Quantitative Volatility Derivatives, OTC Variance Swaps & Vega Convexity
Variance Swap Vega-to-Variance Notional & Convexity Formula
Calculate Variance Notional N_var = N_vega / (2*K_vol), Variance Swap Settlement P&L, Linear Volatility Swap P&L, and Strictly Non-Negative Volatility Convexity Gain.
Why being LONG a Variance Swap beats a linear Volatility Swap whether realized vol crashes OR explodes!
Institutional OTC equity and FX desks quote Variance Swaps in Vega Notional N_vega (currency gained/lost per +1.0 volatility point near the strike K_vol, e.g. 50,000 USD per vol point at K_vol = 20.0%), yet settle cash purely on squared volatility points via Variance Notional N_var = N_vega / (2 * K_vol)! At maturity, Variance Swap Payoff equals P&L_var = N_var * (sigma_realized^2 - K_vol^2), which algebraically decomposes into Linear Volatility Swap P&L = N_vega * (sigma_realized - K_vol) PLUS Convexity Gain = N_var * ((sigma_realized - K_vol)^2) >= 0! Because (sigma_realized - K_vol)^2 is ALWAYS non-negative, long variance wins extra upside when crash volatility spikes AND suffers smaller losses than a linear vol swap when realized volatility collapses—which is why fair variance swap strikes trade at a vol-of-vol convexity premium above ATM implied volatility!
Try this S&P 500 Volatility Desk Settlement Check first—Vega_Notional N_vega = 50,000.00 (B2), Volatility_Strike K_vol = 20.00 vol pts (B3 -> Variance Notional N_var = 50,000 / 40 = 1,250.00 per variance pt in E2), and Realized_Volatility sigma_r = 28.00 vol pts (B4): Variance Swap P&L hits 1,250 * (784 - 400) = +480,000.00 (E3), beating Linear Volatility Swap P&L (+400,000.00 in E4) by +80,000.00 of pure convexity (E5)! On a cloud collaboration risk workbook, desk reviewers using scoped external OAuth tools like the Arte Sheets ChatGPT `0.1.0` connector (currently `In review` / `Not published`; managed in `Profile -> Connected apps`) can inspect `read_range` (`<= 200` cells) raw formulas and the base `revision` without consuming Arte AI credits (`120` MCP calls/min per-account burst limiter; ChatGPT credits never transfer to the metered in-app Pro Assistant).
| OTC Variance Swap KPI (Vega Notional = 50,000, Strike = 20.0, Realized = 28.0) | Formula | Verified Output |
|---|---|---|
| Variance Notional Nvar ($/var pt) | Vega_Notional / (2 * Vol_Strike) | 1,250.00 |
| Variance Swap Settlement P&L | N_var * (Realized_Vol^2 - Strike^2) | +480,000.00 (-320,000 @ 12.0 vol) |
| Linear Volatility Swap P&L | Vega_Notional * (Realized_Vol - Strike) | +400,000.00 (-400,000 @ 12.0 vol) |
| Volatility Convexity Advantage | Var_Swap_PnL - Linear_Vol_Swap_PnL | +80,000.00 (+80,000 @ 12.0 vol too!) |
Vega-to-Variance Notional Conversion & Convexity Decomposition Formula
Validate Vega_Notional != 0 (positive = long variance, negative = short variance—beware uncapped crash tail risk when short!), Vol_Strike > 0, and Realized_Vol >= 0!
=LET(vg_n, B2, kv_l, B3, rv_l, B4, IF(OR(vg_n = 0, kv_l <= 0, rv_l < 0), "Invalid Variance Input", LET(nv_r, vg_n / (2 * kv_l), vp_l, nv_r * ((rv_l ^ 2) - (kv_l ^ 2)), lp_l, vg_n * (rv_l - kv_l), ROUND(vp_l - lp_l, 2))))Symmetric Downside Calm-Market Convexity Cushion Check (Set Realized Vol = 12.00)
Drop Realized_Volatility B4 by 8 vol points below strike to 12.00: Linear Vol Swap P&L E4 loses -400,000.00, whereas Long Variance Swap P&L E3 loses only -320,000.00—delivering the EXACT same +80,000.00 Convexity Gain (E5 = 1,250 * (-8)^2)!
=IF(ISTEXT(E5), E5, LET(nv_r, B2 / (2 * B3), ROUND(nv_r * ((B4 ^ 2) - (B3 ^ 2)), 2)))Try the example
Use the original values or change them to test your understanding.