Enter Values

$
Current price of underlying asset
%
Enter as percentage (e.g., 5 for 5%)
%
Continuous dividend yield
years
Swap maturity (typically 1 month to 2 years)
%
At-the-money implied volatility
%
Positive = put skew (higher OTM put vols)
$
Variance notional ($ per 1.00 variance)
%²×100
Enter as (vol%)² × 100 (e.g., 400 = 20% vol)
Variance Replication
E[V] = (2erT/T) × Σ (ΔK/K²) Q(K)
Q(K) = OTM option price at strike K
Fair variance strike VK = E[V]
Ryan O'Connell, CFA
Calculator by Ryan O'Connell, CFA

Results

Fair Variance Strike

--
E[V] as (vol%)² × 100
Symmetric
Fair Vol Strike
--
Forward Price
--
Swap Value
--

Variance Contribution by Strike

OTM Puts OTM Calls

Reconstructed Volatility Smile

Volatility Smile

Strike K/F0 Vol

Strip Contributions

Strike Type Vol Price Contrib

Formula Breakdown

Model Assumptions

  • Continuous replication: Variance is replicated with a static strip of options (no dynamic hedging).
  • Smile approximation: Linear volatility in log-moneyness from ATM vol and 25Δ skew.
  • Finite strike grid: 21 strikes from 50% to 150% of forward price (truncates tails).
  • Black-Scholes pricing: Options priced using lognormal BS model at smile vols.
  • Variance notional: Payoff = L × (realized variance − fixed variance).

Understanding Variance Swaps

What is a Variance Swap?

A variance swap is an over-the-counter derivative that pays the difference between realized variance and a pre-agreed strike variance, multiplied by a notional amount. Unlike options, variance swaps provide pure exposure to volatility without any delta hedging required during the life of the contract.

Variance Swap Payoff
Payoff = Notional × (Realized Variance − Strike Variance)

Why Replicate with Options?

The key insight from Hull Chapter 26 is that variance can be replicated using a static portfolio of out-of-the-money options. The replication formula weights each option by ΔK/K², which gives more weight to lower strikes. This explains why variance swaps are sensitive to put skew—higher OTM put volatilities translate directly into higher fair variance strikes.

Hull Ch. 26 Replication Formula
E[V] = (2erT/T) × Σi (ΔKi/Ki²) × Q(Ki)
Where Q(K) = put for K < F, call for K ≥ F

Variance vs Volatility Swaps

Variance Swap

Pays on squared returns. Easier to replicate with options. Fair strike = E[V]. This calculator computes this directly.

Volatility Swap

Pays on standard deviation. Due to Jensen's inequality, fair strike is slightly lower than √E[V]. Requires convexity adjustment.

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Frequently Asked Questions

A variance swap is a derivative contract where one party pays the other the difference between realized variance and a fixed variance strike, multiplied by a notional amount. The realized variance is typically calculated from daily log returns over the swap period. Variance swaps provide pure exposure to volatility without delta hedging.

The calculator uses ATM implied volatility and a skew parameter to reconstruct a volatility smile. It then prices OTM puts and calls at each strike using Black-Scholes, and sums the variance contributions using the Hull Chapter 26 replication formula. This approximation avoids requiring a full option chain.

The fair variance strike is the level of fixed variance that makes a variance swap have zero initial value. It equals the expected variance under the risk-neutral measure, calculated by replicating the variance payoff with a strip of OTM options. This is the standard quoting convention for new variance swaps.

For volatility swaps, a convexity adjustment is needed because E[sqrt(V)] is less than sqrt(E[V]) by Jensen's inequality. The fair volatility strike for a volatility swap is approximately sqrt(E[V]) times (1 minus var(V)/(8*E[V]^2)). This calculator shows the simple square root without the convexity adjustment, which is appropriate for variance swaps.

Volatility skew refers to the pattern where OTM put options typically have higher implied volatility than OTM calls (positive skew in equity markets). Higher put skew increases the fair variance strike because the variance replication formula weights OTM options by 1/K^2, giving more weight to lower strikes where puts dominate.

This calculator uses a simplified linear smile model which may not match real market smiles. It uses a finite strike grid (21 strikes over plus/minus 50% of forward) which truncates the tails. Real variance swap pricing uses actual market option prices and may include discrete monitoring adjustments. Use this for educational purposes only.
Disclaimer

This calculator is for educational purposes only. It uses a simplified volatility smile model and finite strike grid. Real variance swap pricing requires actual market option prices and may include discrete monitoring adjustments. Not financial advice.

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