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Variance Swap Replication: From an Option Strip to Variance Exposure

For educational purposes only; not investment advice.

Direct answer

Section titled “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

Section titled “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

Section titled “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 Ask 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

Section titled “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, 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

Section titled “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.
Section titled “Related topics”

Authoritative sources

Section titled “Authoritative sources”