Implied Volatility Surface — SVI Calibration Pipeline

End-to-end pipeline from raw SPY option chain data to a calibrated, arbitrage-verified implied volatility surface using the Gatheral (2004) SVI parametrisation.

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Repository Structure

iv-surface-svi/
├── iv_surface.py          # Per-slice SVI pipeline (clean → IV → SVI → arbitrage)
├── ssvi.py                # Surface SVI: globally arbitrage-free calibration
├── fetch_data.py          # Standalone REAL-data extractor (run in VS Code)
├── requirements.txt       # Pinned dependencies
├── README.md
├── LICENSE
├── .gitignore
├── data/
│   └── spy_chains_YYYY-MM-DD.csv   # Written by fetch_data.py (carries spot)
└── figures/
    ├── fig0_raw_smiles.png          # Raw IV smiles before SVI fit
    ├── fig1_svi_smile_fits.png      # Per-slice SVI fit per maturity
    ├── fig2_svi_surface_3d.png      # 3D SVI surface iv(m, T)
    ├── fig3_atm_term_structure.png  # ATM term structure: model vs raw
    ├── fig4_skew_analysis.png       # Approximate RR and butterfly vs maturity
    ├── fig5_ssvi_vs_svi.png         # SSVI vs per-slice SVI per maturity
    └── fig6_ssvi_surface_3d.png     # 3D SSVI surface (arbitrage-free)

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Installation and Usage

git clone https://github.com/tancredeg/volatility-surface-svi
cd iv-surface-svi
pip install -r requirements.txt

# Step 1 — fetch REAL market data (writes data/spy_chains_<date>.csv)
python fetch_data.py

# Step 2 — per-slice SVI calibration + arbitrage checks + figures
python iv_surface.py

# Step 3 — Surface SVI (globally arbitrage-free) + comparison to per-slice SVI
python ssvi.py

fetch_data.py downloads live SPY option chains, runs realness checks (shared call/put strikes, plausible strike range, open interest present, no synthetic labels), records the underlying spot into the CSV, and writes a dated file to data/. It never fabricates data: on any failure it exits with a diagnostic.

iv_surface.py runs the per-slice pipeline. It attempts a live download; if that fails it loads the most recent CSV in data/. Spot is resolved live, then from the CSV's stored spot — never silently defaulted, because a wrong spot corrupts the forward and the whole surface. It refuses any cached file with synthetic_* labels.

ssvi.py calibrates the Surface SVI parametrisation jointly across maturities, verifies the surface is calendar- and butterfly-arbitrage-free, and prints an RMSE comparison against the per-slice SVI baseline.

Synthetic self-test (opt-in only, never the default):

from iv_surface import run_synthetic_pipeline
run_synthetic_pipeline()   # tests the calibrator against known SVI params

The synthetic path exists solely to verify that calibration recovers known parameters. Its output is labelled DATA_SOURCE = synthetic_with_noise and it never writes to data/.

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Mathematical Background

1. Log-Moneyness

The smile is parametrised in log-moneyness:

m = ln(K / F),   where F = S * exp((r - q) * T)

• S is the spot price, K the strike, T time to expiry in years.
• F is the cost-of-carry forward (continuous dividend yield q, risk-free rate r).
• Log-moneyness is preferred over simple moneyness K/S - 1 because it is symmetric around zero, dimensionless, and consistent with the log-normal distribution underlying Black-Scholes. ATM corresponds to m = 0 exactly.

2. SVI Parametrisation

Gatheral (2004) introduced the Stochastic Volatility Inspired (SVI) parametrisation of total variance w = σ²T:

w(m) = a + b * [ ρ(m - μ) + sqrt((m - μ)² + ξ²) ]

Parameter	Domain	Interpretation
a	(-0.5, 2.0)	Overall variance level. May be slightly negative at short maturities due to microstructure.
b	(0, 2.0)	Slope / curvature. Controls the speed of the wings.
ρ	(-1, 1)	Skew correlation. Negative for equities (put premium).
μ	(-0.5, 0.5)	Shift of the minimum. Near zero for index options.
ξ	(0, 2.0)	ATM smoothness. Larger values produce a flatter ATM region.

The implied volatility is recovered as:

σ(m, T) = sqrt(w(m) / T)

SVI is used in practice because:

• It fits observed smiles accurately with only five parameters.
• It has a closed-form expression, making calibration fast.
• The no-arbitrage conditions are tractable to check and enforce.
• It generalises to a full surface via the SSVI extension (Gatheral & Jacquier 2014).

3. Black-Scholes IV Extraction

Implied volatility is extracted by inverting the Black-Scholes formula (Merton 1973 extension with continuous dividends):

d1 = [ln(S/K) + (r - q + σ²/2) * T] / (σ * sqrt(T))
d2 = d1 - σ * sqrt(T)

Call = S * exp(-qT) * N(d1) - K * exp(-rT) * N(d2)
Put  = K * exp(-rT) * N(-d2) - S * exp(-qT) * N(-d1)
Vega = S * exp(-qT) * φ(d1) * sqrt(T)         (identical for call and put)

Algorithm (Newton-Raphson with Brent fallback):

1. Initialise σ₀ = 0.20.
2. Newton-Raphson step: σ_{n+1} = σ_n - (BS(σ_n) - price) / Vega(σ_n)
3. If Vega(σ_n) < 1e-10 (deep OTM), switch immediately to Brent's method.
   • Brent is slower but guaranteed convergent on a bracketed interval.
4. Brent interval: [1e-4, 10.0]. If no sign change exists (price not bracketed), return NaN with an explicit log warning — never a silent NaN.

NR diverges near the wings because vega approaches zero; the fallback to Brent prevents this failure mode.

4. Butterfly No-Arbitrage Conditions

Lee (2004) — necessary condition:

For no butterfly arbitrage, the SVI parameters must satisfy:

b * (1 + |ρ|) < 4 / T

This is a necessary but not sufficient condition. It bounds the maximum slope of the smile wings.

Durrleman butterfly condition (necessary and sufficient):

A single SVI slice is free of butterfly arbitrage — equivalently, its implied risk-neutral density is non-negative — if and only if the Durrleman function g(m) ≥ 0 for all m:

g(m) = (1 - m·w'(m)/(2·w(m)))²
       - (w'(m)²/4)·(1/w(m) + 1/4)
       + w''(m)/2

with analytic SVI derivatives:

w'(m)  = b·[ρ + (m-μ)/√((m-μ)²+ξ²)]
w''(m) = b·ξ² / ((m-μ)²+ξ²)^{3/2}

The Lee condition is a fast necessary screen; the Durrleman g(m) ≥ 0 is the complete characterisation and is what the pipeline reports in Section 5.

Note on a corrected formula. An earlier version of this project used g(m) = (1 - m·ρ/ξ_eff)² + (ρ²-1)·b²/ξ_eff² and labelled it the "Roper sufficient condition". That expression is not the Durrleman/Gatheral butterfly function: tested against arbitrage-free calibrated surfaces it returned g < 0 (false arbitrage alarms) on 3 of 4 maturities, while the correct Durrleman g was strictly positive everywhere. The formula above is the corrected, standard condition (Gatheral & Jacquier 2014, eq. 2.2–2.3).

Reference: Roper, M. (2010). Arbitrage-Free Implied Volatility Surfaces. Preprint.

Calendar no-arbitrage:

For fixed m, total variance must be non-decreasing in T:

w(m, T1) ≤ w(m, T2)   for T1 < T2

Violations on the raw data are expected due to noise and bid-ask spreads; violations on the calibrated per-slice SVI surface indicate that linear parameter interpolation is insufficient and that SSVI is needed.

5. Surface SVI (SSVI)

SSVI parametrises the entire surface rather than each slice independently:

w(k, θ) = (θ/2) · [ 1 + ρ·φ(θ)·k + √((φ(θ)·k + ρ)² + 1 − ρ²) ]
φ(θ)    = η / ( θ^γ · (1+θ)^(1−γ) )        (power-law form)

where θ = θ_t is the ATM total variance at maturity t (the backbone), and (ρ, η, γ) are three global parameters shared by all maturities. ssvi.py calibrates (ρ, η, γ, θ_1, …, θ_n) jointly by least squares on implied vol, subject to the Gatheral–Jacquier (2014) no-arbitrage conditions enforced as hard constraints:

• Calendar: θ_t non-decreasing in t.
• Butterfly: θ·φ(θ)·(1+|ρ|) < 4 and θ·φ(θ)²·(1+|ρ|) ≤ 4 for all θ.

Each SSVI slice maps to an equivalent raw-SVI slice (b = θφ/2, m = −ρ/φ, ξ = √(1−ρ²)/φ, a = (θ/2)(1−ρ²)), so the same Durrleman g(m) check validates butterfly no-arbitrage on the calibrated surface.

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Key Results

Values below come from the run of 2026-06-04 on SPY (spot 752.18). They illustrate what the pipeline produces; absolute numbers will differ with market conditions on later runs.

ATM Term Structure

Maturity	SVI ATM Vol	SSVI ATM Vol	SSVI θ_t
28d	13.8%	12.9%	0.00127
57d	15.0%	14.1%	0.00311
88d	15.4%	15.1%	0.00552
179d	16.7%	16.7%	0.01368

Term structure is upward-sloping (normal), consistent with long-run uncertainty premium. The two models agree closely on long-dated ATM vol and diverge by ≤1% at 28d, where SSVI's single global ρ trades a small amount of short-end fit against global no-arbitrage.

SVI per-slice Calibration Quality

Maturity	SVI RMSE	Lee butterfly	Durrleman g_min	Verdict
28d	0.0039	52.06 (PASS)	+0.005	PASS (margin tight)
57d	0.0040	25.52 (PASS)	+0.136	PASS
88d	0.0086	16.50 (PASS)	+0.205	PASS (RMSE > 50bp)
179d	0.0037	8.03 (PASS)	+0.274	PASS

PASS threshold: RMSE < 50 bp; Durrleman g_min ≥ 0 (necessary and sufficient for no butterfly arbitrage on the slice).

SSVI Surface (globally arbitrage-free)

Parameter	Value
ρ	−0.494
η	0.832
γ	0.582

Maturity	SSVI RMSE	Durrleman g_min	Calendar
28d	0.0158	+0.260	PASS
57d	0.0099	+0.275	PASS
88d	0.0113	+0.291	PASS
179d	0.0053	+0.335	PASS

SSVI: 0 calendar violations on |k| ≤ 0.35, 4/4 butterfly PASS. The per-slice SVI surface had 152 calendar violations from linear parameter interpolation — SSVI eliminates them by construction.

Approximate Skew Metrics

These use fixed moneyness m = ±0.10 — not true 25-delta strikes. True 25Δ RR requires solving N(d₂(K)) = 0.25 for K; a desk implementation would use the delta-strike mapping.

Maturity	Approx RR (m=±0.10)	Approx BF (m=±0.10)
28d	−10.6%	+7.79%
57d	−9.6%	+3.86%
88d	−7.9%	+3.89%
179d	−6.9%	+1.89%

Negative risk reversal across all maturities confirms equity put premium (market pricing tail risk asymmetrically downward). Skew flattens with maturity, also as expected.

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Known Limitations

1. SVI Parameter Interpolation vs SSVI

The per-slice pipeline (iv_surface.py) fits each maturity independently and interpolates the five SVI parameters linearly across maturities:

p*(T*) = p1 + (T* - T1) / (T2 - T1) * (p2 - p1)

This does not guarantee a calendar-arbitrage-free surface, and in practice it produces calendar violations in the wings (reported in Section 5). This is why ssvi.py is included: the Surface SVI parametrisation (Gatheral & Jacquier 2014) anchors the whole surface on the ATM total-variance term structure with a single correlation, and is free of calendar and butterfly arbitrage by construction. The trade-off is documented below: SSVI fits each slice slightly worse than per-slice SVI (a single global rho cannot match four individually-steeper skews) in exchange for global no-arbitrage.

2. Approximate RR/BF vs True 25Δ

Risk reversal and butterfly are computed at fixed moneyness m = ±0.10. A proper vol desk uses the 25-delta strike:

K_25Δ: solve N(d₂(K)) = 0.25   (call 25Δ)
       solve N(-d₂(K)) = 0.25  (put 25Δ)

This requires numerical root-finding per maturity and is not implemented.

3. Flat Rate and Dividend Curves

The pipeline uses scalar r and q for all maturities. A production implementation uses:

• OIS discount curve for r (bootstrapped from swap rates).
• Dividend term structure for q (bootstrapped from listed forwards or dividend swaps).

The error from flat curves is small for short maturities but material for T > 1y.

4. European Black-Scholes for American SPY Options

SPY options are American. The pipeline uses European Black-Scholes to extract IV. The early exercise premium is negligible near-the-money but non-negligible deep ITM. A desk would use a binomial tree or Barone-Adesi–Whaley approximation for American options.

5. Synthetic Self-Test (not a data source)

The synthetic path uses fixed SVI parameters with additive Gaussian noise to verify that calibration recovers known parameters. It is not a data source and never writes to data/. Two caveats: (a) the chosen parameters imply ATM vols around 30–60% (much higher than real SPY ~12–20%) because the synthetic a/b were illustrative, not calibrated to a realistic level — this is irrelevant since real data replaces them; (b) the synthetic generator assigns each strike to exactly one option type, so the call-put parity test is never exercised by synthetic data. On real chains, where strikes carry both calls and puts, parity runs normally.

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Interview Q&A

Q1: What is implied volatility and how do you extract it?

Implied volatility (IV) is the value of σ that, when substituted into the Black-Scholes formula, reproduces the observed market price of an option. It is the market's risk-neutral expectation of future realised volatility over the life of the option, expressed as an annualised standard deviation. Extraction inverts the BS formula numerically: Newton-Raphson for near-the-money strikes (fast convergence where vega is large), falling back to Brent's method for deep OTM strikes where vega approaches zero and NR diverges.

Q2: What is the SVI model and why is it used in practice?

SVI (Stochastic Volatility Inspired, Gatheral 2004) parametrises total variance w(m) = a + b[ρ(m-μ) + √((m-μ)²+ξ²)] in log-moneyness space. It is used because:

1. Five parameters fit observed smiles with RMSE typically under 50bp.
2. The formula is analytic — calibration via SLSQP is fast (milliseconds per maturity).
3. No-arbitrage conditions are explicit and enforceable.
4. It has the correct asymptotic behaviour: linear wings consistent with Roger Lee's moment formula.
5. The SSVI extension provides a full arbitrage-free surface.

Q3: What is butterfly no-arbitrage and how do you verify it?

Butterfly no-arbitrage requires that the implied risk-neutral density q(K) ≥ 0 everywhere. In terms of the smile, this means the second derivative of the call price with respect to strike is non-negative: ∂²C/∂K² ≥ 0. For SVI, the Lee (2004) necessary condition b(1+|ρ|) < 4/T provides a quick check. The Roper (2010) sufficient condition g(m) ≥ 0 for all m (where g involves the first and second derivatives of w) provides a complete characterisation. This pipeline enforces the Lee condition as a hard constraint in the SLSQP optimiser and reports both conditions in the arbitrage verification section.

Q4: Why does your smile show negative skew for SPY?

The negative skew (implied vol increasing as moneyness decreases, i.e. for puts below the forward) reflects the equity volatility skew: out-of-the-money puts are more expensive relative to Black-Scholes than OTM calls, for three reasons:

1. Demand for portfolio protection: institutions systematically buy OTM puts to hedge equity exposure.
2. Leverage effect: falling equity prices are associated with rising volatility (Black 1976), making put payoffs positively correlated with volatility — puts are a natural hedge for volatility sellers, reducing their required risk premium.
3. Jump risk: the risk-neutral distribution has a fat left tail (crash risk), which is priced into OTM puts.

In the SVI model, negative skew is captured by ρ < 0 (typically -0.5 to -0.8 for SPY). The steepness of the skew is proportional to b|ρ|.

Q5: What would you do differently with more time?

SSVI is already implemented (ssvi.py) — it replaces the linear SVI-parameter interpolation with a globally calendar- and butterfly-arbitrage-free surface. Remaining work, in priority order:

1. eSSVI: extend SSVI to a maturity-dependent correlation ρ(θ) (Hendriks & Martini 2019), recovering per-slice fit quality while remaining arbitrage-free. This addresses SSVI's main weakness — a single global ρ under-fits steep short-dated skews.

2. True delta strikes: replace the fixed-moneyness RR/BF approximation with proper 25-delta and 10-delta strikes, solving N(d₂(K)) = Δ numerically. This is what a desk actually trades.

3. Dupire local volatility: extract the local vol surface from the calibrated SSVI surface for exotic pricing.

4. Heston calibration: calibrate the Heston (1993) stochastic volatility model dσ² = κ(θ-σ²)dt + ν√σ²dW to the surface. Heston gives more stable model dynamics than SVI and is standard for path-dependent exotics.

5. Rate and dividend curves: replace flat r and q with bootstrapped OIS and dividend term structures.

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References

• Black, F. (1976). Studies of stock price volatility changes. Proceedings of the 1976 American Statistical Association.
• Gatheral, J. (2004). A parsimonious arbitrage-free implied volatility parametrization with application to the valuation of volatility derivatives. Presentation at Global Derivatives & Risk Management, Madrid.
• Gatheral, J. & Jacquier, A. (2014). Arbitrage-free SVI volatility surfaces. Quantitative Finance, 14(1), 59–71.
• Heston, S. (1993). A closed-form solution for options with stochastic volatility. Review of Financial Studies, 6(2), 327–343.
• Jaeckel, P. (2006). By Implication. Wilmott Magazine, November 2006.
• Lee, R. (2004). The moment formula for implied volatility at extreme strikes. Mathematical Finance, 14(3), 469–480.
• Merton, R. (1973). Theory of rational option pricing. Bell Journal of Economics, 4(1), 141–183.
• Roper, M. (2010). Arbitrage-free implied volatility surfaces. Preprint, University of Sydney.

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License

MIT