A fast algorithm for simulation of rough volatility models
针对粗糙波动率模型模拟中经典欧拉算法计算量大的问题，提出一种基于指数函数近似弱奇异核的两步迭代快速算法，将单条路径复杂度从O(N²)降至O(N log N)或O(N log² N)，并推广到带机制转换的粗糙Heston模型。
A rough volatility model contains a stochastic Volterra integral with a weakly singular kernel. The classical Euler-Maruyama algorithm is not very efficient for simulating this kind of model because one needs to keep records of all the past path-values and thus the computational complexity is too large. This paper develops a fast two-step iteration algorithm using an approximation of the weakly singular kernel with a sum of exponential functions. Compared to the Euler-Maruyama algorithm, the complexity of the fast algorithm is reduced from O(N2) to O(NlogN) or O(Nlog2N) for simulating one path, where N is the number of time steps. Further, the fast algorithm is developed to simulate rough Heston models with (or without) regime switching, and multi-factor approximation algorithms are also studied and compared. The convergence rates of the Euler-Maruyama algorithm and the fast algorithm are proved. A number of numerical examples are carried out to confirm the high efficiency of the proposed algorithm.