Assume that an agent models a financial asset through a measure ℚ with the goal to price/hedge some derivative or optimise some expected utility. Even if the model ℚ is chosen in the most skilful and sophisticated way, the agent is left with the possibility that ℚ does not provide an exact description of reality. This leads us to the following question: will the hedge still be somewhat meaningful for models in the proximity of ℚ?If we measure proximity with the usual Wasserstein distance (say), the answer is No. Models which are similar with respect to the Wasserstein distance may provide dramatically different information on which to base a hedging strategy.Remarkably, this can be overcome by considering a suitable adapted version of the Wasserstein distance which takes the temporal structure of pricing models into account. This adapted Wasserstein distance is most closely related to the nested distance as pioneered by Pflug and Pichler (SIAM J. Optim. 20:1406–1420, 2009, SIAM J. Optim. 22:1–23, 2012, Multistage Stochastic Optimization, 2014). It allows us to establish Lipschitz properties of hedging strategies for semimartingale models in discrete and continuous time. Notably, these abstract results are sharp already for Brownian motion and European call options.

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调整后的Wasserstein距离和数学金融稳定性

假设代理商通过度量ℚ对金融资产建模，目的是对某些衍生产品进行定价/对冲或优化某些预期效用。即使以最熟练和最熟练的方式选择模型ℚ，也可能会给主体留下that不能提供对现实的准确描述的可能性。这就引出了以下问题：如果我们用通常的Wasserstein距离（比如说）测量接近度，对冲是否对proximity附近的模型仍然有意义？答案是否定的。与Wasserstein相似的模型距离可以提供在其上基部的对冲strategy.Remarkably显着不同的信息，这可以通过考虑一个合适的被克服适于Wasserstein距离的版本，其中考虑了定价模型的时间结构。适应的Wasserstein距离与嵌套距离最紧密相关，如Pflug和Pichler所倡导的（SIAM J. Optim。20：1406–1420，2009； SIAM J. Optim。22：1–23，2012，多阶段随机优化，2014 ）。它使我们能够在离散和连续时间内建立半mart模型的对冲策略的Lipschitz属性。值得注意的是，对于布朗运动和欧式看涨期权，这些抽象结果已经非常明显。