Approximation of martingale couplings on the line in the adapted weak topology

• 成果类型: Article
• 署名作者: Beiglboeck, M.; Jourdain, B.; Margheriti, W.; Pammer, G.
• 署名单位: University of Vienna; Institut Polytechnique de Paris; Ecole Nationale des Ponts et Chaussees; Inria; Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-021-01103-y
• 发表日期: 2022
• 页码: 359-413
• 关键词: optimal transport; PROBABILITY-MEASURES; MARGINALS; Duality; version; plans

摘要

Our main result is to establish stability of martingale couplings: suppose that pi is a martingale coupling with marginals mu, nu. Then, given approximating marginal measures (mu) over tilde approximate to mu, (nu) over tilde approximate to nu in convex order, we show that there exists an approximating martingale coupling (pi) over tilde approximate to pi with marginals (mu) over tilde, (nu) over tilde. In mathematical finance, prices of European call/put option yield information on the marginal measures of the arbitrage free pricing measures. The above result asserts that small variations of call/put prices lead only to small variations on the level of arbitrage free pricing measures. While these facts have been anticipated for some time, the actual proof requires somewhat intricate stability results for the adapted Wasserstein distance. Notably the result has consequences for several related problems. Specifically, it is relevant for numerical approximations, it leads to a new proof of the monotonicity principle of martingale optimal transport and it implies stability of weak martingale optimal transport as well as optimal Skorokhod embedding. On the mathematical finance side this yields continuity of the robust pricing problem for exotic options and VIX options with respect to market data. These applications will be detailed in two companion papers.