Jan Obłój

The Skorokhod embedding problem and its offspring

This is a survey about the Skorokhod embedding problem. It presents all known solutions together with their properties and some applications. Some of the solutions are just described, while others are studied in detail and their proofs are presented. A certain unification of proofs, thanks to real potential theory, is made. Some new facts which appeared in a natural way when different solutions were cross-examined, are reported. Azema and Yors and Roots solutions are studied extensively. A possible use of the latter is suggested together with a conjecture.

Robust pricing and hedging of double no-touch options

Double no-touch options are contracts which pay out a fixed amount provided an underlying asset remains within a given interval. In this work, we establish model-independent 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 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.

Robust Hedging of Double Touch Barrier Options

We consider robust pricing of digital options, which pay out if the underlying asset has crossed both upper and lower barriers. We make only weak assumptions about the underlying process (typically continuity), but assume that the initial prices of call options with the same maturity and all strikes are known. In such circumstances, we are able to give upper and lower bounds on the arbitrage-free prices of the relevant options and show that these bounds are tight. Moreover, pathwise inequalities are derived, which provide the trading strategies with which we are able to realize any potential arbitrages. These super- and subhedging strategies have a simple quasi-static structure, their associated hedging error is bounded below, and in practice they carry low transaction costs. We show that, depending on the risk aversion of the investor, they can outperform significantly the standard delta/vega-hedging in presence of market frictions and/or model misspecification. We make use of embeddings techniques; in particular, we develop two new solutions to the (optimal) Skorokhod embedding problem.

On Local Martingale and its Supremum: Harmonic Functions and beyond

We discuss certain facts involving a continuous local martingale

Maximum Maximum of Martingales Given Marginals

N

Portfolio optimisation under non-linear drawdown constraints in a semimartingale financial model

and its supremum

The Maximality Principle Revisited: On Certain Optimal Stopping Problems

\bar{N}

The Root solution to the multi-marginal embedding problem: an optimal stopping and time-reversal approach

. A complete characterization of

An iterated Azéma–Yor type embedding for finitely many marginals

(N,\bar{N})

Martingale Inequalities for the Maximum via Pathwise Arguments

-harmonic functions is proposed. This yields an important family of martingales, the usefulness of which is demonstrated, by means of examples involving the Skorokhod embedding problem, bounds on the law of the supremum, or the local time at 0, of a martingale with a fixed terminal distribution, or yet in some Brownian penalization problems. In particular we obtain new bounds on the law of the local time at 0, which involve the excess wealth order.