How do you replicate a barrier option and what's the replication cost?

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AcadiFi · Accepted Answer

A down-and-out call pays like a vanilla call but knocks out (expires worthless) if the underlying hits barrier H from above. Static replication uses Carr-Ellis-Gupta approach: the down-and-out call can be replicated by a long vanilla call at strike K minus a 'reflection' portfolio that replicates the payoff beyond the barrier. Specifically, for a down-and-out call with K > H: Value = C(S,K) - (H/S)^(2r/sigma^2 - 1) * C(H^2/S, K). The second term is a scaled call on the reflected spot H^2/S, which mirrors the in-the-barrier region. For Meridian's contract with S = $105, K = $100, H = $95, T = 1, sigma = 25%, r = 4%: vanilla call C(105,100) = $12.80. Reflection: (95/105)^(0.28) * C(85.95, 100) = 0.971 * 1.85 = $1.80. Down-and-out call value = 12.80 - 1.80 = $11.00. ```mermaid flowchart TB V[Vanilla call
long] --> R[Replication
portfolio] M[Mirror call
short] --> R R --> B[Barrier
payoff] ``` Replication cost equals the fair value when markets allow trading at all levels. Discrepancies arise from discrete monitoring (continuous barrier assumption overprices knock-outs by ~sigma*sqrt(dt/2)), jumps, and stochastic volatility. Meridian should add a ~2-3% markup for model risk when selling exotic barriers.

How do you replicate a barrier option and what's the replication cost?

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