Heston SV Model — Nicholas McBride

Calibrated Parameters

Heston (1993) · calibration.json

v₀

Initial Variance

0.05805

σ₀ = 24.09%

κ

Mean Reversion Speed

5.328

t_½ = 0.13 yr

θ

Long-Run Variance

0.04895

σ_∞ = 22.12%

σ_v

Vol of Vol

0.7222

2κθ/σ² = 1.00 ✓

ρ

Spot-Vol Correlation

−1.000

max leverage effect

RMSE (IV)3.11 vol pts

Contracts Used144

Feller Condition✓ Satisfied

Convergence✓ Converged

Methodology

How It Works

01

The Heston Model — Stochastic Variance

The stock price S and its instantaneous variance v are coupled Itô processes. Variance is mean-reverting — pulled toward long-run level θ at speed κ — and correlated to spot via ρ:

d S = r \cdot S \cdot d t + \sqrt v \cdot S \cdot d W 1 d v = κ (θ - v) \cdot d t + σ \cdot \sqrt v \cdot d W 2 d W 1 \cdot d W 2 = ρ d t

The calibrated ρ = −1.00 (maximum leverage effect) explains the steep left-skew in the surface: SPY drops correlate perfectly with vol spikes, elevating OTM put IVs. The Feller condition 2κθ > σ² ensures variance never touches zero.

02

Semi-Analytic Pricing — Gil-Pelaez Fourier Inversion

Heston admits a known characteristic function φ(u) — a complete encoding of the risk-neutral distribution. Option prices are recovered via one-dimensional numerical integration, not Monte Carlo:

C = S \cdot P 1 - K \cdot e - rT \cdot P 2 P j = ½ + (1/ π) \int 0 \infty Re[e - iu log K \cdot φ j (u) / (iu) ] d u

P₂ is the risk-neutral probability the stock finishes above K. P₁ is its stock-measure analogue (the delta). Both follow from integrating φ via adaptive Simpson quadrature.

03

Calibration — 144 Contracts, Two-Stage Optimizer

Market IVs extracted via Newton-Raphson. The five Heston parameters minimize ATM-weighted squared IV error across the full surface:

L (v 0, κ, θ, σ, ρ) = Σ i w i \cdot ( IV model (K i, T i) - IV mkt i) 2

Differential Evolution (global search) escapes local minima in this non-convex landscape; Nelder-Mead refines to convergence in 4,693s across 144 contracts. Result: RMSE = 3.11 vol pts, Feller satisfied.

04

Greeks — Central Finite Differences O(h²)

Greeks are partial derivatives of the option price. Heston has no closed-form Greeks, so each is computed by bumping the input symmetrically by a small h:

\partial P / \partial x \approx [P (x + h) - P (x - h) ] / (2 h) accuracy: O(h²)

Δ = ∂P/∂S (spot sensitivity) · Γ = ∂²P/∂S² (delta convexity) · ν = ∂P/∂σ (vol exposure) · Θ = ∂P/∂T (time decay). Each requires two additional pricing calls.

Implied Volatility Surface

Interactive · Drag to Rotate

Drag to rotate · auto-rotating

—ATM IV

−1.000ρ

3.11 ptsRMSE

Volatility Smile T = 30d

Expiry 30d

Put Skew

OTM puts elevated — ρ = −1.00 perfect leverage

ATM Term Structure

Vol rises toward σ_∞ = 22.1% as T → ∞

Smile Curvature

σ_v = 0.72 vol-of-vol drives wing convexity

ATM skew approximation

∂σ̂/∂k|_k=0 ≈ ρ·σ / (2·√(v̄·T))

Option Pricer

Live Heston Engine

Option Type

Strike (K)

USD

672

Days to Expiry

Days

30

Enter parameters and compute →