In the last decade the role of market incompleteness has been widely recognized in mathematical finance. Important risk factors, such as volatility risk or gap risk caused by sudden jumps of asset prices must not be neglected, but pose challenges to modeling, pricing and hedging. Accordingly, recent mathematical research has focused on describing classes of models, which include these important risk factors. From a practical point of view, computational tractability of models, their ability to be calibrated to market data, and the ease of implementation of hedging strategies are equally important. In the realm of equity and interest rates, models based on affine processes, in particular affine stochastic volatility models (ASVMs) and affine term structure models (ATSMs), have emerged as a class of tractable models that are able to accommodate stochastic volatility, jumps in asset prices, time-varying intensity of jump risk, as well as self-excitement and cross-excitement phenomena between economic factors.
Current research focuses on a systematic analysis and description of affine models as well as on numerical methods to treat them. In many cases the tractable structure of affine processes leads to considerable simplifications and more explicit results, when theory developed for general semi-martingales is applied to them. Methods for statistical estimation, and filtering that are specific to affine processes are currently developed. Affine models are also useful for multivariate modeling, based on the recently introduced class of positive semi-definite matrix-valued affine processes which provide dynamic stochastic models for covariance structures.
- Keller-Ressel, M. (2011), 'Moment explosions and long-term behavior of affine stochastic volatility models', Mathematical Finance 21(1), 73-98.
- Keller-Ressel, M.; Papapantoleon, A. & Teichmann, J. (2009), 'The Affine LIBOR models', arXiv:0904.0555.
- Keller-Ressel, M.; Schachermayer, W. & Teichmann, J. (2011), 'Affine Processes are Regular', Probability Theory and Related Fields 151(3-4), 591-611.
- Keller-Ressel, M. & Steiner, T. (2008), 'Yield Curve Shapes and the Asymptotic Short Rate Distribution in Affine One-Factor Models', Finance and Stochastics 12(2), 149-172.