Recurrent tasks such as pricing, calibration and risk assessment need to be executed accurately and in real time. We concentrate on parametric option pricing (POP) as a generic instance of parametric conditional expectations and show that polynomial interpolation in the parameter space promises to considerably reduce runtimes while maintaining accuracy. The attractive properties of Chebyshev interpolation and its tensorized extension enable us to identify broadly applicable criteria for (sub)exponential convergence and explicit error bounds. The method is most promising when the computation of the prices is most challenging. We therefore investigate its combination with Monte Carlo simulation and analyze the effect of (stochastic) approximations of the interpolation. For a wide and important range of problems, the Chebyshev method turns out to be more efficient than parametric multilevel Monte Carlo. We conclude with a numerical efficiency study.

A recurring theme in the mathematical social sciences is how to select the "most desirable" elements given a binary dominance relation on a set of alternatives. Schwartz's tournament equilibrium set (TEQ) ranks among the most intriguing, but also among the most enigmatic, tournament solutions that have been proposed so far in this context. Due to its unwieldy recursive definition, little is known about TEQ. In particular, its monotonicity remains an open problem up to date. Yet, if TEQ were to satisfy monotonicity, it would be a very attractive tournament solution concept refining both the Banks set and Dutta's minimal covering set. We show that the problem of deciding whether a given alternative is contained in TEQ is NP-hard.

Abstract. We propose an offline-online procedure for Fourier transform based option pricing. The method supports the acceleration of such essential tasks of mathematical finance as model calibration, realtime pricing, and, more generally, risk assessment and parameter risk estimation. We adapt the empirical magic point interpolation method of Barrault et al. [C. R. Math. Acad. Sci. Paris, 339 (2004), pp. 667-672] to parametric Fourier pricing. In the offline phase, a quadrature rule is tailored to the family of integrands of the parametric pricing problem. In the online phase, the quadrature rule then yields fast and accurate approximations of the option prices. Under analyticity assumptions, the pricing error decays exponentially. Numerical experiments in one dimension confirm our theoretical findings and show a significant gain in efficiency, even for examples beyond the scope of the theoretical results.Key words. parametric integration, Fourier pricing, magic point interpolation, empirical interpolation, offlineonline decomposition, calibration, affine processes, Fourier transform, sparse integration AMS subject classifications. 68Q25, 68R10, 65T99, 91G60DOI. 10.1137/16M11013011. Introduction. Most of the option pricing methods based on Fourier transforms are aimed at the evaluation of individual option prices. For this scenario case, existing pricing tools have achieved impressive performance. For real-time applications and those involving repeated evaluations, particularly fast run-times are crucial. Therefore, Fast Fourier transforms (FFTs) have become highly popular in order to reduce computational complexity when prices are required simultaneously for a large set of different strikes, following the seminal works of [8] and [36]. See also the monograph [5]. In this paper we shift the focus from the pricing problem for one strike or several strikes to the full parametric option pricing problem, considering all parameters such as strike, maturity, and model parameters.For a given model and option type, various applications require the evaluation of Fourier pricing routines repeatedly for different parameter constellations. We mention three of those applications: First, during calibration of financial models with Fourier methods, the optimization relies on multiple evaluations of a Fourier integral for varying model and option parameters. In particular, this is the case in modern approaches to model calibration that take into account parameter uncertainty such as are proposed in [25] and [22], as well as in [30] where, additionally, a consistent variation of parameters over time is considered. Second, each (intra-)day recalibration leads to new model parameters for which several option prices and

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