Vera N. Egorova

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.

Constructing positive reliable numerical solution for American call options

A new front-fixing transformation is applied to the Black-Scholes equation for the American call option pricing problem. The transformed non-linear problem involves homogeneous boundary conditions independent of the free boundary. The numerical solution by an explicit finite-difference method is positive and monotone. Stability and consistency of the scheme are studied. The explicit proposed method is compared with other competitive implicit ones from the points of view accuracy and computational cost.

A new efficient numerical method for solving American option under regime switching model

A system of coupled free boundary problems describing American put option pricing under regime switching is considered. In order to build numerical solution firstly a front-fixing transformation is applied. Transformed problem is posed on multidimensional fixed domain and is solved by explicit finite difference method. The numerical scheme is conditionally stable and is consistent with the first order in time and second order in space. The proposed approach allows the computation not only of the option price but also of the optimal stopping boundary. Numerical examples demonstrate efficiency and accuracy of the proposed method. The results are compared with other known approaches to show its competitiveness.

A mixed derivative terms removing method in multi-asset option pricing problems

This work has been partially supported by the European Union in the FP7-PEOPLE-2012-ITN program under Grant Agreement Number 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE-Novel Methods in Computational Finance) and the Ministerio de Economia y Competitividad Spanish grant MTM2013-41765-P.

Finite difference methods for pricing American put option with rationality parameter: Numerical analysis and computing

Abstract In this paper finite difference methods for pricing American option with rationality parameter are proposed. The irrational exercise policy arising in American options is characterized in terms of a rationality parameter. The model is formulated in terms of a new nonlinear Black–Scholes equation that requires specific numerical methods. Although the solution converges to the solution of the classical American option price when the parameter tends to infinity, for finite values of the parameter the classical boundary conditions cannot apply and we propose specific ones. A logarithmic transformation is used to improve properties of the numerical solution that is constructed by explicit finite difference method. Numerical analysis provides stability conditions for the methods and its positivity. Properties of intensity function are studied from the point of view of numerical solution. Concerning the numerical methods for the original problem we propose the θ -method for time discretization, thus including explicit, fully implicit and Crank–Nicolson schemes as particular cases. The nonlinear term is treated by a Newton method. The convergence rate is illustrated by numerical examples.

Moving boundary transformation for American call options with transaction cost: finite difference methods and computing

ABSTRACT The pricing of American call option with transaction cost is a free boundary problem. Using a new transformation method the boundary is made to follow a certain known trajectory in time. The new transformed problem is solved by various finite difference methods, such as explicit and implicit schemes. Broydens and Schuberts methods are applied as a modification to Newtons method in the case of nonlinearity in the equation. An alternating direction explicit method with second-order accuracy in time is used as an example in this paper to demonstrate the technique. Numerical results demonstrate the efficiency and the rate of convergence of the methods.

Numerical valuation of two-asset options under jump diffusion models using Gauss-Hermite quadrature

Abstract In this work a finite difference approach together with a bivariate Gauss–Hermite quadrature technique is developed for partial-integro differential equations related to option pricing problems on two underlying asset driven by jump-diffusion models. Firstly, the mixed derivative term is removed using a suitable transformation avoiding numerical drawbacks such as slow convergence and inaccuracy due to the appearance of spurious oscillations. Unlike the more traditional truncation approach we use 2D Gauss–Hermite quadrature with the additional advantage of saving computational cost. The explicit finite difference scheme becomes consistent, conditionally stable and positive. European and American option cases are treated. Numerical results are illustrated and analyzed with experiments and comparisons with other well recognized methods.

Computing American option price under regime switching with rationality parameter

American put option pricing under regime switching is modelled by a system of coupled partial differential equations. The proposed model combines better the reality of the market by incorporating the regime switching jointly with the emotional behaviour of traders using the rationality parameter approach recently introduced by Tagholt Gad and Lund Petersen to cope with possible irrational exercise policy. The classical rational exercise is recovered as a limit case of the rational parameter. The resulting nonlinear system of PDEs is solved by a weighted finite difference method, also known as ź -method. In order to avoid the need of an iterative method for nonlinear system, the term with rationality parameter and the coupling term are treated explicitly. Next, the resulting linear system is solved by Thomas algorithm. Stability conditions for the numerical scheme are studied by using von Neumann approach. Numerical examples illustrate the efficiency and accuracy of the proposed method.

Conditional full stability of positivity-preserving finite difference scheme for diffusion–advection-reaction models

This work has been partially supported by the Ministerio de Economia y Competitividad Spanish grant MTM2017-89664-P.

Numerical Analysis of Novel Finite Difference Methods

The core target of this chapter is numerical analysis and computing of novel finite difference methods related to several different option pricing models, including jump-diffusion, regime switching and multi-asset options. A special attention is paid to positivity, consistency and stability of the proposed methods. The consideration of jump processes leads to partial integro-differential equation (PIDE) for the European option pricing problem. The problem is solved by using quadrature formulas for the approximation of the integrals and matching the discretization of the integral and differential part of the PIDE problem. More complicated model under assumption that the volatility is a stochastic process derives to a PIDE problem where the volatility is also an independent variable. Such a problem is solved by introducing appropriate change of variables. Moreover, American options are considered proposing various front-fixing transformations to treat a free boundary. This free boundary challenge can be treated also by a recent rationality parameter approach that takes into account the irrational behavior of the market. Dealing with multidimensional problems the core difficulty is the appearance of the cross derivative terms. Appropriate transformations allow eliminating the cross derivative terms and reduce of the computational cost and the numerical instabilities. After using a semidiscretization approach, the time exponential integration method and appropriate quadrature integration formulas, the stability of the proposed method is studied independent to the problem dimension.