In this paper, we analyse some equity-linked contracts that are related to drawdown and drawup events based on assets governed by a geometric spectrally negative Lévy process. Drawdown and drawup refer to the differences between the historical maximum and minimum of the asset price and its current value, respectively. We consider four contracts. In the first contract, a protection buyer pays a premium with a constant intensity p until the drawdown of fixed size occurs. In return, he/she receives a certain insured amount at the drawdown epoch, which depends on the drawdown level at that moment. Next, the insurance contract may expire earlier if a certain fixed drawup event occurs prior to the fixed drawdown. The last two contracts are extensions of the previous ones but with an additional cancellable feature that allows the investor to terminate the contracts earlier. In these cases, a fee for early stopping depends on the drawdown level at the stopping epoch. In this work, we focus on two problems: calculating the fair premium p for basic contracts and finding the optimal stopping rule for the polices with a cancellable feature. To do this, we use a fluctuation theory of Lévy processes and rely on a theory of optimal stopping.

中文翻译：

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具有一般保险和罚金功能的Lévy型提款-提款合同的公平估值

在本文中，我们基于几何负谱的Lévy过程控制的资产，分析了与提款和提款事件相关的一些股票挂钩合同。提用和提用分别是指资产价格的历史最高和最低与其现值之间的差异。我们考虑四个合同。在第一份合同中，保护购买者以恒定强度p支付溢价。直到发生固定尺寸的缩水。作为回报，他/她在提取时期收到一定的保险金额，这取决于当时的提取水平。接下来，如果在固定提取之前发生一定的固定提取事件，则保险合同可能会更早失效。最后两个合同是先前合同的扩展，但具有附加的可取消功能，使投资者可以更早地终止合同。在这些情况下，提早停止的费用取决于停止时期的回撤水平。在这项工作中，我们关注两个问题：计算基本合同的公允溢价p和为具有可取消特征的策略找到最佳停止规则。为此，我们使用Lévy过程的涨落理论，并依赖于最佳停止理论。