2016/02/29 by Thibaut Lux, Lux, Thibaut, Antonis Papapantoleon +1 · 2 citations
Economics, Econometrics and Finance · #60E15 #62H05 #91G20 #Capital Investment and Risk Analysis #FOS: Economics and business #FOS: Mathematics #Market Dynamics and Volatility #Pricing of Securities (q-fin.PR) #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1602.08894
openalex publication_date 2016/02/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We derive upper and lower bounds on the expectation of f(\S) under\ndependence uncertainty, i.e. when the marginal distributions of the random\nvector \S=(S1,\…,Sd) are known but their dependence structure is\npartially unknown. We solve the problem by providing improved FH bounds on the\ncopula of \S that account for additional information. In particular,\nwe derive bounds when the values of the copula are given on a compact subset of\n[0,1]d, the value of a functional of the copula is prescribed or different\ntypes of information are available on the lower dimensional marginals of the\ncopula. We then show that, in contrast to the two-dimensional case, the bounds\nare quasi-copulas but fail to be copulas if d>2. Thus, in order to translate\nthe improved FH bounds into bounds on the expectation of f(\S), we\ndevelop an alternative representation of multivariate integrals with respect to\ncopulas that admits also quasi-copulas as integrators. By means of this\nrepresentation, we provide an integral characterization of orthant orders on\nthe set of quasi-copulas which relates the improved FH bounds to bounds on the\nexpectation of f(\S). Finally, we apply these results to compute\nmodel-free bounds on the prices of multi-asset options that take partial\ninformation on the dependence structure into account, such as correlations or\nmarket prices of other traded derivatives. The numerical results show that the\nadditional information leads to a significant improvement of the option price\nbounds compared to the situation where only the marginal distributions are\nknown.\n