倒向随机方程期权定价模型的一类随机算法

期权定价问题可以转化为对倒向随机微分方程的求解,进而转化为对相应抛物型偏微分方程的求解.为了求解与倒向随机微分方程相应的二阶拟线性抛物型微分方程初值问题,引入一类新的随机算法-分层方法取代传统的确定性数值算法.这种数值方法理论上是通过弱显式欧拉法,离散其相应随机系统解的概率表示而得到.该随机算法的收敛性在文中得到证明,其稳定性是自然的.并构造了易于数值实现的基于插值的算法,实证研究说明这种算法能很好地提供期权定价模型的数值模拟.

A Type of Stochastic Methods to Solve Backward Stochastic Differential Equations for Option Priced Model

To apply backward stochastic differential equations to option evaluation, a partial differential equation system should be numerically solved firstly. Thus a class of new stochastic layer methods rather than the traditional deterministic methods is constructed to solve the Cauchy problem for second-order quasilinear parabolic equations, which is derived by using weak Euler scheme to discretisize probabilistic representation of the solution. The convergence of the new algorithm is proved, and the stability of the algorithm is natural. Correspondingly, the numerical algorithm based on interpolation is proposed. At last, a numerical example is presented.