Real world financial data is often discontinuous and non-smooth. If we attempt to use neural networks to simulate such functions, then accuracy will be a problem. Neural network group models perform this function much better. Both Polynomial Higher Order Neural network Group (PHONG) and Trigonometric polynomial Higher Order Neural network Group (THONG) models are developed. These HONG models are open box, convergent models capable of approximating any kind of piecewise continuous function, to any degree of accuracy. Moreover they are capable of handling higher frequency, higher order non-linear and discontinuous data. Results obtained using a Higher Order Neural network Group financial simulator are presented, which confirm that HONG group models converge without difficulty, and are considerably more accurate than neural network models (more specifically, around twice as good for prediction, and a factor of four improvement in the case of simulation).