Market Presets:
Contract Parameters
Volatility Analytics
OTM Put Strip ($K < F$) |
OTM Call Strip ($K > F$) |
Weight scales as $2 \Delta K / (T \cdot K^2)$
Underlying Spot Path ($S_t$) |
Cumulative Realized Vol |
Fair Strike Level
Variance Swap P&L (Linear in $\sigma_R^2$, Convex in $\sigma_R$) |
Capped Payoff
Fair Variance Strike ($K_{\text{var}}$)
432.50
Equivalent Vol: 20.80%
Variance Risk Premium (VRP)
+32.50
Spread vs ATM²: +0.80% vol
Variance Notional ($N_{\text{var}}$)
$1,201.92
per vol point squared
Swap Net P&L (Long)
+$172,475
Vol P&L: +3.20 vol ptsInstitutional Valuation Architecture
1. Terminal Payoff & Notional Scaling
A variance swap pays the difference between the annualized realized variance $\sigma_R^2$ and the delivery variance strike $K_{\text{var}}$:
Payoff = N_var × (σ_R² - K_var)
N_var = N_vega / (2 × √K_var)
N_var = N_vega / (2 × √K_var)
Realized variance is calculated from $N$ daily closing log returns: $\sigma_R^2 = \frac{252}{N} \sum_{i=1}^N \left(\ln\frac{S_i}{S_{i-1}}\right)^2 \times 10,000$.
2. Static Log-Contract Replication
Following Demeterfi, Derman, Kamal & Zou (1999), continuous variance is replicated without dynamic rehedging by holding a static portfolio of options:
K_var = (2/T) [rT - (F_0/S_* - 1) - ln(S_*/S_0)] + (2/T) e^{rT} [∫_0^{S_*} P(K)/K² dK + ∫_{S_*}^∞ C(K)/K² dK]
Because of negative equity skew, out-of-the-money puts trade at a premium, inflating the $1/K^2$ weighted integral and driving $K_{\text{var}} > \sigma_{\text{ATM}}^2$.
Swap P&L: Realized Vol ($\sigma_R$) vs Strike Vol ($\sqrt{K_{\text{var}}}$)
Fair Variance Strike ($K_{\text{var}}$): Skew Slope vs ATM Vol ($\sigma$)
Quantitative Audit Export:
Sanitized RFC 4180 CSV with formula injection defense.
Variance Swaps & Volatility Mastery Quiz
Frequently Asked Questions
A variance swap is a forward contract on the annualized realized variance of an underlying asset over a specified maturity. At maturity, the buyer receives $N_{\text{var}} \times (\text{Realized Variance} - \text{Fair Variance Strike})$. Variance notional $N_{\text{var}}$ is defined as $\text{Vega Notional} / (2 \cdot \sqrt{\text{Fair Variance Strike}})$.
Following the seminal Demeterfi, Derman, Kamal, and Zou (1999) framework, the log contract $\ln(S_T / S_0)$ is replicated by a static portfolio of out-of-the-money puts and calls with weights inversely proportional to the strike squared ($1 / K^2$). Integrating across the strike surface yields the fair variance strike.
Because equity markets exhibit a pronounced negative volatility skew (expensive out-of-the-money puts due to crash fear), the $1 / K^2$ weighting places disproportionate weight on lower strikes. This skew steepness inflates the integrated fair variance strike above the square of ATM implied volatility.
A variance swap pays linearly on realized variance (volatility squared), whereas a volatility swap pays linearly on realized volatility. By Jensen's Inequality, because the square-root function is strictly concave, the expected volatility is strictly less than the square root of expected variance, creating a convexity premium.