Options Implied Volatility Surface Construction

When to Use

Use this skill when you have discrete exchange-traded option quotes and need the surface $\sigma(K, T)$ they imply, plus evidence that the surface you fitted does not contain a free lunch. Naive interpolation and unconstrained fitting both routinely produce surfaces with static arbitrage: total implied variance that falls with maturity (a calendar spread bought for a negative price) or a call price concave in strike (a butterfly with negative cost, equivalently a negative risk-neutral probability density). A backtest priced off such a surface will find "edge" that is an artifact of the fit.

The engine inverts quoted prices to implied volatilities by bisection, least-squares fits the quadratic smile $\sigma(m) = \sigma_{\text{ATM}} + \alpha(m-1) + \beta(m-1)^2$ per expiration, evaluates the surface on a strike $\times$ expiration grid, and audits both no-arbitrage conditions against the definitions in Gatheral & Jacquier, Arbitrage-free SVI volatility surfaces (arXiv:1204.0646).

When NOT to Use

• To price a backtest off a surface you already trust. This engine calibrates and audits; it has no Greeks and no term-structure decay of the skew. Use options-backtesting-with-realistic-iv-surface, which evaluates a parametric surface with a power-law skew decay and returns analytic Greeks — and which deliberately does not check arbitrage, delegating that here.
• On American option prices without adjustment. Every listed US single-stock option is American. Inverting an American price through a European formula returns a volatility inflated by the early-exercise premium, worst for ITM puts and for calls across an ex-dividend date. See american-vs-european-style-option-exercise-handling.
• Across a discrete cash dividend without adjusting spot. Only a continuous yield $q$ is modelled. Remove the present value of dividends paid before expiry from $S$ yourself, first.
• As a far-wing extrapolator. The quadratic smile violates Lee's moment formula asymptotically (see Common Pitfalls). Do not evaluate it far outside the moneyness range your quotes actually cover.
• As a proof of no arbitrage. The conditions are statements for all $k \in \mathbb{R}$ and all $t > 0$; the engine checks the finite grid you hand it. A clean report is evidence on that grid, not a theorem.
• As raw SVI. This engine fits a quadratic in moneyness, not the five-parameter SVI form $w(k) = a + b{\rho(k-m) + \sqrt{(k-m)^2 + \sigma^2}}$. The arbitrage conditions used here come from the SVI literature; the parameterization does not. Earlier versions of this skill described the model as "SVI / quadratic smile" — that was wrong and has been removed.

Prerequisites

• Option market quotes (strike, tte_years, market_price, option_type), one expiration at a time for calibration, with spot already adjusted for discrete dividends paid before expiry.
• IVSurfaceConfig: spot_price, risk_free_rate (continuously compounded), atm_vol, skew_alpha, smile_beta, dividend_yield.
• Quotes whose vega is large enough to identify a volatility. Deep out-of-the-money short-dated quotes carry almost no volatility information and the engine will say so.

Workflow

1. Invert market prices to implied volatilities — implied_volatility_from_price:
   • Decision point — check the no-arbitrage price bounds before solving. A call must satisfy $\max(Se^{-qt} - Ke^{-rt}, 0) < C < Se^{-qt}$; a put, $\max(Ke^{-rt} - Se^{-qt}, 0) < P < Ke^{-rt}$. A quote at or outside those bounds has no implied volatility. The engine raises rather than returning a clamped number, because a fabricated wing volatility survives into the fit and then into every price derived from it.
   • Decision point — bisection, not Newton-Raphson. The BS price is strictly increasing in $\sigma$, so a bracketed bisection converges unconditionally. Newton's step is $\text{residual}/\text{vega}$, and vega collapses towards zero deep in and out of the money and at very short expiry, where the step explodes.
   • Decision point — a converged solver is not the same as an identified volatility. The engine estimates the resolution as (float64 price resolution) / vega and logs a warning when it exceeds $10^{-6}$. Exclude flagged quotes from the fit; do not let a number that is arbitrary to $\pm 0.0075$ pull the wings.
2. Calibrate the smile per expiration — calibrate_smile_from_quotes:
   • Least-squares fit of $\sigma(x) = \sigma_{\text{ATM}} + \alpha x + \beta x^2$ where $x = K/S - 1$, solved from the $3\times3$ normal equations.
   • Decision point — one expiration per fit. Mixing tenors fits a surface cross-section as if it were a slice; the engine raises. Fewer than three distinct moneyness levels leaves the coefficients unidentified and also raises, rather than returning an arbitrary solution to a singular system.
3. Evaluate the grid and audit static arbitrage — construct_surface_grid:
   • Calendar spread: total implied variance $w(k, \tau) = \sigma^2 \tau$ must be non-decreasing in $\tau$ at fixed log-forward moneyness $k = \ln(K/F_\tau)$, $F_\tau = Se^{(r-q)\tau}$ — Gatheral & Jacquier Lemma 2.1, whose proof compares $K_1/F_{\tau_1} = K_2/F_{\tau_2}$.
   • Decision point — auditing at fixed strike is the wrong comparison. Whenever $r \neq q$ the forward drifts, so the same strike sits at a different $k$ at each expiration. At $r = 5%$ a one-year gap moves the forward ~5%, several strikes on a listed chain. This is a real defect that hides real violations: see Verification for a surface every fixed-strike scan calls monotone and the correct fixed-$k$ scan rejects.
   • Butterfly: for each consecutive strike triple, the spacing-weighted butterfly $w_1 C(K_1) + w_3 C(K_3) - C(K_2)$, with $w_1 = (K_3-K_2)/(K_3-K_1)$ and $w_3 = (K_2-K_1)/(K_3-K_1)$, must be non-negative. This is the tradeable form of $\partial^2 C/\partial K^2 \ge 0$, which by Breeden & Litzenberger (1978) is the risk-neutral density.
   • Decision point — use the spacing weights, not $0.5/0.5$. Listed chains are not equally spaced. On strikes $(90, 95, 150)$ the equal-weighted butterfly is negative on a perfectly benign surface; the correctly weighted one is positive.
   • Decision point — a fit that passes slice-by-slice can still fail as a surface. Independent per-expiration calibration says nothing about calendar monotonicity. Always audit after calibrating.
4. Read the report — IVSurfaceConstructionReport:
   • Decision point — "no violations found" is not "arbitrage-free". A grid with one expiration cannot be audited for calendar arbitrage, and one with two strikes cannot be audited for butterfly arbitrage. calendar_audit_performed / butterfly_audit_performed record what actually ran, the status is UNAUDITED_SURFACE, and is_arbitrage_free is False. Unaudited is not clean.

Full procedure: see references/workflows.md. Standards reference: see references/standards.md. Printable pre-flight checklist: see assets/checklist.md.

Common Pitfalls

• Auditing calendar arbitrage at fixed strike instead of fixed log-forward moneyness: the condition $\partial_\tau w \ge 0$ is stated at constant $K/F$. With $r \neq q$ the forward moves between expirations, so a fixed-strike scan compares two different points on the surface and reports a steeply skewed, genuinely arbitrageable surface as monotone.
• Retrying a quote outside the no-arbitrage bounds with a clamped volatility: below intrinsic there is no solution. Clamping to a floor produces a plausible number that then anchors the whole wing of the fit.
• Trusting an 8-decimal implied volatility from a near-zero-vega quote: a three-month 40%-OTM call at 8% volatility has a total time value of ~$2.6 \times 10^{-15}$; every volatility within roughly $\pm 0.0075$ prices to the same float64 number. Push slightly further out and the price underflows to exactly intrinsic and carries no volatility information at all.
• Equal-weighting an unequally-spaced butterfly: reports arbitrage that does not exist, and can mask arbitrage that does.
• Extrapolating the quadratic smile into the far wings: Lee, The Moment Formula for Implied Volatility at Extreme Strikes (Mathematical Finance 14(3), 2004), proves that absence of arbitrage bounds the tail by $\sigma^2_{\text{BS}} \le \beta|k|/\tau$ with $\beta \le 2$ — total implied variance may grow at most linearly in $|k|$. A quadratic-in-moneyness $\sigma$ grows it like $|k|^4$. The $[0.05, 3.0]$ clamp bounds the damage numerically; it does not make the wings arbitrage-free.
• Treating a clamped wing as an audited one: when the IV clamp binds, the surface there is no longer the parametric one and a wing that would have failed can pass. Every binding clamp is logged; a report carrying clamp warnings is inconclusive in the wings.
• Silently substituting the flat ATM level for a missing tenor in the term structure: that fabricates a term structure and can turn a real calendar violation into a clean report. The engine raises on a missing key.
• Rounding implied volatilities before squaring them into variances: rounding to 4dp can flip a marginal calendar comparison in either direction. Nothing in this engine is rounded; quantize at the presentation layer.

Verification

• Pricing: black_scholes_price("CALL", 100, 100, 1.0, 0.20, 0.05) $= 10.4506$, the standard worked example. Confirm put-call parity $C - P = Se^{-qT} - Ke^{-rT}$ holds to $10^{-12}$ with $q > 0$. Confirm tte=0 and vol=0 return discounted intrinsic, not a clamped time value.
• Smile: with $\sigma_{\text{ATM}}=0.20$, $\alpha=-0.30$, $\beta=0.50$, hand-compute $\sigma(0.90) = 0.20 + 0.030 + 0.005 = 0.2350$ and $\sigma(1.10) = 0.20 - 0.030 + 0.005 = 0.1750$. Version 1.0.0 returned $0.2175$ and $0.1875$ — it scaled the offset by an undocumented $0.5$.
• Inversion: round-trip price$\to$IV$\to$price across strikes $70$–$140$ and tenors one month to two years. Every case where the round trip is worse than $10^{-6}$ must be flagged by the conditioning warning.
• Calendar (regression): with $\alpha = -2.0$, $\beta = 0$, strikes $(98, 100, 102)$ and $\tau \in {0.5, 1.0}$, the fixed-strike scan reports all three strikes monotone. The correct fixed-$k$ scan finds a violation at $k = \ln(102/F_{0.5}) = -0.005197$: $w = 0.012800$ at $\tau=0.5$ falls to $0.011741$ at $\tau=1.0$.
• Calendar (term structure): 40% ATM vol at three months falling to 15% at one year gives $w(0.25) = 0.0400 > w(1.0) = 0.0225$ $\implies$ CALENDAR_ARBITRAGE_VIOLATION.
• Butterfly: with $\alpha = -2.0$, strikes $(95, 100, 105)$ at $\tau = 1$, the smile gives $\sigma = (0.30, 0.20, 0.10)$ and the butterfly prices to $-0.0269$ $\implies$ BUTTERFLY_ARBITRAGE_VIOLATION.
• Unaudited: a one-expiration grid and a two-strike grid must both report UNAUDITED_SURFACE with is_arbitrage_free $=$ False.
• Negative checks: non-positive strike/spot/tte/atm_vol, NaN or Inf anywhere, an unknown option_type such as "C", mixed expirations in one calibration, fewer than three distinct moneyness levels, and a missing atm_vol_by_tte key must all raise ValueError.
• Run python -m unittest discover -s skills/options-implied-volatility-surface-construction/scripts and confirm a 100% pass rate.

Related Skills

• options-backtesting-with-realistic-iv-surface
• options-chain-data-normalization-across-vendors
• options-greeks-real-time-portfolio-aggregation
• american-vs-european-style-option-exercise-handling
• real-time-greeks-recalculation-on-market-moves