$d$ -copula的改进Fréchet $-$ Hoeffding界及其在无模型金融中的应用
Improved Fréchet$-$Hoeffding bounds on $d$-copulas and applications in model-free finance
摘要：
我们推导出$f(\mathbf{S})$的依赖不确定性下的期望上界和下界，即当随机向量$\mathbf{S}=(S_1,\dots,S_d)$的边缘分布是已知的，但其依赖结构是部分未知的。我们解决了这个问题，通过提供改进的\FH边界上的copula的$\mathbf{S}$，占额外的信息。特别是，我们得到的界限时，Copula的值是在$[0,1]^d$的一个紧凑的子集，Copula函数的值是规定的或不同类型的信息是可用的低维边缘的Copula。然后，我们表明，在对比的二维情况下，边界是准copula，但未能copula，如果$d>2$。因此，为了将改进后的\FH界限转化为$f(\mathbf{S})$期望的界限，我们开发了一种多元积分的替代表示，相对于Copula，也承认准Copula作为积分。通过这种表示，我们提供了一个完整的性质的orthant命令的集合上的准copula，涉及到改进的\FH界的期望上的界限$f(\mathbf{S})$。最后，我们应用这些结果来计算多资产期权价格的无模型边界，这些期权考虑了依赖结构上的部分信息，例如其他交易衍生品的相关性或市场价格。数值结果表明，与只知道边缘分布的情形相比，附加信息的引入使得期权价格的边界得到了显著的改善.
英文摘要：
We derive upper and lower bounds on the expectation of $f(\mathbf{S})$ under dependence uncertainty, i.e. when the marginal distributions of the random vector $\mathbf{S}=(S_1,\dots,S_d)$ are known but their dependence structure is partially unknown. We solve the problem by providing improved \FH bounds on the copula of $\mathbf{S}$ that account for additional information. In particular, we derive bounds when the values of the copula are given on a compact subset of $[0,1]^d$, the value of a functional of the copula is prescribed or different types of information are available on the lower dimensional marginals of the copula. We then show that, in contrast to the two-dimensional case, the bounds are quasi-copulas but fail to be copulas if $d>2$. Thus, in order to translate the improved \FH bounds into bounds on the expectation of $f(\mathbf{S})$, we develop an alternative representation of multivariate integrals with respect to copulas that admits also quasi-copulas as integrators. By means of this representation, we provide an integral characterization of orthant orders on the set of quasi-copulas which relates the improved \FH bounds to bounds on the expectation of $f(\mathbf{S})$. Finally, we apply these results to compute model-free bounds on the prices of multi-asset options that take partial information on the dependence structure into account, such as correlations or market prices of other traded derivatives. The numerical results show that the additional information leads to a significant improvement of the option price bounds compared to the situation where only the marginal distributions are known.
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