High-Order Splitting Methods for Forward PDEs and PIDEs

Itkin, Andrey

doi:10.1007/978-1-4939-6792-6_7

Cited by 8 publications

(23 citation statements)

References 26 publications

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“…It is important that all jumps as well as the Brownian motions are correlated. Here we solve just the backward problem (solving the backward Kolmogorov equation, e.g., for pricing derivatives), while the forward problem (solving the forward Kolmogorov equation to find the density of the underlying process) can be treated in a similar way, see Itkin (2015).…”

Section: Discussionmentioning

confidence: 99%

“…( 11) aims to utilize a similar idea, but being transformed to the iterative method. The key point here is that we use a theory of EM-matrices ( see Itkin (2014aItkin ( , 2015 and references therein), and manage to propose a second order approximation of the first derivative which makes our matrices to be real EM-matrices. So again, the inverse of the latter is a positive matrix.…”

Section: Advection-diffusion Problemmentioning

confidence: 99%

“…Note, that the same approach can be applied to the forward equations based on the approach of Itkin (2015). In more detail this will be presented elsewhere.…”

Section: Fully Implicit Schemementioning

confidence: 99%

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LSV models with stochastic interest rates and correlated jumps

Itkin

2016

International Journal of Computer Mathematics

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Pricing and hedging exotic options using local stochastic volatility models drew a serious attention within the last decade, and nowadays became almost a standard approach to this problem. In this paper we show how this framework could be extended by adding to the model stochastic interest rates and correlated jumps in all three components. We also propose a new fully implicit modification of the popular Hundsdorfer and Verwer and Modified Craig-Sneyd finite-difference schemes which provides second order approximation in space and time, is unconditionally stable and preserves positivity of the solution, while still has a linear complexity in the number of grid nodes.

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Section: Discussionmentioning

confidence: 99%

Section: Advection-diffusion Problemmentioning

confidence: 99%

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LSV models with stochastic interest rates and correlated jumps

Itkin

2016

International Journal of Computer Mathematics

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“…We compare the barrier option prices obtained by using our method with those obtained by solving Eq. ( 5) using the FD method described in detail in (Itkin, 2015). This method belongs to a family of so-called ADI (alternative direction implicit) schemes and provides second order of approximation in all dimensions.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Semi-analytical pricing of barrier options in the time-dependent $λ$-SABR model

Itkin,

Muravey

2021

Preprint

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e extend the approach of Carr, Itkin and Muravey, 2021 for getting semianalytical prices of barrier options for the time-dependent Heston model with time-dependent barriers by applying it to the so-called λ-SABR stochastic volatility model. In doing so we modify the general integral transform method (see Itkin, Lipton, Muravey, Generalized integral transforms in mathematical finance, World Scientific, 2021) and deliver solution of this problem in the form of Fourier-Bessel series. The weights of this series solve a linear mixed Volterra-Fredholm equation (LMVF) of the second kind also derived in the paper. Numerical examples illustrate speed and accuracy of our method which are comparable with those of the finite-difference approach at small maturities and outperform them at high maturities even by using a simplistic implementation of the RBF method for solving the LMVF.

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“…Here, we outline the finite difference method used for the PDE solutions, as it is somewhat nonstandard through the use of forward and backward equations. In particular, we extend the adjoint method from Itkin (2015) from two to three dimensions and use it for the forward component. For a more general introduction to finite difference methods for pricing financial derivatives, see for example Tavella and Randall (2000) and, specifically for interest rate derivatives Andersen and Piterbarg (2010).…”

Section: B Numerical Approximation Of the Kolmogorov Pdesmentioning

confidence: 99%

Efficient exposure computation by risk factor decomposition

Graaf

Kandhai

Reisinger

2018

Quantitative Finance

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The focus of this paper is the efficient computation of counterparty credit risk exposure on portfolio level. Here, the large number of risk factors rules out traditional PDE-based techniques and allows only a relatively small number of paths for nested Monte Carlo simulations, resulting in large variances of estimators in practice. We propose a novel approach based on Kolmogorov forward and backward PDEs, where we counter the high dimensionality by a generalisation of anchored-ANOVA decompositions. By computing only the most significant terms in the decomposition, the dimensionality is reduced effectively, such that a significant computational speed-up arises from the high accuracy of PDE schemes in low dimensions compared to Monte Carlo estimation. Moreover, we show how this truncated decomposition can be used as control variate for the full high-dimensional model, such that any approximation errors can be corrected while a substantial variance reduction is achieved compared to the standard simulation approach. We investigate the accuracy for a realistic portfolio of exchange options, interest rate and cross-currency swaps under a fully calibrated ten-factor model.

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High-Order Splitting Methods for Forward PDEs and PIDEs

Cited by 8 publications

References 26 publications

LSV models with stochastic interest rates and correlated jumps

LSV models with stochastic interest rates and correlated jumps

Semi-analytical pricing of barrier options in the time-dependent $λ$-SABR model

Efficient exposure computation by risk factor decomposition

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