Black-Scholes 方程的拉普拉斯变换方法

评估线性和非线性Black-Scholes方程最优执行边界近似方法的变换技术

Transformation methods for evaluating approximations to the optimal exercise boundary for linear and nonlinear Black-Scholes equations

本文提出一种变换方法，将美式期权提前执行边界的自由边界问题转化为固定域上的非线性非局部抛物型方程，并设计迭代数值方案以近似求解边界，同时分析了线性和非线性Black-Scholes方程中边界对模型参数的依赖。

Analytical and numerical results for American style of perpetual put options through transformation into nonlinear stationary Black-Scholes equations

We analyze and calculate the early exercise boundary for a class of stationary generalized Black-Scholes equations in which the volatility function depends on the second derivative of the option price itself. A motivation for studying the nonlinear Black Scholes equation with a nonlinear volatility arises from option pricing models including, e.g., non-zero transaction costs, investors preferences, feedback and illiquid markets effects and risk from unprotected portfolio. We present a method how to transform the problem of American style of perpetual put options into a solution of an ordinary …

Symmetries of the Black-Scholes equation

We determine the algebra of isovectors for the Black--Scholes equation. As a consequence, we obtain some previously unknown families of transformations on the solutions.

A Method of Reducing Dimension of Space Variables in Multi-dimensional Black-Scholes Equations

We study a method of reducing space dimension in multi-dimensional Black-Scholes partial differential equations as well as in multi-dimensional parabolic equations. We prove that a multiplicative transformation of space variables in the Black-Scholes partial differential equation reserves the form of Black-Scholes partial differential equation and reduces the space dimension. We show that this transformation can reduce the number of sources of risks by two or more in some cases by giving remarks and several examples of financial pricing problems. We also present that the invariance of the form…

Pricing American Call Options by the Black-Scholes Equation with a Nonlinear Volatility Function

In this paper we investigate a nonlinear generalization of the Black-Scholes equation for pricing American style call options in which the volatility term may depend on the underlying asset price and the Gamma of the option. We propose a numerical method for pricing American style call options by means of transformation of the free boundary problem for a nonlinear Black-Scholes equation into the so-called Gamma variational inequality with the new variable depending on the Gamma of the option. We apply a modified projective successive over relaxation method in order to construct an effective nu…

Comparison of Two Numerical Methods for Computation of American Type of the Floating Strike Asian Option

We present a numerical approach for solving the free boundary problem for the Black-Scholes equation for pricing American style of floating strike Asian options. A fixed domain transformation of the free boundary problem into a parabolic equation defined on a fixed spatial domain is performed. As a result a nonlinear time-dependent term is involved in the resulting equation. Two new numerical algorithms are proposed. In the first algorithm a predictor-corrector scheme is used. The second one is based on the Newton method. Computational experiments, confirming the accuracy of the algorithms are…