Calibrating Option Pricing Models with Heuristics

Calibrating option pricing models to market prices often leads to optimisation problems to which standard methods (like such based on gradients) cannot be applied. We investigate two models: Heston’s stochastic volatility model, and Bates’s model which also includes jumps. We discuss how to price options under these models, and how to calibrate the parameters of the models with heuristic techniques.

Accuracy and Stability of Computing High-Order Derivatives of Analytic Functions by Cauchy Integrals

High-order derivatives of analytic functions are expressible as Cauchy integrals over circular contours, which can very effectively be approximated, e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius of convergence is equal, numerical stability strongly depends on r. We give a comprehensive study of this effect; in particular we show that there is a unique radius that minimizes the loss of accuracy caused by round-off errors. For large classes of functions, though not for all, this radius actually gives about full accuracy; a remarkable fact that we explain by the theory of Hardy spaces, by the Wiman-Valiron and Levin-Pfluger theory of entire functions, and by the saddle-point method of asymptotic analysis. Many examples and non-trivial applications are discussed in detail.Comment: Version 4 has some references and a discussion of other quadrature rules added; 57 pages, 7 figures, 6 tables; to appear in Found. Comput. Mat

Computing Classical Power Indices For Large Finite Voting Games.

Voting Power Indices enable the analysis of the distribution of power in a legislature or voting body in which different members have different numbers of votes. Although this approach to the measurement of power, based on co-operative game theory, has been known for a long time its empirical application has been to some extent limited, in part by the difficulty of computing the indices when there are many players. This paper presents new algorithms for computing the classical power indices, those of Shapley and Shubik (1954) and of Banzhaf (1963), which are essentially modifications of approximation methods due to Owen, and have been shown to work well in real applications.VOTING ; INDEXES ; GAMES

Fast, reliable and unrestricted iterative computation of Gauss-Hermite and Gauss-Laguerre quadratures

Methods for the computation of classical Gaussian quadrature rules are described which are effective both for small and large degree. These methods are reliable because the iterative computation of the nodes has guaranteed convergence, and they are fast due to their fourth-order convergence and its asymptotic exactness for an appropriate selection of the variables. For Gauss?Hermite and Gauss?Laguerre quadratures, local Taylor series can be used for computing efficiently the orthogonal polynomials involved, with exact initial values for the Hermite case and first values computed with a continued fraction for the Laguerre case. The resulting algorithms have almost unrestricted validity with respect to the parameters. Full relative precision is reached for the Hermite nodes, without any accuracy loss and for any degree, and a mild accuracy loss occurs for the Hermite and Laguerre weights as well as for the Laguerre nodes. These fast methods are exclusively based on convergent processes, which, together with the high order of convergence of the underlying iterative method, makes them particularly useful for high accuracy computations. We show examples of very high accuracy computations (of up to 1000 digits of accuracy)

Just SLaQ When You Approximate: Accurate Spectral Distances for Web-Scale Graphs

Graph comparison is a fundamental operation in data mining and information retrieval. Due to the combinatorial nature of graphs, it is hard to balance the expressiveness of the similarity measure and its scalability. Spectral analysis provides quintessential tools for studying the multi-scale structure of graphs and is a well-suited foundation for reasoning about differences between graphs. However, computing full spectrum of large graphs is computationally prohibitive; thus, spectral graph comparison methods often rely on rough approximation techniques with weak error guarantees. In this work, we propose SLaQ, an efficient and effective approximation technique for computing spectral distances between graphs with billions of nodes and edges. We derive the corresponding error bounds and demonstrate that accurate computation is possible in time linear in the number of graph edges. In a thorough experimental evaluation, we show that SLaQ outperforms existing methods, oftentimes by several orders of magnitude in approximation accuracy, and maintains comparable performance, allowing to compare million-scale graphs in a matter of minutes on a single machine.Comment: To appear at TheWebConf (WWW) 202