In this paper, we focus on model-free pricing and robust hedging of options depending on the local time , consistent with Vanilla options. This problem is classically approached by means of the Skorokhod embedding problem (SEP), which consists in representing a given probability on the real line as the distribution of a Brownian motion stopped at a chosen stopping time . By using the stochastic control approach initiated by Galichon, Henry-Labordere and Touzi, we recover the optimal hedging strategies and the corresponding prices given by Vallois ' embeddings to the SEP . Furthermore, we extend the analysis to the two-marginal case . We provide a construction of two-marginal embedding and some examples for which the robust superhedging problem is solved . Finally, a special multi-marginal case is studied, where we construct a Markov martingale and compute its explicit generator . In particular, we provide a new example of fake Brownian motion.
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 Vallois solutions 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.
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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
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