Static replication of Barrier-type options via integral equations

This study provides a systematic and unified approach for constructing exact and static replications for exotic options, using the theory of integral equations. In particular, we focus on barrier-type options including standard, double and sequential barriers. Our primary approach to static options replication is the DEK method proposed by Derman et al. (1995). However, our solution approach is novel in the sense that we study its continuous-time version using integral equations. We prove the existence and uniqueness of hedge weights under certain conditions. Further, if the underlying dynamics is time-homogeneous, then hedge weights can be explicitly found via Laplace transforms. Based on our framework, we propose an improved version of the DEK method. This method is applicable under general Markovian diffusion with killing.