  A heat kernel approach is proposed for the development of a general,
flexible, and mathematically tractable asset pricing framework in finite time.
The pricing kernel, giving rise to the price system in an incomplete market, is
modelled by weighted heat kernels which are driven by multivariate Markov
processes and which provide enough degrees of freedom in order to calibrate to
relevant data, e.g. to the term structure of bond prices. It is shown how, for
a class of models, the prices of bonds, caplets, and swaptions can be computed
in closed form. The dynamical equations for the price processes are derived,
and explicit formulae are obtained for the short rate of interest, the risk
premium, and for the stochastic volatility of prices. Several of the
closed-form asset price models presented in this paper are driven by
combinations of Markovian jump processes with different probability laws. Such
models provide a rich basis for consistent applications in several sectors of a
financial market including equity, fixed-income, commodities, and insurance.
The flexible, multidimensional and multivariate structure, on which the asset
price models are constructed, lends itself well to the transparent modelling of
dependence across asset classes. As an illustration, the impact on prices by
spiralling debt, a typical feature of a financial crisis, is modelled
explicitly, and contagion effects are readily observed in the dynamics of asset
returns.
