A Feynman–Kac-type formula for Lévy processes with discontinuous killing rates

Glau, Kathrin

doi:10.1007/s00780-016-0301-7

Cited by 20 publications

(28 citation statements)

References 47 publications

(71 reference statements)

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“…We give a short list of examples below. For further examples we refer to Glau (2016a) and Glau (2016b).…”

Section: The Symbol Methodsmentioning

confidence: 99%

“…These have proven to apply to a large set of option types and models. We refer to Eberlein and Glau (2014) and Glau (2016b), where particularly Feynman-Kac type formulas have been derived linking European and path-dependent options to weak solutions of Kolmogorov equations in Sobolev-Slobodeckii spaces.…”

Section: Solution Spacesmentioning

confidence: 99%

“…This well-known identity has already proven highly beneficial for the analysis of the variational solutions of the Komogorov equations, compare e.g. Hilber et al (2013), Glau (2016a) and Glau (2016b). Here we exploit this representation for the numerical implementation.…”

Section: Fourier Approach To the Kolmogorov Equationmentioning

confidence: 99%

“…In academia, a series of publications by Cont and Voltchkova (2005b), Hilber, Reich, Schwab and Winter (2009), Salmi, Toivanen and Sydow (2014), Itkin (2015) and Glau (2016b) and the monograph of Hilber, Reichmann, Schwab and Winter (2013) have opened the theory to include even more sophisticated models of Lévy type, resulting in Partial Integro Differential Equations (PIDEs). The theoretical results have been validated by sophisticated numerical studies.…”

Section: Introductionmentioning

confidence: 99%

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A Flexible Galerkin Scheme for Option Pricing in Lévy Models

Gaß¹,

Glau²

2018

SIAM J. Finan. Math.

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One popular approach to option pricing in Lévy models is through solving the related partial integro differential equation (PIDE). For the numerical solution of such equations powerful Galerkin methods have been put forward e.g. by Hilber et al. (2013). As in practice large classes of models are maintained simultaneously, flexibility in the driving Lévy model is crucial for the implementation of these powerful tools. In this article we provide such a flexible finite element Galerkin method. To this end we exploit the Fourier representation of the infinitesimal generator, i.e. the related symbol, which is explicitly available for the most relevant Lévy models. Empirical studies for the Merton, NIG and CGMY model confirm the numerical feasibility of the method.

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“…We give a short list of examples below. For further examples we refer to Glau (2016a) and Glau (2016b).…”

Section: The Symbol Methodsmentioning

confidence: 99%

Section: Solution Spacesmentioning

confidence: 99%

Section: Fourier Approach To the Kolmogorov Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Flexible Galerkin Scheme for Option Pricing in Lévy Models

Gaß¹,

Glau²

2018

SIAM J. Finan. Math.

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“…, M . Thus, taking into account the error stemming from truncating the integration domain to Ω, (20) induces an approximation of the option price by a linear combination of snapshot prices,…”

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confidence: 99%

Magic Points in Finance: Empirical Integration for Parametric Option Pricing

Gaß¹,

Glau²,

Mair³

2017

SIAM J. Finan. Math.

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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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Chebyshev interpolation for parametric option pricing

et al. 2018

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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.

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A Feynman–Kac-type formula for Lévy processes with discontinuous killing rates

Abstract: The challenge to fruitfully merge state-of-the-art tech-

Cited by 20 publications

References 47 publications

A Flexible Galerkin Scheme for Option Pricing in Lévy Models

A Flexible Galerkin Scheme for Option Pricing in Lévy Models

Magic Points in Finance: Empirical Integration for Parametric Option Pricing

Chebyshev interpolation for parametric option pricing

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