Magic Points in Finance: Empirical Integration for Parametric Option Pricing

Abstract

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

Gaussian process regression for derivative portfolio modeling and application to credit valuation adjustment computations

“…GPs can cope with noisy data but they are also interpolating in the noise-free limit. As opposed to Chebyshev interpolation, which uses a deterministic node location imposed by the scheme (in conjunction with suitable interpolation weights, see Gaß et al (2017)), GPs can use an arbitrary, possibly unstructured (e.g. stochastically simulated) grid of observations.…”

Chebyshev interpolation for parametric option pricing

“…for a large set of strikes of European plain vanillas. A method that tailors Fourier pricing to the whole parametric family of integrands has recently been developed in [17]. On the other hand, numerical experiments have shown a promising gain in efficiency of reduced basis methods when an accurate PDE solver is readily available.…”

Strong convergence rates for markovian representations of fractional processes

“…This can be verified by substituting k = ǫ −s in Table 1. Using exponentially converging quadrature rules such as Chebychev [22,21], one could at best hope to reduce the complexity of our method from kǫ −1/r down to k log ǫ −1 . In the important special case k = ǫ −s with s = 1/H, this would result in exactly the same complexity ǫ −1/H log ǫ −1 as the hybrid scheme [12] and the circulant embedding method [18].…”