Computational efficiency is essential for enhancing the accuracy and practicality of pricing complex financial derivatives. In this paper, we discuss Isogeometric Analysis (IGA) for valuing financial derivatives, modeled by two nonlinear Black-Scholes PDEs: the Leland model for European call with transaction costs and the AFV model for convertible bonds with default options. We compare the solutions of IGA with finite difference methods (FDM) and finite element methods (FEM). In particular, very accurate solutions can be numerically calculated on far less mesh (knots) than FDM or FEM, by using non-uniform knots and weighted cubic NURBS, which in turn reduces the computational time significantly.

翻译：计算效率对于提高复杂金融衍生品定价的准确性和实用性至关重要。本文探讨了等几何分析（IGA）在金融衍生品估值中的应用，这些衍生品由两个非线性Black-Scholes偏微分方程建模：用于含交易成本的欧式看涨期权的Leland模型，以及用于含违约期权的可转换债券的AFV模型。我们将IGA的求解结果与有限差分法（FDM）和有限元法（FEM）进行了比较。特别地，通过使用非均匀节点和加权三次NURBS，可以在远少于FDM或FEM所需的网格（节点）数量上数值计算出非常精确的解，从而显著减少了计算时间。