This paper presents an algorithm for a complete and efficient calibration of the Heston stochastic volatility model. We express the calibration as a nonlinear least squares problem. We exploit a suitable representation of the Heston characteristic function and modify it to avoid discontinuities caused by branch switchings of complex functions. Using this representation, we obtain the analytical gradient of the price of a vanilla option with respect to the model parameters, which is the key element of all variants of the objective function. The interdependency between the components of the gradient enables an efficient implementation which is around ten times faster than a numerical gradient. We choose the Levenberg-Marquardt method to calibrate the model and do not observe multiple local minima reported in previous research. Two-dimensional sections show that the objective function is shaped as a narrow valley with a flat bottom. Our method is the fastest calibration of the Heston model developed so far and meets the speed requirement of practical trading.

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Let S (respectively, S′) be a finite subset of a compact connected Riemann surface X (respectively, X′) of genus at least two. Let [Mscr ] (respectively, [Mscr ]′) denote a moduli space of parabolic stable bundles of rank 2 over X (respectively, X′) with fixed determinant of degree 1, having a nontrivial quasi-parabolic structure over each point of S (respectively, S′) and of parabolic degree less than 2. It is proved that [Mscr ] is isomorphic to [Mscr ]′ if and only if there is an isomorphism of X with X′ taking S to S′.

The systematic trading of equities forms the basis of the asset management industry. Analysts are trying to outperform a passive investment in an index such as the S&P 500 Index. However, statistics have shown that most analysts fail to consistently beat the index. A number of Neural Network based methods for detecting trading opportunities on Futures contracts on the S&P 500 Index have been published in the literature. However, such methods have generally been unable to demonstrate sustained performance over a significant period of time. The authors of this paper show, through the application of over ten years of experience in quantitative modelling and trading, a different type of Neural Network approach to beating the S&P 500 Index. Rather than trading Futures contracts, it is shown that by using Neural Networks to intelligently select just a handful of stocks a performance significantly in excess of a buy and hold position on the S&P 500 Index could have been achieved over a seven year period. The effect of transaction costs is also considered.

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