This paper studies the efficient methods for option pricing under multivariate rough volatility models. The characteristic functions of the asset log-price, which play important role in the option pricing under the multivariate rough volatility models, are determined by a system of parametric nonlinear fractional Riccati equations. This paper obtains the results on the existence, uniqueness and regularity of the solutions to the parametric nonlinear fractional Riccati equations, proposes a high-order scheme to solve the system and proves the high-order convergence. The option pricing problem is solved by the Fourier-cosine formula with the fast approximation of the characteristic functions. Numerical examples are carried out to confirm the theoretical results and show efficiency of the methods.

本文研究了多元粗糙波动率模型下期权定价的有效方法。在多元粗糙波动率模型下，对期权定价起重要作用的资产对数价格特征函数由参数非线性分数Riccati方程组确定。得到了参数非线性分数阶Riccati方程解的存在性、唯一性和正则性的结果，提出了求解该系统的高阶格式，并证明了高阶收敛性。期权定价问题通过快速逼近特征函数的傅里叶余弦公式来解决。通过数值算例验证了理论结果并展示了方法的有效性。