How is a variance swap replicated using options, and why is this replication significant for volatility trading?

Variance swap replication is one of the most important results in derivatives theory, connecting the options market to the volatility market. It's why the VIX exists.

Variance Swap Basics:

• Long variance: Pays off if realized variance > strike variance
• Short variance: Pays off if realized variance < strike variance
• Payoff = Notional x (Realized Variance - Strike Variance)

The Key Insight: Static Replication with Options

A variance swap can be replicated by a portfolio of options across ALL strikes, weighted inversely by the square of the strike price (1/K^2 weighting).

Specifically:

Variance = (2/T) x integral of [Price(K)/K^2] dK for all strikes K from 0 to infinity

In practice, this means:

• Buy OTM puts at all available strikes below the forward price
• Buy OTM calls at all available strikes above the forward price
• Weight each option by 1/K^2

Why 1/K^2 Weighting?

A log contract on the stock payoff = ln(S_T/S_0) has a payoff that exactly equals the negative of half the realized variance (plus a linear term). And a log contract is replicated by the 1/K^2 weighted strip of options. This is a mathematical result from the Breeden-Litzenberger theorem.

Discrete Approximation (How It Works in Practice):

Sum of contributions (x 2/T, annualized) = Implied Variance = Strike Variance for the variance swap.

Why This Matters:

1. VIX Calculation — The VIX index is literally the square root of this replication formula applied to S&P 500 options. The VIX IS the variance swap strike, expressed as volatility.

1. Model-Free — Unlike Black-Scholes IV (which assumes a specific model), the variance swap replication is model-free. It uses market prices directly.

1. OTM Put Dominance — Because of the 1/K^2 weighting, low-strike OTM puts get the highest weights. This is why the VIX is so sensitive to put skew and crash risk.

1. Hedging — Variance swap dealers use this strip of options as their hedge. They're not making a directional bet — they're arbitraging between the variance swap price and the options market.

Exam Tip: Know that variance swaps can be replicated by a strip of OTM options weighted by 1/K^2, understand this is the basis of VIX calculation, and recognize that this replication is model-free. The exam may ask why OTM puts have outsized influence on the variance swap strike.

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