Joint Distribution of Passage Times of an Ornstein-Uhlenbeck Diffusion and Real-Time Computational Methods for Financial Sensitivities

This thesis analyses two broad problems: the computation of financial sensitivities, which is a computationally expensive exercise, and the evaluation of barriercrossing probabilities which cannot be approximated to reach a certain precision in certain circumstances. In the former case, we consider the computation of the parameter sensitivities of large portfolios and also valuation adjustments. The traditional approach to compute sensitivities is by the finite-difference approximation method, which requires an iterated implementation of the original valuation function. This leads to substantial computational costs, no matter whether the valuation was implemented via numerical partial differential equation methods or Monte Carlo simulations. However, we show that the adjoint algorithmic differentiation algorithm can be utilised to calculate these price sensitivities reliably and orders of magnitude faster compared to standard finite-difference approaches. In the latter case, we consider barrier-crossing problems of Ornstein-Uhlenbeck diffusions. Especially in the case where the barrier is difficult to reach, the problem turns into a rare event occurrence approximation problem. We prove that it cannot be estimated accurately and robustly with direct Monte Carlo methods because of the irremovable bias and Monte Carlo error. Instead, we adopt a partial differential equation method alongside the eigenfunction expansion, from which we are able to calculate the distribution and the survival functions for the maxima of a homogeneous Ornstein-Uhlenbeck process in a single interval. By the conditional independence property of Markov processes, the results can be further extended to inhomogeneous cases and multiple period barrier-crossing problems, both of which can be efficiently implemented by quadrature and Monte Carlo integration methods

Multiple barrier-crossings of an Ornstein-Uhlenbeck diffusion in consecutive periods

We investigate the joint distribution and the multivariate survival functions for the maxima of an Ornstein-Uhlenbeck (OU) process in consecutive time-intervals. A PDE method, alongside an eigenfunction expansion is adopted, with which we first calculate the distribution and the survival functions for the maximum of a homogeneous OU-process in a single interval. By a deterministic time-change and a parameter translation, this result can be extended to an inhomogeneous OU-process. Next, we derive a general formula for the joint distribution and the survival functions for the maxima of a continuous Markov process in consecutive periods. With these results, one can obtain semi-analytical expressions for the joint distribution and the multivariate survival functions for the maxima of an OU-process, with piecewise constant parameter functions, in consecutive time periods. The joint distribution and the survival functions can be evaluated numerically by an iterated quadrature scheme, which can be implemented efficiently by matrix multiplications. Moreover, we show that the computation can be further simplified to the product of single quadratures if the filtration is enlarged. Such results may be used for the modelling of heatwaves and related risk management challenges.Comment: 38 pages, 10 figures, 2 table

Exact Covering Systems in Number Fields

It is well known that in an exact covering system in $\mathbb{Z}$, the biggest modulus must be repeated. Very recently, Kim gave an analogous result for certain quadratic fields, and Kim also conjectured that it must hold in any algebraic number field. In this paper, we prove Kim's conjecture. In other words, we prove that exact covering systems in any algebraic number field must have repeated moduli.Comment: 13 page