In this paper, American options on a discount bond are priced under the Cox-Ingrosll-Ross (CIR) model. The linear complementarity problem of the option value is solved numerically by a penalty method. The problem is transformed into a nonlinear partial differential equation (PDE) by adding a power penalty term. The solution of the penalized problem converges to the one of the original problem. To numerically solve this nonlinear PDE, we use the horizontal method of lines to discretize the temporal variable and the spatial variable by means of trapezoidal method and a cubic spline collocation method, respectively. We show that this full discretization scheme is second order convergent, and hence the convergence of the numerical solution to the viscosity solution of the continuous problem is guaranteed. Numerical results are presented and compared with other collocation methods given in the literature.

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In this paper, we develop a new numerical method, game theory and option pricing to compute a bankruptcy triggering asset value. we will draw our attention to determining a the numerical asset value, or price of a share, at which a bankruptcy is triggered. This paper develops and analyze a cubic spline collocation method for approximating solutions of the problem. This method converges quadratically. In addition, this article also provides with a real-life case study of the investment bank, and the optimal bankruptcy strategy in this particular case. As we will observe, the bankruptcy trigger computed in this example could have served as a good guide for predicting fall of this investment bank.

This paper presents the Generalized Poisson Jump Process to describe the effect of radical technological innovation on the market. It defines the market impact index of radical technological innovation, and constructs a real option game model for radical technological innovation. Besides, this paper develops a numerical method in the model to show that the market impact index of radical technological innovation and the Generalized Poisson Process parameter both have a good influence on both the investment value and investment limit.

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