High-Order Compact Finite Difference Schemes for Option Pricing in Stochastic Volatility Models on Non-Uniform Grids

Düring, Bertram; Fournié, Michel; Heuer, Christof

doi:10.2139/ssrn.2295581

Cited by 14 publications

(24 citation statements)

References 25 publications

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“…In the early 1970s, Black, Scholes and Merton introduced the popular Black-Scholes-Merton (BSM) model [2,5]. Under the consideration, stock prices are assumed to follow geometric Brownian motion, while the volatility of the stock prices is fixed and no sudden jumps occur.…”

Section: Introductionmentioning

confidence: 99%

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An Exploration of a Balanced Up-downwind Scheme for Solving Heston Volatility Model Equations on Variable Grids

Sun¹,

Sheng²

2018

Preprint

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This paper studies an effective finite difference scheme for solving two-dimensional Heston stochastic volatility option pricing model problems. A dynamically balanced up-downwind strategy for approximating the cross-derivative is implemented and analyzed. Semi-discretized and spatially nonuniform platforms are utilized. The numerical method comprised is simple, straightforward with reliable first order overall approximations. The spectral norm is used throughout the investigation and numerical stability is proven. Simulation experiments are given to illustrate our results.

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Section: Introductionmentioning

confidence: 99%

“…These have motivated our approaches. In this paper, we are particularly interested in computations based on a Heston put option model [4,5,8,10,12,22].…”

Section: Introductionmentioning

confidence: 99%

An Exploration of a Balanced Up-downwind Scheme for Solving Heston Volatility Model Equations on Variable Grids

Sun¹,

Sheng²

2018

Preprint

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“…in computational fluid dynamics [18,16,15,8] and computational finance [5,6,22,2,4], an even wider breakthrough of the high-order compact methodology has been hampered by the algebraic complexity that is inherent to this approach. The derivation of high-order compact schemes is algebraically demanding, hence these schemes are often taylor-made for a specific application or a rather smaller class of problems (with some notable exceptions as, for example Lele's paper [14]).…”

Section: Introductionmentioning

confidence: 99%

High-Order Compact Schemes for Parabolic Problems with Mixed Derivatives in Multiple Space Dimensions

Düring¹,

Heuer²

2015

SIAM J. Numer. Anal.

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We present a high-order compact finite difference approach for a class of parabolic partial differential equations with time and space dependent coefficients as well as with mixed secondorder derivative terms in n spatial dimensions. Problems of this type arise frequently in computational fluid dynamics and computational finance. We derive general conditions on the coefficients which allow us to obtain a high-order compact scheme which is fourth-order accurate in space and second-order accurate in time. Moreover, we perform a thorough von Neumann stability analysis of the Cauchy problem in two and three spatial dimensions for vanishing mixed derivative terms, and also give partial results for the general case. The results suggest unconditional stability of the scheme. As an application example we consider the pricing of European Power Put Options in the multidimensional Black-Scholes model for two and three underlying assets. Due to the low regularity of typical initial conditions we employ the smoothing operators of Kreiss et al. to ensure high-order convergence of the approximations of the smoothed problem to the true solution.

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“…We transform the PIDE (1) into a new PIDE without mixed spatial derivative before the discretization, following the idea of [20], and avoiding the above quoted drawbacks. Furthermore, this strategy has additional computa-50 tional advantage of the reduction of the stencil scheme points, from nine [21,22] or seven [13,17] to just five.…”

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confidence: 99%

Positive finite difference schemes for a partial integro-differential option pricing model

Fakharany

Jódar

2014

Applied Mathematics and Computation

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This paper provides a numerical analysis for European options under partial integro-differential Bates model. An explicit finite difference scheme has been used for the differential part, while the integral part has been approximated using the four-points open type formula. The stability and consistency have been studied. Moreover, conditions guaranteing positivity of the solutions are provided. Illustrative numerical examples are included.

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High-Order Compact Finite Difference Schemes for Option Pricing in Stochastic Volatility Models on Non-Uniform Grids

Cited by 14 publications

References 25 publications

An Exploration of a Balanced Up-downwind Scheme for Solving Heston Volatility Model Equations on Variable Grids

An Exploration of a Balanced Up-downwind Scheme for Solving Heston Volatility Model Equations on Variable Grids

High-Order Compact Schemes for Parabolic Problems with Mixed Derivatives in Multiple Space Dimensions

Positive finite difference schemes for a partial integro-differential option pricing model

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