3 AN NEW ALGORITHM FOR PRICING SHOUT OPTIONS
Another innovation is that this paper proposes a new competitive algorithm to choose two aspects for this mathematical model. One is employing high-order difference for integral and partial derivative terms, the other is using Howard algorithm (also called policy iteration) for the complementarity problem.
Let\(\ x=\text{logS},\tau=T-t\), we obtain
\begin{equation} -\frac{\partial V}{\partial\tau}+\left(r-\text{βλ}-\frac{1}{2}\sigma^{2}\right)\frac{\partial V}{\partial x}+\frac{1}{2}\sigma^{2}\frac{\partial^{2}V}{\partial x^{2}}+\lambda\left[\int_{0}^{+\infty}{V\left(T-\tau,ye^{x}\right)f\left(y\right)\text{dy}-V\left(T-\tau,e^{x}\right)}\right]-\text{rV}\leq 0.\ \ \ \ \ \ (3.1)\nonumber \\ \end{equation}
In order to calculate conveniently, it is necessary to limit \(x\) to the region\(\ \Omega=\left[-\varphi,+\varphi\right]\). Considering the uniform grid with interval\(\ h=2\varphi/M\ \) that the grid points are given by\(\ x_{m}=-\varphi+\text{mh},m\in\left[0,M\right]\), and \(V_{m}\left(\tau\right)=V\left(x_{m},\tau\right)\ \)is defined.