A practical finite difference method for the three-dimensional Black-Scholes equation

Junseok Kim, Taekkeun Kim, Jaehyun Jo, Yongho Choi, Seunggyu Lee, Hyeongseok Hwang, Minhyun Yoo, Darae Jeong

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

In this paper, we develop a fast and accurate numerical method for pricing of the three-asset equity-linked securities options. The option pricing model is based on the Black-Scholes partial differential equation. The model is discretized by using a non-uniform finite difference method and the resulting discrete equations are solved by using an operator splitting method. For fast and accurate calculation, we put more grid points near the singularity of the nonsmooth payoff function. To demonstrate the accuracy and efficiency of the proposed numerical method, we compare the results of the method with those from Monte Carlo simulation in terms of computational cost and accuracy. The numerical results show that the cost of the proposed method is comparable to that of the Monte Carlo simulation and it provides more stable hedging parameters such as the Greeks.

Original languageEnglish
Pages (from-to)183-190
Number of pages8
JournalEuropean Journal of Operational Research
Volume252
Issue number1
DOIs
Publication statusPublished - 2016 Jul 1

Fingerprint

Black-Scholes Equation
Finite difference method
Difference Method
Finite Difference
Monte Carlo Simulation
Numerical Methods
Operator Splitting Method
Three-dimensional
Black-Scholes
Nonsmooth Function
Hedging
Option Pricing
Equity
Discrete Equations
Pricing
Computational Cost
Costs
Numerical methods
Partial differential equation
Singularity

Keywords

  • Black-Scholes partial differential equation
  • Equity-linked securities
  • Non-uniform grid
  • Operator splitting method
  • Option pricing

ASJC Scopus subject areas

  • Management Science and Operations Research
  • Modelling and Simulation
  • Information Systems and Management

Cite this

A practical finite difference method for the three-dimensional Black-Scholes equation. / Kim, Junseok; Kim, Taekkeun; Jo, Jaehyun; Choi, Yongho; Lee, Seunggyu; Hwang, Hyeongseok; Yoo, Minhyun; Jeong, Darae.

In: European Journal of Operational Research, Vol. 252, No. 1, 01.07.2016, p. 183-190.

Research output: Contribution to journalArticle

Kim, Junseok ; Kim, Taekkeun ; Jo, Jaehyun ; Choi, Yongho ; Lee, Seunggyu ; Hwang, Hyeongseok ; Yoo, Minhyun ; Jeong, Darae. / A practical finite difference method for the three-dimensional Black-Scholes equation. In: European Journal of Operational Research. 2016 ; Vol. 252, No. 1. pp. 183-190.
@article{20fa24b51c11489e9344c058d14db0f6,
title = "A practical finite difference method for the three-dimensional Black-Scholes equation",
abstract = "In this paper, we develop a fast and accurate numerical method for pricing of the three-asset equity-linked securities options. The option pricing model is based on the Black-Scholes partial differential equation. The model is discretized by using a non-uniform finite difference method and the resulting discrete equations are solved by using an operator splitting method. For fast and accurate calculation, we put more grid points near the singularity of the nonsmooth payoff function. To demonstrate the accuracy and efficiency of the proposed numerical method, we compare the results of the method with those from Monte Carlo simulation in terms of computational cost and accuracy. The numerical results show that the cost of the proposed method is comparable to that of the Monte Carlo simulation and it provides more stable hedging parameters such as the Greeks.",
keywords = "Black-Scholes partial differential equation, Equity-linked securities, Non-uniform grid, Operator splitting method, Option pricing",
author = "Junseok Kim and Taekkeun Kim and Jaehyun Jo and Yongho Choi and Seunggyu Lee and Hyeongseok Hwang and Minhyun Yoo and Darae Jeong",
year = "2016",
month = "7",
day = "1",
doi = "10.1016/j.ejor.2015.12.012",
language = "English",
volume = "252",
pages = "183--190",
journal = "European Journal of Operational Research",
issn = "0377-2217",
publisher = "Elsevier",
number = "1",

}

TY - JOUR

T1 - A practical finite difference method for the three-dimensional Black-Scholes equation

AU - Kim, Junseok

AU - Kim, Taekkeun

AU - Jo, Jaehyun

AU - Choi, Yongho

AU - Lee, Seunggyu

AU - Hwang, Hyeongseok

AU - Yoo, Minhyun

AU - Jeong, Darae

PY - 2016/7/1

Y1 - 2016/7/1

N2 - In this paper, we develop a fast and accurate numerical method for pricing of the three-asset equity-linked securities options. The option pricing model is based on the Black-Scholes partial differential equation. The model is discretized by using a non-uniform finite difference method and the resulting discrete equations are solved by using an operator splitting method. For fast and accurate calculation, we put more grid points near the singularity of the nonsmooth payoff function. To demonstrate the accuracy and efficiency of the proposed numerical method, we compare the results of the method with those from Monte Carlo simulation in terms of computational cost and accuracy. The numerical results show that the cost of the proposed method is comparable to that of the Monte Carlo simulation and it provides more stable hedging parameters such as the Greeks.

AB - In this paper, we develop a fast and accurate numerical method for pricing of the three-asset equity-linked securities options. The option pricing model is based on the Black-Scholes partial differential equation. The model is discretized by using a non-uniform finite difference method and the resulting discrete equations are solved by using an operator splitting method. For fast and accurate calculation, we put more grid points near the singularity of the nonsmooth payoff function. To demonstrate the accuracy and efficiency of the proposed numerical method, we compare the results of the method with those from Monte Carlo simulation in terms of computational cost and accuracy. The numerical results show that the cost of the proposed method is comparable to that of the Monte Carlo simulation and it provides more stable hedging parameters such as the Greeks.

KW - Black-Scholes partial differential equation

KW - Equity-linked securities

KW - Non-uniform grid

KW - Operator splitting method

KW - Option pricing

UR - http://www.scopus.com/inward/record.url?scp=84960338682&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84960338682&partnerID=8YFLogxK

U2 - 10.1016/j.ejor.2015.12.012

DO - 10.1016/j.ejor.2015.12.012

M3 - Article

AN - SCOPUS:84960338682

VL - 252

SP - 183

EP - 190

JO - European Journal of Operational Research

JF - European Journal of Operational Research

SN - 0377-2217

IS - 1

ER -