical differential operators. Applications to noncommutative gauge field theories and phase space reduction of star products are discussed.

  • Noncommutative Field Theories from a Deformation Point of View
    Quantum Field Theory, 2009
    Co-Authors: Stefan Waldmann
    Abstract:

    In this review we discuss the global geometry of noncommutative field theories from a deformation point of view: The space-times under consideration are deformations of classical space-time manifolds using star products. Then matter fields are encoded in deformation quantizations of vector Bundles over the classical space-time. For gauge theories we establish a notion of deformation quantization of a Principal Fibre Bundle and show how the deformation of associated vector Bundles can be obtained.