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Martingale Optimal Transport and Robust Hedging in Continuous Time
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 has the 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
Pathwise super-replication via Vovk's outer measure
Since Hobson's seminal paper [D. Hobson: Robust hedging of the lookback
option. In: Finance Stoch. (1998)] the connection between model-independent
pricing and the Skorokhod embedding problem has been a driving force in robust
finance. We establish a general pricing-hedging duality for financial
derivatives which are susceptible to the Skorokhod approach.
Using Vovk's approach to mathematical finance we derive a model-independent
super-replication theorem in continuous time, given information on finitely
many marginals. Our result covers a broad range of exotic derivatives,
including lookback options, discretely monitored Asian options, and options on
realized variance.Comment: 18 page
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