Volatility Surface

Fit an implied volatility surface from real SPY option chains, analyze the smile and term structure, and check for no-arbitrage violations.

Live demo

Try it here: https://volatility-surface-3qwrckrlldquymzv6xnwzo.streamlit.app/

The deployed app loads a precomputed surface snapshot (so it works offline) and lets you drag to rotate the 3D surface, adjust filters, and compare smile curves across expiries.

What this shows

Black-Scholes assumes volatility is a single number. Real markets disagree: implied volatility varies systematically across strike (the smile/skew) and maturity (the term structure). This project constructs the full implied volatility surface from a real SPY option chain and analyzes it.

Two things matter:

1. The shape — equities have a pronounced skew: OTM puts trade richer than OTM calls. This is the market's crash-insurance premium.
2. The validity — a fitted surface must be arbitrage-free. We check butterfly convexity, calendar monotonicity, and vertical spread bounds.

Key findings

The smile has a pronounced skew

For the 30-day expiry, implied vol ranges from ~0.25 at ATM up to ~0.50 at deep OTM puts — a classic equity-index skew.

Short-dated smiles are steeper

The same smile across 5 maturities. Short expiries have a steeper smile (higher gamma, sharper risk premium); longer expiries flatten out.

The 3D surface

Strike on X, days to expiry on Y, implied vol on Z.

No-arbitrage checks

Check	What it detects	Result
Butterfly	∂²C/∂K² < 0	Small violations (~1e-3), expected on real data
Calendar	σ²·T decreases with maturity	Occasional violations at stale strikes
Vertical bounds	Call prices not monotone in strike	Clean across all expiries

Vertical-bound violations being zero is the strongest signal: it confirms the IV solver is producing sensible prices. Small butterfly and calendar violations are normal for raw market data.

Method

1. Fetch chain via yfinance for 5+ SPY expiries
2. Clean quotes — drop zero-bid, crossed, zero-OI strikes
3. Combine — OTM puts below spot, OTM calls above (avoids illiquid ITM)
4. Invert IV — Newton-Raphson with brentq fallback
5. Filter — IVs outside [0.02, 1.5] and mid price < $0.05 dropped
6. Stack — assemble (T, K, IV) grid
7. Validate — butterfly, calendar, vertical-bound checks

Project structure

volatility-surface/
├── src/
│   ├── data.py           # yfinance chain fetch
│   ├── black_scholes.py  # Reused from options-pricer
│   ├── implied_vol.py    # Reused IV solver
│   ├── iv_grid.py        # Per-strike IV, smile filtering
│   ├── surface.py        # Multi-expiry surface construction
│   └── no_arbitrage.py   # Butterfly / calendar / vertical checks
├── notebooks/analysis.ipynb
├── results/              # surface_data.csv and PNG plots
├── app.py                # Streamlit + Plotly 3D dashboard
└── requirements.txt

Reproduce locally

pip install -r requirements.txt
streamlit run app.py

Or open notebooks/analysis.ipynb to step through the analysis.

Limitations and next steps

• Discrete strike grid — no interpolation between strikes
• No parametric fit — SVI or SABR would give a genuinely arbitrage-free surface
• Single snapshot — no dynamics, no sticky-strike analysis

Natural extensions:

• SVI parameterization — 5-parameter fit per expiry with arbitrage constraints
• Heston calibration — fit stochastic vol parameters to the surface
• Surface dynamics — track daily surfaces, estimate skew beta to spot

Skills demonstrated

• Derivatives pricing — Black-Scholes, IV inversion, put-call parity
• Data engineering — live option chains, quote cleaning
• Numerical methods — Newton-Raphson, brentq
• Visualization — matplotlib 3D, Plotly, Streamlit
• Financial reasoning — no-arbitrage, skew, term structure