Early exercise European option and early termination American option pricing models

The maximum relative error between continuous-time American option pricing model and binomial tree model is very small. In order to improve the European and American options in trade course, the thesis tried to build early exercise European option and early termination American option pricing models. Firstly, the authors reviewed the characteristics of American option and European option, then there was compares between them. Base on continuous-time American option pricing model, this research analyzed the value of these options.

A Fixed Point Method for the Linear Complementarity Problem Arising from American Option Pricing

For American option pricing, the Black-Scholes-Merton model can be discretized as a linear comple- mentarity problem （LCP） by using some finite difference schemes. It is well known that the Projected Successive Over Relaxation （PSOR） has been widely applied to solve the resulted LCP. In this paper, we propose a fixed point iterative method to solve this type of LCPs, where the splitting technique of the matrix is used. We show that the proposed method is globally convergent under mild assumptions. The preliminary numerical results are reported, which demonstrate that the proposed method is more accurate than the PSOR for the problems we tested.

Early exercise premium method for pricing American options under the J-model

Background:This study develops a new model called J-am for pricing American options and for determining the related early exercise boundary(EEB).This model is based on a closed-form solution J-formula for pricing European options,defined in the study by Jerbi(Quantitative Finance,15:2041-2052,2015).The J-am pricing formula is a solution of the Black&Scholes(BS)PDE with an additional function called f as a second member and with limit conditions adapted to the American option context.The aforesaid function f represents the cash flows resulting from an early exercise of the option.Methods:This study develops the theoretical formulas of the early exercise premium value related to three American option pricing models called J-am,BS-am,and Heston-am models.These three models are based on the J-formula by Jerbi(Quantitative Finance,15:2041-2052,2015),BS model,and Heston(Rev Financ Stud,6:327-343,1993)model,respectively.This study performs a general algorithm leading to the EEB and to the American option price for the three models.Results:After implementing the algorithms,we compare the three aforesaid models in terms of pricing and the EEB curve.In particular,we examine the equivalence between J-am and Heston-am as an extension of the equivalence studied by Jerbi(Quantitative Finance,15:2041-2052,2015).This equivalence is interesting since it can reduce a bi-dimensional model to an equivalent uni-dimensional model.Conclusions:We deduce that our model J-am exactly fits the Heston-am one for certain parameters values to be optimized and that all the theoretical results conform with the empirical studies.The required CPU time to compute the solution is significantly less in the case of the J-am model compared with to the Heston-am model.

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.