The problem of model uncertainty has recently received widespread attention in financial mathematics. On the application side, this is due to the recent financial crisis where overconfidence in models played an important role. On the mathematical side, the reason is that an abundance of interesting connections to other areas have emerged; in particular, optimal transport, stochastic analysis, nonlinear partial differential equations, Skorokhod embeddings, nonlinear expectations, decision theory, and quasi-sure analysis. In this project, the investigator studies how model uncertainty influences the fundamental tasks of pricing, hedging and investment in financial markets. Students are included in the work of the project.
The first part of this project is concerned with the pricing and hedging of financial derivatives under model uncertainty. If both the underlying security and liquid options are used as hedging instruments, the superreplication principle yields sharp and robust bounds for derivatives prices that are consistent with the market data, without making excessively strong assumptions about model dynamics. Moreover, this approach yields a robust hedging strategy to manage the associated risk. In a mild idealization where a continuum of call options can be traded, superreplication is intimately linked to a Monge-Kantorovich optimal transport problem, namely, a transport between the marginal laws of the security. The no-arbitrage principle of finance imposes a probabilistic structure on this transport, which leads to the so-called martingale optimal transport problem that is studied vigorously in this project. The investigation leads to the computation of worst-case scenarios and hedging strategies, and is also of great interest to probability theory and analysis. A second part of the project is dedicated to understanding the impact of model uncertainty on the optimal portfolio choice for an investor such as a retirement fund. A third part, again related to the pricing of derivatives, studies a problem of financial engineering: How does one algorithmically construct market models that are consistent with quoted option prices as observed in financial markets? More precisely, the project shows how to build a risk-neutral model that is calibrated to a given set of liquidly traded instruments, not necessarily of plain vanilla type. Students are included in the work of the project.
