发表机构
Louisiana State University（路易斯安那州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种B样条伽辽金方法求解CGMY过程下期权定价的tempered分数阶偏微分方程，通过修正傅里叶符号构造实现快速计算，并证明稳定性与误差估计，在S&P 500数据上校准表现优异。
AI 中文摘要
我们针对 CGMY 过程下欧式和美式期权价格所满足的 tempered 分数阶偏微分方程，开发并分析了一种 B 样条尺度函数伽辽金方法。空间生成元耦合了阶数 $Y\in(0,2)$ 的指数 tempered 左、右 Riemann--Liouville 导数，其非局部性和不对称性给离散化带来了挑战。我们推导了 tempered 连接系数矩阵的修正傅里叶符号构造，其内部块为 Toeplitz 结构，保持精确的分数阶缩放，并通过快速傅里叶变换实现 $\mathcal{O}(N\log N)$ 的矩阵-向量乘积。在 $Y\in(1,2)$ 区间内，我们通过零延拓到全直线符号，证明了全直线和有限域上的 Gårding 不等式，当无风险利率为正时具有强制性。在此基础之上，我们建立了椭圆投影、半离散和全离散误差估计，空间阶为 $\mathcal{O}(h^{p-Y/2})$，同时给出了强制 Crank--Nicolson 格式的无条件移位能量稳定性估计，其常数与网格无关。对于正利率，允许的移位消失，从而精确控制实际实现的普通格式。一次反向求解即可在固定到期日对整个执行价条带进行定价，并给出平滑、解析微分的 Greeks。相同的离散化通过投影线性互补求解对美式看跌期权进行定价，提前行权溢价和行权边界通过与解析确定性参考和 Longstaff--Schwartz 蒙特卡洛基准的对比得到验证。针对 S\\&P~500 期权曲面在低波动率、危机和复苏状态下的校准，在良好识别的曲面上产生接近两个波动率点的隐含波动率误差，而与稳定的 Grünwald--Letnikov 基线的精度匹配比较则精确界定了高阶变分结构何时发挥作用。
英文摘要
We develop and analyze a B-spline scaling-function Galerkin method for the tempered fractional partial differential equation governing European and American option prices under the CGMY process. The spatial generator couples exponentially tempered left- and right-sided Riemann--Liouville derivatives of order $Y\in(0,2)$, whose nonlocality and asymmetry present discretization challenges. We derive a corrected Fourier-symbol construction of the tempered connection-coefficient matrices whose interior blocks are Toeplitz, maintain exact fractional scaling, and enable $\mathcal{O}(N\log N)$ matrix--vector products via the fast Fourier transform. In the regime $Y\in(1,2)$, we prove full-line and finite-domain Gårding inequalities, coercive whenever the risk-free rate is positive, via zero-extension to the full-line symbol. On this foundation, we establish elliptic-projection, semidiscrete, and fully discrete error estimates with spatial rate $\mathcal{O}(h^{p-Y/2})$, alongside an unconditional shifted-energy stability estimate for the forced Crank--Nicolson scheme with mesh-independent constants. For positive rates the admissible shift vanishes, governing the plain scheme exactly as implemented. A single backward solve prices an entire strike strip at fixed maturity with smooth, analytically differentiated Greeks. The same discretization prices American puts through a projected linear complementarity solve, with early-exercise premiums and exercise boundaries validated against a resolved deterministic reference and a Longstaff--Schwartz Monte Carlo benchmark. Calibration to S\&P~500 option surfaces across low-volatility, crisis, and recovery regimes yields implied-volatility errors near two vol-points on well-identified surfaces, while an accuracy-matched comparison with a stabilized Grünwald--Letnikov baseline delineates precisely when the high-order variational structure pays off.
