How does discretization error affect path-dependent option pricing?

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AcadiFi · Accepted Answer

Path-dependent options (lookbacks, Asians, barriers) have payoffs that depend on the full trajectory, so discrete-time simulation misses extrema and averages that would occur in continuous time. For a lookback call paying max(S_T - min S_t, 0), sampling at N discrete points systematically overestimates the minimum (misses troughs between samples), which underprices the option. The bias shrinks as O(sqrt(dt)) for extrema—quite slow. For Harbor's 1-year lookback with S = $50, sigma = 30%: with 50 time steps vs 500, Monte Carlo price jumps from $7.80 to $8.45 (+8.3%). Broadie-Glasserman continuity correction shifts the discrete barrier/extremum by 0.5826 * sigma * sqrt(dt) to match continuous-time expectations, nearly eliminating bias. ```mermaid flowchart LR D[Discrete
sampling] --> B[Biased
extremum] B --> C[Broadie
correction] C --> U[Unbiased
price] ``` For Asians, discretization bias is smaller since averages are less sensitive to sampling. However, convergence is still O(sqrt(dt)) without variance reduction. Best practices: (1) use Broadie-Glasserman for barriers/lookbacks, (2) use bridge sampling for intermediate points, (3) apply control variates using geometric Asian as exact solution. Harbor should target at least 250 time steps per year for lookbacks and validate with the correction formula before marking positions.

How does discretization error affect path-dependent option pricing?

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