Discrete Time Term Structure Theory and Consistent Recalibration Models

Abstract

We develop a theory and applications of forward characteristic processes in discrete time following the ideas of a seminal paper of Kallsen and Kruhner [Finance Stoch., 19 (2015), pp. 583--615]. Particular emphasis is given to the dynamics of volatility surfaces which can be readily formulated and implemented from our chosen discrete point of view. In mathematical terms we provide an algorithmic answer to the following question: describe a rich, still tractable class of discrete time stochastic processes, whose marginal distributions are given at an initial time and which are free of arbitrage. In terms of mathematical finance we can construct models with predescribed (implied) volatility surface at initial time and a quite general volatility surface dynamics along time. In terms of the works of Carmona and Nadtochiy [Finance Stoch., 13 (2009), pp. 1--48; Finance Stoch., 16 (2012), pp. 63--104] we analyze the dynamics of tangent affine models. We believe that the discrete approach due to its technical sim...