In this paper, we focus on model-free pricing and robust hedging of options depending on the local time when one or more marginals of the underlying price process are known. By using the stochastic control approach initiated in Galichon, Henry-Labord\`ere and Touzi, we identify the optimal hedging strategies and the corresponding prices in the one-marginal case . As a by-product, we recover the property that the Valloissolutions to the Skorokhod embedding problem (SEP) maximize and minimize the expectation of any convex function of the local time. Furthermore, we extend the analysis to the two-marginal case, where we provide candidates for the optimal hedging strategies, and, we construct a new solution to the two-marginal SEP as a generalization of the Vallois embedding . Finally, a special multi-marginal case is studied, where the stopping times given by Vallois are well-ordered . In the n-marginal case, we solve the robust hedging problem as it essentially reduces to the one-marginal case. In the full marginal setting, we construct a remarkable Markov martingale and compute its generator explicitly. In particular, this provides a new example of fake Brownian motion.
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.
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focus on model-free pricing and robust hedging
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
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when one or more marginals of the underlying price process are known. By using the
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initiated in Galichon, Henry-Labord\`ere and Touzi, we
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and the corresponding prices in the one-marginal case
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we recover the property that the Valloissolutions to the Skorokhod embedding problem (SEP) maximize and minimize the expectation of any convex function of the local time. Furthermore, we extend the analysis to the two-marginal case, where we provide candidates for the optimal hedging strategies, and, we construct
the optimality of Vallois' Skorokhod embeddings is recovered. In addition, under appropriate conditions, we derive
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a special multi-marginal case is studied,
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. In the n-marginal case, we solve the robust hedging problem as it essentially reduces to the one-marginal case. In the full marginal setting, we
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