SIAM Journal on Financial Mathematics, Volume 12, Issue 1, Page 487-549, January 2021.
Efficient sampling for the conditional time integrated variance process in the Heston stochastic volatility model is key to the simulation of the stock price based on its exact distribution. We construct a new series expansion for this integral in terms of double infinite weighted sums of particular independent random variables through a change of measure and the decomposition of squared Bessel bridges. When approximated by series truncations, this representation has exponentially decaying truncation errors. We propose feasible strategies to largely reduce the implementation of the new series to simulations of simple random variables that are independent of any model parameters. We further develop direct inversion algorithms to generate samples for such random variables based on Chebyshev polynomial approximations for their inverse distribution functions. These approximations can be used under any market conditions. Thus, we establish a strong, efficient, and almost exact sampling scheme for the Heston model.

中文翻译：

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Heston 模型的级数展开和直接反演

SIAM 金融数学杂志，第 12 卷，第 1 期，第 487-549 页，2021 年 1 月。
对 Heston 随机波动率模型中的条件时间积分方差过程进行有效采样是基于股票价格的精确分布模拟股票价格的关键。我们通过测量的变化和平方贝塞尔桥的分解，根据特定独立随机变量的双无穷加权和为这个积分构建了一个新的级数展开式。当用系列截断近似时，这种表示具有指数衰减的截断误差。我们提出了可行的策略，以在很大程度上减少新系列的实现，以模拟独立于任何模型参数的简单随机变量。我们进一步开发了直接反演算法，以基于其逆分布函数的切比雪夫多项式近似为此类随机变量生成样本。这些近似值可用于任何市场条件。因此，我们为 Heston 模型建立了一个强大、高效且几乎精确的采样方案。