Mathematical Finance and Probability Seminars (Since covid these events are taking place online.)

Long dated derivatives require a flexible modelling framework. The econometrics challenge is to embed historical and cross-sectional estimations into derivative calibration. The engineering challenge is to structure a model agnostic pricing engine whose performance depends only on the model size but not on the process specification. The mathematical and numerical challenge is to understand and use the smoothing mechanisms behind diffusion equations.

We illustrate through examples an efficient framework of this sort based on direct kernel manipulations and operator algebraic methods. We find that fully explicit discretization schemes provide a robust, low-noise numerical valuation method for fundamental solutions of diffusion equations and their derivatives. Path dependent options are associated to an operator algebra and can be classified into Abelian and non-Abelian: block-diagonalizations and moment methods apply to the first and block-factorization to the latter. Direct kernel manipulations also allow one to correlate lattice models by means of dynamic conditioning across even hundreds of factors without incurring into the curse of dimensionality. Thanks to the internal smoothing mechanisms, calculations are best executed in single precision floating point arithmetics and staggering performance can be achieved by invoking BLAS Level-3 routines on massively parallel chipsets such as GPUs and the Cell BE.

Examples to be discussed include the swaption volatility cube calibration, CMSs and CMS spreads, snowballs, synthetic and bespoke CDOs, long dated equity structures and volatility derivatives. The list is long, but the model-agnostic math is just the same.

Speaker: Claudio Albanese, Level 3 Finance