A reduced basis method for parabolic partial differential equations with parameter functions and application to option pricing

Mayerhofer, Antonia; Urban, Karsten

doi:10.21314/jcf.2016.323

Cited by 6 publications

(5 citation statements)

References 37 publications

(46 reference statements)

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“…The POD with Galerkin projection is the most known and frequently used reduced-order modeling method. POD based reduced-order modeling is investigated for pricing European option and American option wherein different from European option there exist variational inequalities [5,30,31] to be handled, under Black-Scholes and Heston model [2,4,20,21]. Reduced-order models based on POD are optimal in terms of energy content.…”

Section: Introductionmentioning

confidence: 99%

Reduced-Order Modeling for Heston Stochastic Volatility Model

KOZPINAR,

UZUNCA,

KARASÖZEN

2024

Hacettepe Journal of Mathematics and Statistics

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In this paper, we compare the intrusive proper orthogonal decomposition (POD) with Galerkin projection and the data-driven dynamic mode decomposition (DMD), for Heston's option pricing model. The full order model is obtained by discontinuous Galerkin discretization in space and backward Euler in time. Numerical results for butterfly spread, European and digital call options reveal that in general DMD requires more modes than the POD modes for the same level of accuracy. However, the speed-up factors are much higher for DMD than POD due to the non-intrusive nature of the DMD.

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Section: Introductionmentioning

confidence: 99%

Reduced-Order Modeling for Heston Stochastic Volatility Model

KOZPINAR,

UZUNCA,

KARASÖZEN

2024

Hacettepe Journal of Mathematics and Statistics

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“…First results for RBM can be found in [16,48,49]. In [43], RBM was applied to more complex models, e.g., with parameter functions as an initial condition. While these references focus on the simplest case of European options, American options, which are described by parabolic variational inequalites, are considered in [12,10,5].…”

Section: Introductionmentioning

confidence: 99%

Complexity reduction for calibration to American options

Burkovska¹,

Glau²,

Mahlstedt³

et al. 2019

JCF

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American put options are among the most frequently traded single stock options, and their calibration is computationally challenging since no closed-form expression is available. Due to the higher flexibility in comparison to European options, the mathematical model involves additional constraints, and a variational inequality is obtained. We use the Heston stochastic volatility model to describe the price of a single stock option. In order to speed up the calibration process, we apply two model reduction strategies. Firstly, a reduced basis method (RBM) is used to define a suitable low-dimensional basis for the numerical approximation of the parameter-dependent partial differential equation (µPDE) model. By doing so the computational complexity for solving the µPDE is drastically reduced, and applications of standard minimization algorithms for the calibration are significantly faster than working with a high-dimensional finite element basis. Secondly, so-called de-Americanization strategies are applied. Here, the main idea is to reformulate the calibration problem for American options as a problem for European options and to exploit closed-form solutions. Both reduction techniques are systematically compared and tested for both synthetic and market data sets.

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“…For instance the field of reduced basis methods to efficiently solve parametric partial differential equations (PDEs) has experienced a tremendous development over the last decade, see, e.g., [23,40,41] and the references therein. Pioneered by [43,10] the potential of reduced basis methods is also increasingly exploited for problems in finance; see [8,37,7] for examples. These methods can be viewed as high-dimensional interpolation methods that are trained in an offline step to solve a specific class of parametric PDEs.…”

Section: Introductionmentioning

confidence: 99%

Low-rank tensor approximation for Chebyshev interpolation in parametric option pricing

Glau¹,

Kreßner²,

Statti³

2019

Preprint

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Treating high dimensionality is one of the main challenges in the development of computational methods for solving problems arising in finance, where tasks such as pricing, calibration, and risk assessment need to be performed accurately and in real-time. Among the growing literature addressing this problem, Gass et al. [14] propose a complexity reduction technique for parametric option pricing based on Chebyshev interpolation. As the number of parameters increases, however, this method is affected by the curse of dimensionality. In this article, we extend this approach to treat high-dimensional problems: Additionally exploiting low-rank structures allows us to consider parameter spaces of high dimensions. The core of our method is to express the tensorized interpolation in tensor train (TT) format and to develop an efficient way, based on tensor completion, to approximate the interpolation coefficients. We apply the new method to two model problems: American option pricing in the Heston model and European basket option pricing in the multi-dimensional Black-Scholes model. In these examples we treat parameter spaces of dimensions up to 25. The numerical results confirm the low-rank structure of these problems and the effectiveness of our method compared to advanced techniques.

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A reduced basis method for parabolic partial differential equations with parameter functions and application to option pricing

Cited by 6 publications

References 37 publications

Reduced-Order Modeling for Heston Stochastic Volatility Model

Reduced-Order Modeling for Heston Stochastic Volatility Model

Complexity reduction for calibration to American options

Low-rank tensor approximation for Chebyshev interpolation in parametric option pricing

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