26 research outputs found
Root to Kellerer
We revisit Kellerer's Theorem, that is, we show that for a family of real
probability distributions which increases in convex
order there exists a Markov martingale s.t.\ .
To establish the result, we observe that the set of martingale measures with
given marginals carries a natural compact Polish topology. Based on a
particular property of the martingale coupling associated to Root's embedding
this allows for a relatively concise proof of Kellerer's theorem.
We emphasize that many of our arguments are borrowed from Kellerer
\cite{Ke72}, Lowther \cite{Lo07}, and Hirsch-Roynette-Profeta-Yor
\cite{HiPr11,HiRo12}.Comment: 8 pages, 1 figur
Robust pricing and hedging of double no-touch options
Double no-touch options, contracts which pay out a fixed amount provided an
underlying asset remains within a given interval, are commonly traded,
particularly in FX markets. In this work, we establish model-free bounds on the
price of these options based on the prices of more liquidly traded options
(call and digital call options). Key steps are the construction of super- and
sub-hedging strategies to establish the bounds, and the use of Skorokhod
embedding techniques to show the bounds are the best possible.
In addition to establishing rigorous bounds, we consider carefully what is
meant by arbitrage in settings where there is no {\it a priori} known
probability measure. We discuss two natural extensions of the notion of
arbitrage, weak arbitrage and weak free lunch with vanishing risk, which are
needed to establish equivalence between the lack of arbitrage and the existence
of a market model.Comment: 32 pages, 5 figure