The present article deals with intra‐horizon risk in models with jumps. Our general understanding of intra‐horizon risk is along the lines of the approach taken in Boudoukh et al. (2004); Rossello (2008); Bhattacharyya et al. (2009); Bakshi and Panayotov (2010); and Leippold and Vasiljević (2020). In particular, we believe that quantifying market risk by strictly relying on point‐in‐time measures cannot be deemed a satisfactory approach in general. Instead, we argue that complementing this approach by studying measures of risk that capture the magnitude of losses potentially incurred at any time of a trading horizon is necessary when dealing with (m)any financial position(s). To address this issue, we propose an intra‐horizon analogue of the expected shortfall for general profit and loss processes and discuss its key properties. Our intra‐horizon expected shortfall is well‐defined for (m)any popular class(es) of Lévy processes encountered when modeling market dynamics and constitutes a coherent measure of risk, as introduced in Cheridito et al. (2004). On the computational side, we provide a simple method to derive the intra‐horizon risk inherent to popular Lévy dynamics. Our general technique relies on results for maturity‐randomized first‐passage probabilities and allows for a derivation of diffusion and single jump risk contributions. These theoretical results are complemented with an empirical analysis, where popular Lévy dynamics are calibrated to the S&P 500 index and Brent crude oil data, and an analysis of the resulting intra‐horizon risk is presented.

中文翻译：

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跳跃模型中的IntraHorizo​​n预期不足和风险结构

本文讨论带跳跃模型的地平线内风险。我们对地平线内风险的一般理解与Boudoukh等人所采用的方法类似。（2004）；Rossello（2008）；Bhattacharyya等。（2009）；Bakshi和Panayotov（2010）; Leippold和Vasiljević（2020）。特别是，我们认为，严格地依靠时间点度量来量化市场风险通常不能被认为是令人满意的方法。相反，我们认为，在处理任何（多个）财务状况时，有必要通过研究捕获交易时间范围内任何时间可能发生的损失幅度的风险度量来补充这种方法。为了解决这个问题，我们提出了一般损益过程中预期缺口的水平内模拟方法，并讨论了其关键特性。正如Cheridito等人所介绍的，对于在建模市场动态时遇到的所有常见的Lévy过程类别，我们的地平线内预期缺口是明确定义的，并且构成了连贯的风险度量。（2004）。在计算方面，我们提供了一种简单的方法来得出流行的Lévy动力学固有的地平线内风险。我们的通用技术依赖于成熟度随机的第一遍概率的结果，并允许推导扩散和单跳风险贡献。这些理论结果得到了经验分析的补充，在该分析中，流行的Lévy动力学已根据S＆P 500指数和Brent原油数据进行了校准，并对由此产生的水平内风险进行了分析。