Variance Swap Calculator: Fair Variance Strike & Swap Valuation

Enter Values

Spot Price (S)

Current price of underlying asset

Risk-Free Rate (r)

Enter as percentage (e.g., 5 for 5%)

Dividend Yield (q)

Continuous dividend yield

Time to Maturity (T)

years

Swap maturity (typically 1 month to 2 years)

ATM Implied Volatility

At-the-money implied volatility

Skew (25Δ Put − 25Δ Call Vol)

Positive = put skew (higher OTM put vols)

Variance Notional (L_var)

Variance notional ($ per 1.00 variance)

Fixed Variance Rate (V_K)

%²×100

Enter as (vol%)² × 100 (e.g., 400 = 20% vol)

Variance Replication

E[V] = (2e^rT/T) × Σ (ΔK/K²) Q(K)

Q(K) = OTM option price at strike K
Fair variance strike V_K = E[V]

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.

The payoff at maturity is: 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.

Variance vs Volatility Swaps

Variance swaps pay on realized variance (squared returns), while volatility swaps pay on realized volatility (standard deviation). Due to Jensen's inequality, the fair volatility strike is slightly lower than the square root of the fair variance strike. This calculator shows the simple square root, appropriate for variance swap pricing.

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.

Variance Swap Calculator

Enter Values

Variance Replication

Results

Fair Variance Strike

Variance Contribution by Strike

Reconstructed Volatility Smile

Volatility Smile

Strip Contributions

Formula Breakdown

Model Assumptions

Understanding Variance Swaps

What is a Variance Swap?

Why Replicate with Options?

Variance vs Volatility Swaps

Frequently Asked Questions

What is a variance swap?

How does this calculator approximate the option strip?

What is the fair variance strike?

Why is the volatility strike lower than the square root of the variance strike?

What does skew mean for variance swaps?

What are the limitations of this calculator?

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