SINH-acceleration for B-spline projection with Option Pricing Applications

Boyarchenko, Svetlana; Levendorskiu{i}, Sergei; Kirkby, Justin; Cui, Zhenyu

doi:10.48550/arxiv.2109.08738

Cited by 5 publications

(11 citation statements)

References 54 publications

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“…Lewis-Lipton formula is the standard Fourier inversion formula with the prefixed line of integration; the choice of the line is non-optimal in almost all cases and increases the CPU time. See [14,16,8,24,39,29,30,17,20] for details and numerical examples.…”

Section: Discussionmentioning

confidence: 99%

“…In many cases of interest, the integrand decays slowly at infinity, and a very large number of terms of the truncated sum (simplified trapezoid rule) is needed to satisfy even a moderate error tolerance. However, in the case of standard European options, and in the case of piece-wise polynomial approximations of complicated payoffs [7,24,40], Ĝ is meromorphic with a finite number of simple poles; in [20], approximations with infinite number of poles appear. If X is SINH-regular of order (ν ′ , ν) with ν ′ > 0, one can use an appropriate conformal deformation and the corresponding change of variables to reduce calculations to the case of an integrand which is analytic in a strip around the line of integration and decays at infinity faster than exponentially.…”

Section: Discussionmentioning

confidence: 99%

“…The deformations in all integrals must be in a certain agreement. See [16,36,15,24,8,39,17,19,20] for details and comparison with other methods; note that in [19], triple and quadruple integrals are efficiently calculated.…”

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confidence: 99%

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Lévy models amenable to efficient calculations

Boyarchenko¹,

Levendorskiı̌²

2022

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In our previous publications (IJTAF 2019, Math. Finance 2020), we introduced a general class of SINH-regular processes and demonstrated that efficient numerical methods for the evaluation of the Wiener-Hopf factors and various probability distributions (prices of options of several types) in Lévy models can be developed using only a few general properties of the characteristic exponent ψ. Essentially all popular Lévy processes enjoy these properties. In the present paper, we define classes of Stieltjes-Lévy processes (SL-processes) as processes with completely monotone Lévy densities of positive and negative jumps, and signed Stieltjes-Lévy processes (sSL-processes) as processes with densities representable as differences of completely monotone densities. We demonstrate that 1) all crucial properties of ψ are consequences of the representationwhere ST (G) is the Stieltjes transform of the (signed) Stieltjes measure G and a ± j ≥ 0; 2) essentially all popular processes other than Merton's model and Meixner processes are SLprocesses; 3) Meixner processes are sSL-processes; 4) under a natural symmetry condition, essentially all popular classes of Lévy processes are SL-or sSL-subordinated Brownian motion.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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Lévy models amenable to efficient calculations

Boyarchenko¹,

Levendorskiı̌²

2022

Preprint

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“…As applications of the general theorems, in Section 3.3, we derive explicit formulas for the cumulative distribution function (cpdf) of the Lévy process and its maximum, and for the option to exchange e XT for the power e βX T . In Section 4, we demonstrate how the sinh-acceleration technique used in [30] to price European options and applied in [32,31,34] to pricing barrier options, evaluation of special functions and the coefficients in BPROJ method respectively can be applied to greatly decrease the sizes of grids and the CPU time needed to satisfy the desired error tolerance. This feature makes the method of the paper more efficient than methods that use the fast inverse Fourier transform, fast convolution or fast Hilbert transform.…”

Section: Introductionmentioning

confidence: 99%

Efficient evaluation of expectations of functions of a Lévy process and its extremum

Boyarchenko¹,

Levendorskiı̌²

2022

Preprint

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We prove a simple general formula for the expectation of a function of a Lévy process and its running extremum, which is more convenient for applications than general formulas in the first version of the paper. The derivation of explicit formulas in applications significantly simplifies. Under additional conditions, we derive analytical formulas using the Fourier/Laplace inversion and Wiener-Hopf factorization, and discuss efficient numerical methods for realization of these formulas. As applications, the cumulative probability distribution function of the process and its running maximum and the price of the option to exchange the maximum of a stock price for a power of the price are calculated. The most efficient numerical methods use the sinh-acceleration technique and simplified trapezoid rule. The program in Matlab running on a Mac with moderate characteristics achieves the precision E-7 and better in several milliseconds, and E-14 -in a fraction of a second.

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“…See [22] for the explicit calculation of the coni of analyticity in popular models. Therefore, the sinh-acceleration technique used in [18] to price European options and applied in [20,19,23] to price barrier options and evaluate special functions and the coefficients in BPROJ method respectively can be applied to greatly decrease the sizes of grids and the CPU time needed to satisfy the desired error tolerance. The changes of variables must be in a certain agreement as in [16,42,21].…”

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confidence: 99%