Some Results on Skorokhod Embedding and Robust Hedging with Local Time

TL;DR

This paper explores Skorokhod embedding with local time and applies it to robust option hedging, deriving new solutions and revealing market extremal models, including a novel fake Brownian motion example.

Contribution

It introduces a generalized Skorokhod embedding solution relaxing monotonicity, applies stochastic control to robust hedging, and constructs a new Markov martingale example.

Findings

Recovered the optimality of Vallois' embeddings.

Derived a new two-marginal Skorokhod embedding solution.

Constructed a new fake Brownian motion example.

Abstract

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.