Martingale optimal transport and robust hedging in continuous time

Abstract

The duality between the robust (or equivalently, model independent) hedging of path dependent European options and a martingale optimal transport problem is proved. The financial market is modeled through a risky asset whose price is only assumed to be a continuous function of time. The hedging problem is to construct a minimal super-hedging portfolio that consists of dynamically trading the underlying risky asset and a static position of vanilla options which can be exercised at the given, fixed maturity. The dual is a Monge-Kantorovich type martingale transport problem of maximizing the expected value of the option over all martingale measures that have a given marginal at maturity. In addition to duality, a family of simple, piecewise constant super-replication portfolios that asymptotically achieve the minimal super-replication cost is constructed.

Mini-Flash Crashes, Model Risk, and Optimal Execution

“…The seminal book by Hansen & Sargent offers a comprehensive discussion ( [69]). There have been many more recent developments as well (see [51], [90], [36], [33], [29], [27], [28], [30], [14], and the references therein).…”

Model uncertainty and the pricing of American options

“…It might be expected that very little can be said with this approach, but the model-free literature ( [18,6,20,8,13,19,22,21,23,3,15,1]) shows that if we take the prices of vanilla options as given, and only consider martingale models which are consistent with these prices, then we can find non-trivial bounds on the prices of exotic derivatives. In this literature the idea is to not specify a model, or even a probability space or filtration.…”

“…Kahalé [23] describes a general approach via convex programming for pricing and hedging European path-dependent claims in the presence of European options, using a set-up which is similar to that in the main part of this paper. Beiglböck, Henry-Labordère and Penkner [3] and Dolinsky and Soner [15] use arguments from the mass-transportation literature to find bounds on general path-dependent options in the presence of European option prices; they show that the dual problem can be interpreted as a robust hedge, as also does Acciaio et al [1]. Hobson [19] provides a survey and relates the problem to the Skorokhod embedding problem.…”

“…trading strategies that are defined over paths (eg Definition 1.1 in Acciaio et al [1], and Section 2.3 of Dolinsky and Soner [15]). But with an American claim the situation is more subtle because the exercise decision may be altered by the arrival of new information.…”

Convex Duality with Transaction Costs

“…For the convenience of the reader we briefly review this assumption, but refer to [12] for an extended discussion and its connection with the Skorokhod metric. In particular, all options on the running maximum and Asian type options satisfy it.…”