20 research outputs found
Robust pricing--hedging duality for American options in discrete time financial markets
We investigate pricing-hedging duality for American options in discrete time
financial models where some assets are traded dynamically and others, e.g. a
family of European options, only statically. In the first part of the paper we
consider an abstract setting, which includes the classical case with a fixed
reference probability measure as well as the robust framework with a
non-dominated family of probability measures. Our first insight is that by
considering a (universal) enlargement of the space, we can see American options
as European options and recover the pricing-hedging duality, which may fail in
the original formulation. This may be seen as a weak formulation of the
original problem. Our second insight is that lack of duality is caused by the
lack of dynamic consistency and hence a different enlargement with dynamic
consistency is sufficient to recover duality: it is enough to consider
(fictitious) extensions of the market in which all the assets are traded
dynamically. In the second part of the paper we study two important examples of
robust framework: the setup of Bouchard and Nutz (2015) and the martingale
optimal transport setup of Beiglb\"ock et al. (2013), and show that our general
results apply in both cases and allow us to obtain pricing-hedging duality for
American options.Comment: 29 page
Robust Superhedging with Jumps and Diffusion
We establish a nondominated version of the optional decomposition theorem in
a setting that includes jump processes with nonvanishing diffusion as well as
general continuous processes. This result is used to derive a robust
superhedging duality and the existence of an optimal superhedging strategy for
general contingent claims. We illustrate the main results in the framework of
nonlinear L\'evy processes.Comment: Forthcoming in 'Stochastic Processes and their Applications
Robust bounds for the American Put
We consider the problem of finding a model-free upper bound on the price of
an American put given the prices of a family of European puts on the same
underlying asset. Specifically we assume that the American put must be
exercised at either or and that we know the prices of all vanilla
European puts with these maturities. In this setting we find a model which is
consistent with European put prices and an associated exercise time, for which
the price of the American put is maximal. Moreover we derive a cheapest
superhedge. The model associated with the highest price of the American put is
constructed from the left-curtain martingale transport of Beiglb\"{o}ck and
Juillet.Comment: 31 pages, 14 figure
On Arbitrage and Duality under Model Uncertainty and Portfolio Constraints
We consider the fundamental theorem of asset pricing (FTAP) and hedging
prices of options under non-dominated model uncertainty and portfolio
constrains in discrete time. We first show that no arbitrage holds if and only
if there exists some family of probability measures such that any admissible
portfolio value process is a local super-martingale under these measures. We
also get the non-dominated optional decomposition with constraints. From this
decomposition, we get duality of the super-hedging prices of European options,
as well as the sub- and super-hedging prices of American options. Finally, we
get the FTAP and duality of super-hedging prices in a market where stocks are
traded dynamically and options are traded statically.Comment: Final version. To appear in Mathematical Financ