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

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.