31,649 research outputs found
Can You hear the Shape of a Market? Geometric Arbitrage and Spectral Theory
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 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
Interest Rates and Information Geometry
The space of probability distributions on a given sample space possesses
natural geometric properties. For example, in the case of a smooth parametric
family of probability distributions on the real line, the parameter space has a
Riemannian structure induced by the embedding of the family into the Hilbert
space of square-integrable functions, and is characterised by the Fisher-Rao
metric. In the nonparametric case the relevant geometry is determined by the
spherical distance function of Bhattacharyya. In the context of term structure
modelling, we show that minus the derivative of the discount function with
respect to the maturity date gives rise to a probability density. This follows
as a consequence of the positivity of interest rates. Therefore, by mapping the
density functions associated with a given family of term structures to Hilbert
space, the resulting metrical geometry can be used to analyse the relationship
of yield curves to one another. We show that the general arbitrage-free yield
curve dynamics can be represented as a process taking values in the convex
space of smooth density functions on the positive real line. It follows that
the theory of interest rate dynamics can be represented by a class of processes
in Hilbert space. We also derive the dynamics for the central moments
associated with the distribution determined by the yield curve.Comment: 20 pages, 3 figure
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