Abstracts

We present a model for the market of covariance swaps with a particular emphasis on calibration and (consistent) re-calibration. For this we start with a particularly tractable class of time-homogeneous Markov processes and introduce a dynamic version of the Hull-White extension such that consistent re-calibration is possible without losing too much of the tractability of the model.
In more detail, we parametrize the market by means of a stochastic volatility model such that the log-prices and the instantaneous spot-covariation are jointly polynomial. Polynomial processes were recently introduced as time-homogeneous Markov processes with the property that the computation of moments reduces to the computation of matrix-exponentials. This class generalizes the widely used class of affine processes and contain popular non-affine models such as the Pearson diffusion and (jump-) diffusion limits of GARCH models. We slightly generalize this class to processes which takes their values in the cone of positive semi-definite matrices. The corresponding forward co-variation processes inherit the polynomial property and are thus tailor-made for calibration. However, a perfect fit to an arbitrary initial matrix valued curve of forward covariations is not to be expected.
Similar problems arise within the interest rates world, where affine diffusion models for the short-rate lack the ability to fit an arbitrary initial forward curve. Here the popular notion of Hull-White extensions provide a somewhat minimal extension towards a time-inhomogeneous affine diffusion such that this lack is fixed. However, as for re-calibration it is not obvious how to apply this notion repeatedly in a consistent way.
We build on a model which was recently introduced for the forward characteristics of affine processes in discrete time and apply it to our setting. For this we discretize the above mentioned model and provide the necessary extension such that consistent re-calibration is possible. The model is realized as a weak solution of a (discrete time) SPDE. Finally we consider the continuous time limit of this model and provide an algorithm for calibration and simulation. We discuss some applications which include risk management, portfolio optimization and pricing of relevant derivatives.