# Nonstandard Finite Difference Schemes with Application to Finance: Option Pricing

• Volume: 35, Issue: 1, page 75-88
• ISSN: 1310-6600

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## Abstract

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2000 Mathematics Subject Classification: 65M06, 65M12.The paper is devoted to pricing options characterized by discontinuities in the initial conditions of the respective Black-Scholes partial differential equation. Finite difference schemes are examined to highlight how discontinuities can generate numerical drawbacks such as spurious oscillations. We analyze the drawbacks of the Crank-Nicolson scheme that is most frequently used numerical method in Finance because of its second order accuracy. We propose an alternative scheme that is free of spurious oscillations and satisfy the positivity requirement, as it is demanded for the financial solution of the Black-Scholes equation.

## How to cite

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Milev, Mariyan, and Tagliani, Aldo. "Nonstandard Finite Difference Schemes with Application to Finance: Option Pricing." Serdica Mathematical Journal 35.1 (2010): 75-88. <http://eudml.org/doc/281413>.

@article{Milev2010,
abstract = {2000 Mathematics Subject Classification: 65M06, 65M12.The paper is devoted to pricing options characterized by discontinuities in the initial conditions of the respective Black-Scholes partial differential equation. Finite difference schemes are examined to highlight how discontinuities can generate numerical drawbacks such as spurious oscillations. We analyze the drawbacks of the Crank-Nicolson scheme that is most frequently used numerical method in Finance because of its second order accuracy. We propose an alternative scheme that is free of spurious oscillations and satisfy the positivity requirement, as it is demanded for the financial solution of the Black-Scholes equation.},
author = {Milev, Mariyan, Tagliani, Aldo},
journal = {Serdica Mathematical Journal},
keywords = {Black-Scholes Equation; Finite Difference Schemes; Jacobi Matrix; M-Matrix; Nonsmooth Initial Conditions; Positivity-Preserving; Black-Scholes equation; finite difference schemes; Jacobi matrix; M-matrix; nonsmooth initial conditions; positivity-preserving; option pricing; Crank-Nicolson scheme},
language = {eng},
number = {1},
pages = {75-88},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {Nonstandard Finite Difference Schemes with Application to Finance: Option Pricing},
url = {http://eudml.org/doc/281413},
volume = {35},
year = {2010},
}

TY - JOUR
AU - Milev, Mariyan
AU - Tagliani, Aldo
TI - Nonstandard Finite Difference Schemes with Application to Finance: Option Pricing
JO - Serdica Mathematical Journal
PY - 2010
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 35
IS - 1
SP - 75
EP - 88
AB - 2000 Mathematics Subject Classification: 65M06, 65M12.The paper is devoted to pricing options characterized by discontinuities in the initial conditions of the respective Black-Scholes partial differential equation. Finite difference schemes are examined to highlight how discontinuities can generate numerical drawbacks such as spurious oscillations. We analyze the drawbacks of the Crank-Nicolson scheme that is most frequently used numerical method in Finance because of its second order accuracy. We propose an alternative scheme that is free of spurious oscillations and satisfy the positivity requirement, as it is demanded for the financial solution of the Black-Scholes equation.
LA - eng
KW - Black-Scholes Equation; Finite Difference Schemes; Jacobi Matrix; M-Matrix; Nonsmooth Initial Conditions; Positivity-Preserving; Black-Scholes equation; finite difference schemes; Jacobi matrix; M-matrix; nonsmooth initial conditions; positivity-preserving; option pricing; Crank-Nicolson scheme
UR - http://eudml.org/doc/281413
ER -

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