The subject of this paper is the static replication of barrier options with vanilla options. The authors establish analytically the connection between the two presently known methods for static replication: fixed-strike variable-maturity for the vanillas (the so-called 'horizontal' replication) and variable-strike fixed-maturity ('vertical' replication). As a corollary, the continuous limit of the 'horizontal' replication strategy is derived. Static replication strategies are also demonstrated for arbitrary (continuous) barrier payoffs, nonflat barriers, and time-dependent parameters (i.e. drift and volatility).
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