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Variance Swap Replication: From an Option Strip to Variance Exposure

See how a strip of out-of-the-money options approximates a log contract and fair variance, calculate strike weights, and identify replication errors.

Updated

For educational purposes only; not investment advice. Investing may result in loss.

Scope: this article explains an illustrative fair-variance approximation for European-style options under stated continuous-path and market assumptions. It does not specify any particular Cboe index, OTC variance swap, listed option, broker account, margin arrangement, or tax treatment. Product terms, data rules, eligibility, protections, and legal or tax consequences vary by venue, account, jurisdiction, and date; consult the current governing documents and qualified advisers for a specific case.

Direct answer

Variance swap replication connects realized return variance to a portfolio containing a log payoff, a dynamically adjusted forward or underlying position, and cash. The terminal log payoff can in turn be represented by a continuum of European Calls and Puts across strikes. This leads to a practical approximation of fair variance from an option strip.

The key insight is not “buy every option and variance appears.” Under idealized assumptions, out-of-the-money Puts below a reference strike and Calls above it receive weights proportional to 1/K²; the option strip supplies terminal convexity, while dynamic trading along the path links that payoff to realized variance.

Real markets offer only discrete strikes, finite tails, Bid/Ask quotes, discrete hedging, and contract-specific settlement. The result is an approximation with explicit replication and execution risk, not a guaranteed arbitrage.

From the log contract to the option strip

For a continuous diffusion, Itô’s formula applied to the logarithm of price relates cumulative squared returns to a dynamically traded position and a terminal log payoff. A static-replication identity then decomposes a sufficiently smooth terminal payoff into cash, a forward, and options across all strikes. For the log payoff, the option density scales as 1/K².

A widely used discrete fair-variance approximation is:

σ² ≈ (2/T) × Σᵢ [ΔKᵢ/Kᵢ² × e^(RT) × Q(Kᵢ)] − (1/T) × (F/K₀ − 1)²

Here T is years to expiry, F is the option-implied forward, K₀ is the first strike at or below F, R is the matched risk-free rate, and Q(Kᵢ) is the selected option price. Under the Cboe-style construction, use OTM Puts for Kᵢ<K₀, OTM Calls for Kᵢ>K₀, and an average of the Put and Call at K₀. For an interior strike, ΔKᵢ=(Kᵢ₊₁−Kᵢ₋₁)/2; endpoints use their adjacent interval.

The 1/K² factor is only one part of the weight. Strike spacing ΔK, time, rates, price, forward adjustment, and option-selection rules all matter. The result is annualized variance; taking its square root and multiplying by 100 produces a volatility-style number only if the variance is nonnegative and the conventions match.

VIX methodology is an observable implementation related to variance-swap mathematics, but VIX itself is an index with prescribed option-selection, interpolation, timing, and dissemination rules. It is not identical to every OTC variance-swap contract or its realized leg.

One-strike contribution example

Consider one OTM Put in a 30-day strip. Let T=30/365, R=4%, strike K=90, neighboring-strike interval ΔK=5, and selected option price Q(K)=$0.60 per index unit. Its contribution to the summation after the leading annualization factor is:

(2/T) × (ΔK/K²) × e^(RT) × Q(K)

= [2/(30/365)] × (5/90²) × e^(0.04×30/365) × 0.60 ≈ 0.00904

That is about 90.4 variance basis points, because one variance basis point is 0.0001. It is not the whole fair variance and √0.00904≈9.51% is not the strip’s final volatility. Every eligible strike must be added and the forward-adjustment term must then be subtracted.

Suppose a stale quote makes this Put’s selected price $0.90 instead of $0.60. Its calculated contribution rises by 50% to roughly 0.01356, illustrating why deep-tail quotes, selection rules, and liquidity can materially affect an estimate even though the formula is applied correctly.

For listed equity options, quoted premiums are usually per share and a standard contract commonly has a 100-share multiplier. That contract cash conversion is separate from the normalized index or variance formula; never insert the multiplier unless the methodology calls for it.

Replication gaps and controls

  • Use Calls, Puts, forward, rates, and timestamps from the same expiry and observation time; asynchronous quotes can create a false surface.
  • Derive F and K₀ according to one documented method rather than substituting spot mechanically.
  • Apply exact strike-selection, zero-bid/zero-ask, midpoint, endpoint, and interpolation rules; these choices alter the strip.
  • Inspect both Bid and Ask. A smooth midpoint calculation does not prove that the entire portfolio can trade simultaneously.
  • Quantify tail truncation. Missing low-strike Puts and high-strike Calls leave the theoretical continuum incomplete.
  • Compare several strike grids and quote filters to expose discretization and stale-price sensitivity.
  • Match the realized-variance definition: log or simple returns, observation calendar, sampling frequency, annualization, disruption rules, and settlement price.
  • Continuous-path identities can incur jump and discrete-sampling error; dynamic hedging also introduces gap, liquidity, and transaction-cost risk.
  • Corporate actions, dividends, American exercise, settlement style, taxes, margin, and contract adjustments can separate listed-option execution from the idealized European derivation.
  • A variance notional can create nonlinear dollar exposure. Read the contract’s payoff cap, variance strike, notional convention, and settlement terms.

Common misconceptions

  • “The 1/K² rule alone gives the weight.” ΔK, option price, discounting, time, selection, and forward adjustment are also required.
  • “Only at-the-money options determine variance.” The construction uses a broad cross-section of OTM Puts and Calls.
  • “The option strip by itself exactly pays realized variance.” The theoretical identity also contains dynamic trading and cash terms.
  • “VIX is a variance swap.” It is an index calculated under its own methodology; contract payoffs and realized legs must be checked separately.
  • “More distant strikes can be ignored because their premiums are small.” The tail helps represent extreme states, and truncation creates error.
  • “A model-free formula has no assumptions.” It reduces dependence on a chosen stochastic-volatility model but still relies on market, path, sampling, and implementation conditions.
  • “A correct spreadsheet ensures an executable hedge.” Wide spreads, stale quotes, partial fills, margin, and rebalancing costs remain.

Authoritative sources

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