# 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.

Canonical: https://wiki.fcontext.com/options/variance-swap-replication/
Fact checked: 2026-07-22

> For educational purposes only; not investment advice.

<a id="answer"></a>

## 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.

<a id="mechanism"></a>

## 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.

<a id="example"></a>

## 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.

<a id="risks"></a>

## 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.

<a id="misconceptions"></a>

## 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.

<a id="related"></a>

## Related topics

- [Variance Risk Premium](/options/variance-risk-premium/)
- [Implied Volatility](/options/implied-volatility/)
- [Implied and Realized Volatility](/options/iv-and-rv/)
- [Option Model Risk](/options/option-model-risk/)

<a id="sources"></a>

## Authoritative sources

- [A Guide to Volatility and Variance Swaps](https://doi.org/10.3905/jod.1999.319129) — Kresimir Demeterfi, Emanuel Derman, Michael Kamal, and Joseph Zou
- [Variance Risk Premiums](https://doi.org/10.1093/rfs/hhn038) — Peter Carr and Liuren Wu
- [Cboe Volatility Index Mathematics Methodology](https://cdn.cboe.com/resources/indices/Cboe_Volatility_Index_Mathematics_Methodology.pdf) — Cboe Global Indices
- [Characteristics and Risks of Standardized Options](https://www.theocc.com/company-information/documents-and-archives/options-disclosure-document) — Options Clearing Corporation