Derivative pricing as a transport problem: MPDATA solutions to Black–Scholes-type equations

Arabas, Sylwester; Farhat, Ahmad

doi:10.1016/j.cam.2019.05.023

Cited by 2 publications

(2 citation statements)

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“…1 Arabas and Farhat discussed the correspondence between transport phenomena and financial models and developed a framework for solving Black-Scholes-type equations by transforming them into advection-diffusion problems. 2 Vesa et al study the determinants of WTI crude oil call option prices with a special emphasis on the relationship between implied volatility and moneyness. 3 Black-Scholes model 4 is a mathematical tool for calculating the fair price of a European option in absence of arbitrage.…”

Section: Introductionmentioning

confidence: 99%

“…Xiaoli et al proposed a hybrid optimization algorithm based on the relation between Black–Scholes model and heat equation, which dealt with numerical solution for stock option in water conservancy finance 1 . Arabas and Farhat discussed the correspondence between transport phenomena and financial models and developed a framework for solving Black–Scholes‐type equations by transforming them into advection–diffusion problems 2 . Vesa et al study the determinants of WTI crude oil call option prices with a special emphasis on the relationship between implied volatility and moneyness 3 .…”

Section: Introductionmentioning

confidence: 99%

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Haar wavelet transform and variational iteration method for fractional option pricing models

Liu

Meng

Xing

et al. 2022

Math Methods in App Sciences

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Comparing with the linear Black-Scholes model, the fractional option pricing models are constructed by taking account some more parameters like, for example, the transaction cost, so that it becomes more difficult to find the exact analytical solution. In this paper, we analyze a nonlinear fractional Black and Scholes model, and we find the solution by using a novel numerical method, based on a mixture of efficient techniques. In particular, we combine (1) Haar wavelet integration method which transforms the PDEs into a system of algebraic equations, (2) the homotopy perturbation method in order to linearize the problem, and (3) the variational iteration method which will be used to solve the large system of algebraic equations efficiently. We will also show that, in comparison with other popular methods, our coupling technique has a higher efficiency and calculation precision.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Haar wavelet transform and variational iteration method for fractional option pricing models

Liu

Meng

Xing

et al. 2022

Math Methods in App Sciences

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PyMPDATA v1: Numba-accelerated implementation of MPDATA with examples in Python, Julia and Matlab

Bartman¹,

Banaśkiewicz²,

Drenda³

et al. 2022

JOSS

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Convection-diffusion problems arise across a wide range of pure and applied research, in particular in geosciences, aerospace engineering, and financial modelling (for an overview of applications, see, e.g., section 1.1 in Morton (1996)). One of the key challenges in numerical solutions of problems involving advective transport is sign preservation of the advected field (for an overview of this and other aspects of numerical solutions to advection problems, see, e.g., Røed (2019)). The Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) is a robust, explicit-in-time, and sign-preserving solver introduced in Smolarkiewicz (1983) and Smolarkiewicz (1984). MPDATA has been subsequently developed into a family of numerical schemes with numerous variants and solution procedures addressing a diverse set of problems in geophysical fluid dynamics and beyond. For reviews of MPDATA applications and variants, see, e.g., and Smolarkiewicz (2006).The PyMPDATA project introduced herein constitutes a high-performance and multi-threaded implementation of structured-mesh MPDATA in Python. PyMPDATA is aimed to address several aspects which steepen the learning curve and limit collaborative usage and development of existing C++ (e.g., Jaruga et al., 2015) and Fortran (e.g., Kühnlein et al., 2019) implementations of MPDATA. Performance on par with compiled-language implementations is targeted by employing just-in-time (JIT) compilation using Numba (Lam et al., 2015), which translates Python code into fast machine code using the Low Level Virtual Machine (LLVM, https://llvm.org/) compiler infrastructure (for a discussion of another JIT implementation of MPDATA using PyPy, see Arabas et al., 2014). PyMPDATA is engineered aiming at both performance and usability, the latter encompassing research user's, developer's and maintainer's perspectives. From researcher's perspective, PyMPDATA offers hassle-free installation on Linux, macOS, and Windows. It also eliminates the compilation stage from the perspective of the user. From developer's and maintainer's perspectives, PyMPDATA offers a suite of unit tests, multi-platform continuous integration setup, seamless integration with Python development tools including debuggers, profilers, and code analysers.

show abstract

Derivative pricing as a transport problem: MPDATA solutions to Black–Scholes-type equations

Cited by 2 publications

References 30 publications

Haar wavelet transform and variational iteration method for fractional option pricing models

Haar wavelet transform and variational iteration method for fractional option pricing models

PyMPDATA v1: Numba-accelerated implementation of MPDATA with examples in Python, Julia and Matlab

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