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

Gulen S., Sari M.

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, vol.45, no.2, pp.899-913, 2022 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 45 Issue: 2
  • Publication Date: 2022
  • Doi Number: 10.1002/mma.7821
  • Journal Name: MATHEMATICAL METHODS IN THE APPLIED SCIENCES
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Social Sciences Citation Index (SSCI), Scopus, Academic Search Premier, Aerospace Database, Applied Science & Technology Source, Communication Abstracts, Compendex, INSPEC, MathSciNet, Metadex, zbMATH, Civil Engineering Abstracts
  • Page Numbers: pp.899-913
  • Keywords: Frechet derivative, hedge cost, illiquid markets, linearization, Newton iteration, nonlinear Black-Scholes equation, BLACK-SCHOLES EQUATION, NUMERICAL-SOLUTION, REPLICATION, TRANSACTIONS, SCHEME, PRICES, COST
  • Istanbul Technical University Affiliated: Yes

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 Frechet 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.