How do I simulate path-dependent exotics like Asian and lookback options?

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AcadiFi · Accepted Answer

Path-dependent options require simulating full price trajectories and computing payoff from path statistics. For an arithmetic Asian call on Kilgore Semiconductors strike $88, 6-month maturity, you simulate N paths with daily steps (126 steps). At each step update S(t+dt) under GBM, accumulate the daily close into a running sum, and at maturity compute average = sum/126. Payoff = max(average - 88, 0). Discount and average across paths. For a floating-strike lookback put, track the running maximum S_max along each path and at maturity compute payoff = max(S_max - S(T), 0). Continuous-monitoring lookbacks have closed-form solutions, but discrete monitoring (daily fixings) requires simulation. Key numerical issue: arithmetic Asian options have no closed form because the sum of lognormals isn't lognormal. Geometric Asian does have closed form, which enables the control variate technique: simulate both, use geometric's analytic price to correct arithmetic's simulated price, reducing variance 90%+. ```mermaid graph LR A[Simulate
path] --> B[Track
statistic] B --> C[Compute
payoff] C --> D[Average
discount] ```

How do I simulate path-dependent exotics like Asian and lookback options?

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