Constructing positive reliable numerical solution for American call options: a new front-fixing approach

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 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

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

Pricing pension plans under jump–diffusion models for the salary

[Abstract] In this paper we consider the valuation of a defined benefit pension plan in the presence of jumps in the underlying salary and including the possibility of early retirement. We will consider that the salary follows a jump–diffusion model, thus giving rise to a partial integro-differential equation (PIDE). After posing the model, we propose the appropriate numerical methods to solve the PIDE problem. These methods mainly consists of Lagrange–Galerkin discretizations combined with augmented Lagrangian active set techniques and with the explicit treatment of the integral term. Finally, we compare the numerical results with those ones obtained with Monte Carlo techniques.This paper has been partially funded by MCINN (Project MTM2010-21135-C02-01 and MTM2013-47800-C2-1-P) and by Xunta de Galicia (Ayuda GRC2014/044, partially funded with FEDER funds).Xunta de Galicia; GRC2014/04

Double Discretization Difference Schemes for Partial Integrodifferential Option Pricing Jump Diffusion Models

A new discretization strategy is introduced for the numerical solution of partial integrodifferential equations appearing in option pricing jump diffusion models. In order to consider the unknown behaviour of the solution in the unbounded part of the spatial domain, a double discretization is proposed. Stability, consistency, and positivity of the resulting explicit scheme are analyzed. Advantages of the method are illustrated with several examples.This paper was supported by the Spanish M.E.Y.C. Grant DPI2010-20891-C02-01.Casabán Bartual, MC.; Company Rossi, R.; Jódar Sánchez, LA.; Romero Bauset, JV. (2012). Double Discretization Difference Schemes for Partial Integrodifferential Option Pricing Jump Diffusion Models. Abstract and Applied Analysis. 2012:1-20. https://doi.org/10.1155/2012/120358S120201

Penalty methods for the numerical solution of American multi-asset option problems

AbstractWe derive and analyze a penalty method for solving American multi-asset option problems. A small, non-linear penalty term is added to the Black–Scholes equation. This approach gives a fixed solution domain, removing the free and moving boundary imposed by the early exercise feature of the contract. Explicit, implicit and semi-implicit finite difference schemes are derived, and in the case of independent assets, we prove that the approximate option prices satisfy some basic properties of the American option problem. Several numerical experiments are carried out in order to investigate the performance of the schemes. We give examples indicating that our results are sharp. Finally, the experiments indicate that in the case of correlated underlying assets, the same properties are valid as in the independent case