Before revision: 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 .
After revision: In this paper, we focus on model-free pricing and robust hedging of options depending on the local time by using the recently introduced 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 .