论文标题

通过快速检测最坏情况来计算预期短缺

Computation of Expected Shortfall by fast detection of worst scenarios

论文作者

Bouchard, Bruno, Reghai, Adil, Virrion, Benjamin

论文摘要

我们考虑了用于计算历史预期短缺的多步算法,例如巴塞尔最低资本风险的最低资本要求所定义。在算法的每个步骤中,我们都会使用蒙特卡洛模拟来减少潜在属于最坏情况的历史场景的数量。仿真数量随着候选场景的数量减少而增加,并且它们之间的距离减少。对于最幼稚的方案,我们表明,预期不足的估计器的l p-error是由每个步骤中有利和不利场景的反转概率的线性组合以及与每种情况相关的最后一步蒙特卡洛误差的概率。通过使用浓度不等式,我们表明,对于亚伽马定价误差,反转概率以指数率在模拟路径的数量中收敛。然后,我们提出了一个适应性版本,其中算法逐步改善了对未知参数的知识:蒙特卡洛估计量的均值和差异。可以通过使用可以离线求解的动态编程算法来优化这两种方案。据我们所知,这些是此类估计量的第一个非反应界限。我们的假设足够弱,可以根据相同的随机变量在不同的情况和步骤中使用估计量,实际上,这大大减少了计算工作。进行了第一个数值测试。

We consider a multi-step algorithm for the computation of the historical expected shortfall such as defined by the Basel Minimum Capital Requirements for Market Risk. At each step of the algorithm, we use Monte Carlo simulations to reduce the number of historical scenarios that potentially belong to the set of worst scenarios. The number of simulations increases as the number of candidate scenarios is reduced and the distance between them diminishes. For the most naive scheme, we show that the L p-error of the estimator of the Expected Shortfall is bounded by a linear combination of the probabilities of inversion of favorable and unfavorable scenarios at each step, and of the last step Monte Carlo error associated to each scenario. By using concentration inequalities, we then show that, for sub-gamma pricing errors, the probabilities of inversion converge at an exponential rate in the number of simulated paths. We then propose an adaptative version in which the algorithm improves step by step its knowledge on the unknown parameters of interest: mean and variance of the Monte Carlo estimators of the different scenarios. Both schemes can be optimized by using dynamic programming algorithms that can be solved off-line. To our knowledge, these are the first non-asymptotic bounds for such estimators. Our hypotheses are weak enough to allow for the use of estimators for the different scenarios and steps based on the same random variables, which, in practice, reduces considerably the computational effort. First numerical tests are performed.

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