Variance Swap And Volatility Derivative Pricing

When to Use

Use this skill when you need a fair variance strike, an accrued realized variance, or a mark on a seasoned variance swap — pricing a new trade off a listed option chain, marking a live position for variation margin, or checking a dealer's quote.

The engine provides:

• Fair variance strike $K_{\text{var}}$ by static log-contract replication over an OTM option strip (Demeterfi, Derman, Kamal & Zou, More Than You Ever Wanted To Know About Volatility Swaps, Goldman Sachs, March 1999 — "DDKZ" — Equation 27).
• Annualized realized variance from a price history, zero-mean convention.
• Fair volatility strike $K_{\text{vol}}$ with the convexity correction, given a vol-of-vol input.
• Notional conversion $N_{\text{var}} = N_{\text{vega}} / (2 K_{\text{vol}})$.
• Seasoned MTM: the accrued variance leg blended with the forward leg priced off today's strip.

Units. Everything is in volatility points squared. A 20% volatility is K_vol = 20.0 and K_var = 400.0 — never 0.20 / 0.04. Vega notional is dollars per volatility point; variance notional is dollars per variance point.

When NOT to Use

• To settle a contract. calculate_realized_variance divides by the number of returns actually observed. A term sheet divides by the expected observation count fixed at inception and specifies market-disruption handling for missing days. Use this for accrual-to-date monitoring; settle from the confirmation.
• To strike a volatility swap at sqrt(K_var). There is no static replication of a volatility swap. sqrt(K_var) is DDKZ's naive estimate (Equation 44) and is a strict upper bound — striking there means the variance swap dominates the volatility swap at every realized volatility. Supply vol_of_vol_points. price_variance_swap_mtm refuses a VOLATILITY_SWAP contract for the same reason.
• On a jumpy underlying, without a haircut. Replication is exact only for a continuous path. A single downward jump of size $J$ leaves a residual whose leading term is cubic, $\frac{2}{3T}J^3$ (DDKZ Equations 40–42) — a 10% one-day gap is worth ~7.2 variance points on a one-year swap (DDKZ Table 5).
• On a thin or one-sided chain. See the truncation pitfall below. The engine raises on a one-sided strip rather than returning a number.
• For VIX futures or listed volatility ETPs. Those are forwards on an index, not OTC swaps — see vix-and-volatility-index-derivative-strategies.

Prerequisites

• Python 3.10+, standard library only (math, dataclasses, enum, logging).
• An option chain for the swap's maturity: strike, CALL/PUT, and a price per quote. A full two-sided chain is fine — ITM quotes are discarded and each strike contributes once.
• The continuously compounded risk-free rate to maturity, and the spot as of the valuation date (not inception).

Workflow

1. Fix the units before anything else. Confirm whether the desk quotes $K_{\text{vol}}$ in points (20.0) or decimals (0.20), and whether the notional on the term sheet is vega or variance. At a 20% strike these differ by a factor of 40 — see the first pitfall.

2. Build the contract. VarianceSwapContract(...) with strike_vol_pct, vega_notional_usd, t_years, and the inception spot_price / risk_free_rate. Read variance_notional_usd rather than dividing by hand.

3. Replicate the fair variance strike. Call calculate_fair_strikes(spot, r, t_years, option_strip). The engine picks $S^* = K_0$, the largest available strike at or below the forward $F = S_0 e^{rT}$, uses puts below it, calls above it, and the average of the put and call at $K_0$ (the Cboe convention). It then applies DDKZ Equation 27:

   $$K_{\text{var}} = \frac{2}{T}\left[rT - \left(\frac{F}{S^} - 1\right) - \ln\frac{S^}{S_0}\right] + \frac{2}{T}e^{rT}\sum_i \frac{\Delta K_i}{K_i^2}Q(K_i)$$

   The bracketed term is not zero in general. It vanishes only when $S^* = F$ exactly, which a discrete strike grid almost never delivers.

4. Read the replication diagnostics, don't just take the number. Check reference_strike, min_strike, max_strike, and num_options_used. If the engine logged a truncation warning, treat $K_{\text{var}}$ as a lower bound and decide whether the missing wings are material at this maturity — they are not the same size at three months and at one year.

5. Decide the volatility strike deliberately. For a variance swap, stop at step 4. For a volatility swap, pass vol_of_vol_points — the standard deviation of realized volatility in volatility points, from a model or a VIX-of-VIX style market. Under DDKZ's Appendix D normal-volatility assumption $K_{\text{var}} = K_{\text{vol}}^2 + \operatorname{Var}(\sigma_R)$ exactly, so $K_{\text{vol}} = \sqrt{K_{\text{var}} - \text{vol-of-vol}^2}$ and convexity_adjustment_pct is that variance. Leaving the default of 0.0 gives the naive upper bound, not a tradeable strike.

6. Compute the accrued variance. calculate_realized_variance(price_history) on the official closing prices named in the confirmation, with the term sheet's annualization factor (DDKZ use 260; the default here is 252). The sample mean is deliberately not subtracted.

7. Mark the seasoned contract. price_variance_swap_mtm(..., current_spot=..., current_risk_free_rate=...). Pass both. They default to the inception values with a warning, which puts the forward — and with it the put/call boundary — in the wrong place. The blend is exact because variance is additive in time:

   $$V_{\text{exp}} = \frac{t}{T}\sigma^2_{\text{realized}} + \frac{T-t}{T}K_{\text{var,rem}}, \qquad \text{MTM} = e^{-r(T-t)}N_{\text{var}}\left(V_{\text{exp}} - K_{\text{var,strike}}\right)$$

8. Re-mark on a schedule, not on a move. Both legs move: the accrued leg grows with each new close, and the forward leg re-prices with the strip.

Common Pitfalls

• Quoting variance notional as if it were vega notional. P&L is linear in variance, not volatility. $N_{\text{var}} = N_{\text{vega}} / (2K_{\text{vol}})$, so a $100,000 vega-notional trade at a 20% strike is a $2,500 variance notional — a factor of 40. Sizing the variance notional at $100,000 is a 40x over-exposure, and the error only shows up once realized volatility moves.
• Passing a full option chain and trusting the $\Delta K$ grid. A two-sided chain quotes a put and a call at every strike. If $\Delta K_i$ is computed over the raw list, every interior spacing is halved and $K_{\text{var}}$ is understated by roughly half. Collapse to one OTM price per strike first, then build the grid. This engine does; a hand-rolled sum usually does not.
• Dropping the $S^*$ anchor term because "it's zero at the forward". It is zero when $S^* = F$. Once you anchor on a traded strike $K_0 \ne F$ — which you must, to avoid a gap at the boundary — the term is live. It reduces to Cboe's $-\frac{1}{T}\left(\frac{F}{K_0}-1\right)^2$ to second order. Omitting it, or substituting $\frac{1}{T}\left(\frac{F}{S_0} - 1 - \ln\frac{F}{S_0}\right)$, biases $K_{\text{var}}$ by ~13 variance points at $r = 5%$, $T = 1$.
• Reading a truncated strip as a fair price. A finite strike range always understates the fair variance, and the shortfall grows with maturity: DDKZ Table 4 prices a flat-25%-vol underlying at $(25.0)^2$ from a 50%–200% strike range but at only $(23.0)^2$ from a 75%–125% range at one year — two full volatility points. At three months the same narrow range costs only 0.1 points. Never treat a narrow-strip $K_{\text{var}}$ as a fair mid.
• Accepting a one-sided strip. A calls-only chain silently contributes nothing below the forward; the arithmetic still returns a number, and it is badly low. Check that both wings are present before integrating — this engine raises instead.
• Substituting the strike for missing data. If a seasoned contract has accrued time but no price history, or remaining time but no strip, there is no mark. Filling the gap with $K_{\text{var,strike}}$ produces a mark of exactly zero P&L on the missing leg, which reads as "flat" rather than "unknown" on a risk blotter.
• "Fixing" the zero-mean convention into a sample variance. DDKZ (page 2) note the zero-mean method "is theoretically preferable, because it corresponds most closely to the contract that can be replicated by options portfolios". Subtracting the sample mean makes a trending underlying look calm.

Verification

python -m unittest discover -s skills/variance-swap-and-volatility-derivative-pricing/scripts

The suite checks the engine against sources outside itself, not against its own arithmetic:

• A dense, wide strip of flat-volatility Black-Scholes prices must return $K_{\text{var}} = \sigma^2$ — DDKZ Figure 5 states the theoretical value is exactly $(20)^2 = 400$ at $\Delta K \to 0$.
• The DDKZ Table 1 worked example (page 21: $S_0 = 100$, $r = 5%$, $T = 0.25$, strikes 50–150 spaced 5 apart, 20% ATM vol with a 1-point-per-5-strike skew) must reproduce the paper's $K_{\text{var}} = (20.467)^2$.
• The DDKZ Table 4 truncation cases must reproduce $(25.0)^2$ wide versus $(24.9)^2$ and $(23.0)^2$ narrow, at three months and one year respectively.
• A full two-sided chain and the equivalent OTM-only strip must agree.
• $K_{\text{var}} - K_{\text{vol}}^2$ must equal the supplied vol-of-vol variance exactly, and $K_{\text{vol}} < \sqrt{K_{\text{var}}}$ strictly.

Then work assets/checklist.md before a trade goes out.

Related Skills

• vix-and-volatility-index-derivative-strategies — listed volatility forwards, where the same $\Delta K / K^2$ weighting appears as the published index methodology.
• options-implied-volatility-surface-construction — build and clean the strip this skill integrates over.
• options-greeks-real-time-portfolio-aggregation — aggregate the option-strip hedge alongside the swap.
• tail-risk-hedging-with-options — the downside-skew exposure that lifts $K_{\text{var}}$ above ATM implied variance.
• quanto-options-and-cross-currency-derivative-structures — the cross-currency analogue when the underlying and the settlement currency differ.
• total-return-swap-synthetic-exposure — the other major OTC swap wrapper.
• warrants-and-structured-product-integration — structured wrappers that embed variance exposure.