This special issue on computational and algorithmic finance showcases contemporary developments ranging from advanced numerical methods to machine learning techniques and efficient parallel implementations in finance and insurance. This, in particular, includes: calibration of various asset pricing models (local volatility, stochastic volatility, jumps) to market data; development of new approaches in constructing efficient finite difference and radial basis function methods; study of models and machine learning techniques, like Bayesian and neural networks, for asset liability management and limit order books; analysis of bond quote inconsistencies; and also implementation issues on GPU of a Monte Carlo insurance balance sheet projection. (C) 2017 Published by Elsevier B.V.
Adaptivity in space and time is introduced to control the error in the numerical solution of hyperbolic partial differential equations. The equations are discretised by a finite volume method in space and an implicit linear multistep method in time. The computational grid is refined in blocks. At the boundaries of the blocks, there may be jumps in the step size. Special treatment is needed there to ensure second order accuracy and stability. The local truncation error of the discretisation is estimated and is controlled by changing the step size and the time step. The global error is obtained by integration of the error equations. In the implicit scheme, the system of linear equations at each time step is solved iteratively by the GMRES method. Numerical examples executed on a parallel computer illustrate the method.
We develop and analyse a numerical method for solving the Ross recovery problem for a diffusion problem with unbounded support, with a transition independent pricing kernel. Asset prices are assumed to only be available on a bounded subinterval B=[−N,N]. Theoretical error bounds on the recovered pricing kernel are derived, relating the convergence rate as a function of NN to the rate of mean reversion of the diffusion process. Our suggested numerical method for finding the pricing kernel employs finite differences, and we apply Sturm–Liouville theory to make use of inverse iteration on the resulting discretized eigenvalue problem. We numerically verify the derived error bounds on a test bench of three model problems.