Some Results on Skorokhod Embedding and Robust Hedging with Local Time

In this paper, we provide some results on Skorokhod embedding with local time and its applications to the robust hedging problem in finance. First we investigate the robust hedging of options depending on the local time by using the recently introduced stochastic control approach, in order to identify the optimal hedging strategies, as well as the market models that realize the extremal no-arbitrage prices. As a by-product, the optimality of Vallois' Skorokhod embeddings is recovered. In addition, under appropriate conditions, we derive a new solution to the two-marginal Skorokhod embedding as a generalization of the Vallois solution. It turns out from our analysis that one needs to relax the monotonicity assumption on the embedding functions in order to embed a larger class of marginal distributions. Finally, in a full-marginal setting where the stopping times given by Vallois are well-ordered, we construct a remarkable Markov martingale which provides a new example of fake Brownian motion

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

The maximum maximum of a martingale with given nn marginals

We obtain bounds on the distribution of the maximum of a martingale with fixed marginals at finitely many intermediate times. The bounds are sharp and attained by a solution to $n$-marginal Skorokhod embedding problem in Ob{\l}\'oj and Spoida [An iterated Az\'ema-Yor type embedding for finitely many marginals (2013) Preprint]. It follows that their embedding maximizes the maximum among all other embeddings. Our motivating problem is superhedging lookback options under volatility uncertainty for an investor allowed to dynamically trade the underlying asset and statically trade European call options for all possible strikes and finitely-many maturities. We derive a pathwise inequality which induces the cheapest superhedging value, which extends the two-marginals pathwise inequality of Brown, Hobson and Rogers [Probab. Theory Related Fields 119 (2001) 558-578]. This inequality, proved by elementary arguments, is derived by following the stochastic control approach of Galichon, Henry-Labord\`ere and Touzi [Ann. Appl. Probab. 24 (2014) 312-336].Comment: Published at http://dx.doi.org/10.1214/14-AAP1084 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Model-independent pricing with insider information: a Skorokhod embedding approach

In this paper, we consider the pricing and hedging of a financial derivative for an insider trader, in a model-independent setting. In particular, we suppose that the insider wants to act in a way which is independent of any modelling assumptions, but that she observes market information in the form of the prices of vanilla call options on the asset. We also assume that both the insider's information, which takes the form of a set of impossible paths, and the payoff of the derivative are time-invariant. This setup allows us to adapt recent work of Beiglboeck, Cox and Huesmann (2016) to prove duality results and a monotonicity principle, which enables us to determine geometric properties of the optimal models. Moreover, we show that this setup is powerful, in that we are able to find analytic and numerical solutions to certain pricing and hedging problems

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

Optimal Skorokhod embedding under finitely-many marginal constraints

The Skorokhod embedding problem aims to represent a given probability measure on the real line as the distribution of Brownian motion stopped at a chosen stopping time. In this paper, we consider an extension of the optimal Skorokhod embedding problem to the case of finitely-many marginal constraints. Using the classical convex duality approach together with the optimal stopping theory, we obtain the duality results which are formulated by means of probability measures on an enlarged space. We also relate these results to the problem of martingale optimal transport under multiple marginal constraints

On the monotonicity principle of optimal Skorokhod embedding problem

In this paper, we provide an alternative proof of the monotonicity principle for the optimal Skorokhod embedding problem established by Beiglb\"ock, Cox and Huesmann. This principle presents a geometric characterization that reflects the desired optimality properties of Skorokhod embeddings. Our proof is based on the adaptation of the Monge-Kantorovich duality in our context together with a delicate application of the optional cross-section theorem and a clever conditioning argument