A consistent stable numerical scheme for a nonlinear option pricing model in illiquid markets

Markets liquidity is an issue of very high concern in financial risk management. In a perfect liquid market the option pricing model becomes the well-known linear Black-Scholes problem. Nonlinear models appear when transaction costs or illiquid market effects are taken into account. This paper deals with the numerical analysis of nonlinear Black-Scholes equations modeling illiquid markets when price impact in the underlying asset market affects the replication of a European contingent claim. Numerical analysis of a nonlinear model is necessary because disregarded computations may waste a good mathematical model. In this paper we propose a finite-difference numerical scheme that guarantees positivity of the solution as well as stability and consistency. © 2011 IMACS. Published by Elsevier B.V. All rights reserved.This paper has been supported by the Spanish Department of Science and Education grant TRA2007-68006-C02-02 and the Generalitat Valenciana grant GVPRE/20081092.Company Rossi, R.; Jódar Sánchez, LA.; Pintos Taronger, JR. (2012). A consistent stable numerical scheme for a nonlinear option pricing model in illiquid markets. Mathematics and Computers in Simulation. 82(10):1972-1985. https://doi.org/10.1016/j.matcom.2010.04.026S19721985821

Nonlinear Parabolic Equations arising in Mathematical Finance

This survey paper is focused on qualitative and numerical analyses of fully nonlinear partial differential equations of parabolic type arising in financial mathematics. The main purpose is to review various non-linear extensions of the classical Black-Scholes theory for pricing financial instruments, as well as models of stochastic dynamic portfolio optimization leading to the Hamilton-Jacobi-Bellman (HJB) equation. After suitable transformations, both problems can be represented by solutions to nonlinear parabolic equations. Qualitative analysis will be focused on issues concerning the existence and uniqueness of solutions. In the numerical part we discuss a stable finite-volume and finite difference schemes for solving fully nonlinear parabolic equations.Comment: arXiv admin note: substantial text overlap with arXiv:1603.0387

Solving American option pricing models by the front fixing method: numerical analysis and computing

This paper presents an explicit finite-difference method for nonlinear partial differential equation appearing as a transformed Black-Scholes equation for American put option under logarithmic front fixing transformation. Numerical analysis of the method is provided. The method preserves positivity and monotonicity of the numerical solution. Consistency and stability properties of the scheme are studied. Explicit calculations avoid iterative algorithms for solving nonlinear systems. Theoretical results are confirmed by numerical experiments. Comparison with other approaches shows that the proposed method is accurate and competitive

An ETD method for American options under the Heston model

A numerical method for American options pricing on assets under the Heston stochastic volatility model is developed. A preliminary transformation is applied to remove the mixed derivative term avoiding known numerical drawbacks and reducing computational costs. Free boundary is treated by the penalty method. Transformed nonlinear partial differential equation is solved numerically by using the method of lines. For full discretization the exponential time differencing method is used. Numerical analysis establishes the stability and positivity of the proposed method. The numerical convergence behaviour and effectiveness are investigated in extensive numerical experiments.This work has been supported by the Spanish Ministerio de Economía, Industria y Competitividad (MINECO), the Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P