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
