Implied Volatility Surface SPX Options

Calibration of implied volatility surfaces from live SPX options data using the SVI (Stochastic Volatility Inspired), SSVI (Surface SVI), and Heston stochastic volatility models. The project covers the full pipeline: data acquisition, OTM filtering, calibration, and 3D surface visualisation.

---

Data Pipeline

Options data is fetched live from Yahoo Finance via yfinance. For each expiry in the option chain, OTM calls and puts are selected and merged into a unified DataFrame.

Filtering rules applied:

• Expiry range: configurable [min_days, max_days] window (e.g. 2 days to 2 months)
• Strike range: within ±10% of spot by default
• OTM selection: calls with K >= 0.99 * S, puts with K <= 1.01 * S
• IV cleanup: 0.10 < IV < 0.50 to strip stale or illiquid quotes

For SVI and SSVI, strikes are converted to log-moneyness k = ln(K/F) where F = S * exp(r * T) is the forward price, and total variance w = IV^2 * T is used as the fitting target. This is the natural space for SVI and makes no-arbitrage conditions easier to enforce.

For Heston, raw strikes and market IVs are used directly, as the model prices in strike space via numerical integration.

---

Model 1: SVI (Stochastic Volatility Inspired)

Parametrisation

The raw SVI model fits total variance as a function of log-moneyness:

w(k) = a + b * ( rho * (k - m) + sqrt((k - m)^2 + sigma^2) )

Parameter interpretation:

Parameter	Role
a	Overall variance level (vertical shift)
b >= 0	Slope / wings steepness
rho in (-1, 1)	Skew — left/right asymmetry
m	Horizontal shift (ATM location)
sigma > 0	Curvature / smile smoothness

• Each expiry slice is calibrated independently. The optimisation minimises squared total-variance residuals using non-linear least squares.

• Uses PCHIP (Piecewise Cubic Hermite Interpolation Polynomial)

Results

---

Model 2: SSVI (Surface SVI)

Parametrisation

SSVI extends SVI by directly modelling the entire surface with a single globally consistent parametrisation. Total variance is:

w(k, theta) = (theta / 2) * ( 1 + rho * phi(theta) * k + sqrt((phi(theta) * k + rho)^2 + (1 - rho^2)) )

where the power-law wings function is:

phi(theta) = eta / ( theta^gamma * (1 + theta)^(1 - gamma) )

and theta(T) is the ATM total variance at each maturity, extracted via linear interpolation.

Parameter interpretation:

Parameter	Role
rho in (-1, 1)	Global skew
eta > 0	Overall vol-of-vol level
gamma in (0, 1)	Controls how wings decay with maturity

• Unlike SVI, there is no per-slice fitting. rho, eta, and gamma are calibrated once across the entire surface simultaneously.
• Optimisation uses a two-phase approach: differential evolution for global search followed by SLSQP for local refinement.
• No-arbitrage is enforced via explicit constraints: calendar eta * (1 + |rho|) <= 2 and butterfly theta * phi^2 * (1 + |rho|)^2 <= 4.

Results

---

Model 3: Heston Stochastic Volatility (In Progress)

Model Dynamics

The Heston model specifies asset price and variance jointly:

dS =  mu * S dt + sqrt(v) * S dW1
dv = kappa * (theta - v) dt + sigma * sqrt(v) dW2
corr(dW1, dW2) = rho

Parameter interpretation:

Parameter	Role
v0	Initial (spot) variance
kappa	Mean-reversion speed of variance
theta	Long-run mean variance
sigma	Vol-of-vol (volatility of variance)
rho	Spot-vol correlation — primary skew driver

• The Feller condition 2 * kappa * theta > sigma^2 ensures variance stays strictly positive.

• Call prices are computed using the Heston characteristic function

• IV is calculated from Heston price using brentq

Results

---