This paper develops efficient importance sampling schemes for a class of jump–diffusion processes that are commonly used for modeling stock prices. For such financial models, related option pricing problems are often difficult, especially when the option under study is out-of-the-money and there are multiple underlying assets. Even though analytical pricing formulas do exist in a few very simple cases, often analysts must resort to numerical methods or Monte Carlo simulation. We demonstrate that efficient and easy-to-implement importance sampling schemes can be constructed via the method of cross-entropy combined with the expectation–maximization algorithm, when the alternative sampling distributions are chosen from the family of exponentially tilted distributions or their mixtures. Theoretical justification is given by characterizing the limiting behavior of the cross-entropy algorithm under appropriate scaling. Numerical experiments on vanilla options, path-dependent options and rainbow options are also performed to illustrate the use of this technology.

中文翻译：

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通过交叉熵对跳跃扩散进行重要性采样

本文为一类通常用于模拟股票价格的跳跃扩散过程开发了有效的重要性抽样方案。对于这样的金融模型，相关的期权定价问题往往很困难，尤其是当所研究的期权是价外的并且有多个标的资产时。尽管在一些非常简单的情况下确实存在分析定价公式，但分析师通常必须求助于数值方法或蒙特卡罗模拟。我们证明，当替代抽样分布选自指数倾斜分布族或其混合物时，可以通过交叉熵方法结合期望最大化算法构建有效且易于实现的重要性抽样方案。通过表征适当缩放下交叉熵算法的限制行为，给出了理论依据。还对普通选项、路径相关选项和彩虹选项进行了数值实验，以说明该技术的使用。