How is a variance swap replicated using a strip of options, and why is this replication important?

A variance swap is a forward contract on realized variance. The buyer pays a fixed variance strike (K_var) and receives the realized variance of the underlying over the contract period. Replication using options is the foundation for how dealers price and hedge these instruments.\n\nWhy Options Replicate Variance:\n\nThe key mathematical insight is that the log return of a stock can be decomposed using a Taylor expansion into a portfolio of options across all strikes. Specifically:\n\nRealized variance = (2/T) x [integral from 0 to infinity of (1/K^2) x O(K) dK]\n\nwhere O(K) is the price of an out-of-the-money option at strike K (puts for K < S, calls for K > S).\n\nDiscretized Replication:\n\nIn practice, you approximate the continuous integral with a finite strip of options:\n\n| Strike | Type | Weight (1/K^2) | Contribution |\n|---|---|---|---|\n| $70 | OTM Put | 1/4900 = 0.000204 | Small |\n| $80 | OTM Put | 1/6400 = 0.000156 | Medium |\n| $90 | OTM Put | 1/8100 = 0.000123 | Medium |\n| $100 | ATM (both) | 1/10000 = 0.000100 | Largest |\n| $110 | OTM Call | 1/12100 = 0.000083 | Medium |\n| $120 | OTM Call | 1/14400 = 0.000069 | Small |\n| $130 | OTM Call | 1/16900 = 0.000059 | Small |\n\nThe 1/K^2 weighting means lower strikes get more weight, which captures the skew in the volatility surface.\n\nWorked Example:\n\nGranville Index at 1,000. A dealer sells a 3-month variance swap at K_var = (22%)^2 = 0.0484.\n\nTo hedge, the dealer buys the following option strip (simplified to 5 strikes):\n\n- 2 x 900-strike puts at $8.50 each = $17.00\n- 3 x 950-strike puts at $14.20 each = $42.60\n- 4 x 1000-strike straddle at $45.00 each = $180.00\n- 3 x 1050-strike calls at $12.80 each = $38.40\n- 2 x 1100-strike calls at $6.90 each = $13.80\n\nTotal replication cost: $291.80\n\nThis portfolio gains value when realized volatility exceeds implied (the stock moves more than expected in either direction), perfectly offsetting the dealer's variance swap liability.\n\nWhy This Matters:\n\n1. Pricing: The fair variance strike is determined by options prices across the surface, not by historical volatility\n2. VIX Calculation: The CBOE VIX index is calculated using exactly this options strip methodology\n3. Model-Free: The replication doesn't depend on Black-Scholes or any particular pricing model\n4. Truncation Risk: In practice, far OTM strikes may not have liquid markets, creating replication error\n\nLearn more about volatility derivatives in our CFA Derivatives resources.