A heat kernel approach is proposed for the development of a novel method for asset pricing over a finite time horizon. We work in an incomplete market setting and assume the existence of a pricing kernel that determines the prices of financial instruments. The pricing kernel is modeled by a weighted heat kernel driven by a multivariate Markov process. The heat kernel is chosen so as to provide enough freedom to ensure that the resulting model can be calibrated to appropriate data, e.g. to the initial term structure of bond prices. A class of models is presented for which 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 models presented are driven by combinations of Markovian jump processes with different probability laws. Such models provide a basis for consistent applications in various market sectors, including equity markets, fixed-income markets, commodity markets, and insurance. The flexible multidimensional and multivariate structure on which the resulting price models are based lends itself well to the modeling of dependence across asset classes. As an illustration, the impact of spiraling debt, a typical feature of a financial crisis, is modeled explicitly, and the contagion effects can be readily observed in the dynamics of the associated asset returns.