A Fréchet derivative-based novel approach to option pricing models in illiquid markets

Seda Gulen*, Murat Sari

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

Nonlinear option pricing models have been increasingly concerning in financial industries since they build more accurate values by regarding more realistic assumptions such as transaction cost, market liquidity, or uncertain volatility. This study defines a nonclassical numerical method to effectively capture the behavior of the nonlinear option pricing model in illiquid markets where the implementation of a dynamic hedging strategy affects the price of the underlying asset. Unlike the conventional numerical approaches, this study describes a numerical scheme based on the Newton iteration technique and the Fréchet derivative for linearization of the model. The linearized time-dependent PDE is then discretized by a sixth-order finite difference scheme in space and a second-order trapezoidal rule in time. The computations revealed that the current approach appears to be somewhat more effective to some extent and at the same time economical for illustrative examples compared to the existing competitors. In addition, this method helps to prevent considering the convergence issues of the Newton approach applied to the nonlinear algebraic system.

Original languageEnglish
Pages (from-to)899-913
Number of pages15
JournalMathematical Methods in the Applied Sciences
Volume45
Issue number2
DOIs
Publication statusPublished - 30 Jan 2022
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2021 John Wiley & Sons, Ltd.

Keywords

  • Fréchet derivative
  • hedge cost
  • illiquid markets
  • linearization
  • Newton iteration
  • nonlinear Black–Scholes equation

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