Price dynamics
dS_t = mu S_t dt + sqrt(v_t) S_t dW_t^SThe asset's instantaneous volatility changes with the variance state.
Research & Systems
Stochastic Volatility (Heston) and Characteristic Functions
Heston couples price and variance dynamics; its characteristic function gives semi-closed-form prices.
Mental model
Heston lets variance move randomly, giving the model a mechanism for volatility smiles.
Heston models stochastic variance with mean reversion, vol-of-vol, and correlation with the price process.
Its characteristic function allows Fourier-based option pricing and fast calibration to smiles.
Explore the mechanism
How do mean reversion, volatility of variance, and spot-variance correlation shape option smiles?
Heston couples the price and variance dynamics: volatility itself moves over time and can be correlated with price shocks. Here you see a single simulated path and the implied-volatility smile it induces.
Numbers
Long-run vol σ̄ ≈ 20.0%
Average path vol ≈ 11.0%
Base smile vol ≈ 11.0%
Vol-of-vol η ≈ 60.0%
Correlation ρ ≈ -0.70 (downward (left skew))
Interpretation
Increasing vol-of-vol makes the green volatility path move more and deepens the curvature of the smile. Changing ρ tilts the smile: negative correlation (price down when vol up) produces the familiar equity-style left skew, while positive ρ pushes mass to the right.
The top panel shows how price and volatility co-move over time (scaled to the same frame). The bottom panel summarises this in one number per strike: the implied volatility that would be consistent with these stochastic-volatility dynamics at a fixed maturity.
UNDERSTAND
What to notice
Parameter combinations control smile shape: skew, curvature, and term structure of implied volatility.
The quantitative structure
Notation
S_t- asset price
v_t- instantaneous variance
kappa- variance mean-reversion speed
theta- long-run variance
xi- volatility of variance
rho- spot-variance Brownian correlation
Variance dynamics
dv_t = kappa(theta - v_t)dt + xi sqrt(v_t)dW_t^vVariance mean-reverts while receiving state-dependent random shocks.
Correlation
dW_t^S dW_t^v = rho dtCorrelation links price moves to variance shocks and shapes skew.
This model assumes
- Parameters are constant over the modeled horizon.
- The variance follows a square-root diffusion and the selected parameter set yields a valid process.
Where it appears
- pricing and calibrating equity and FX options
- generating state-dependent volatility paths
- benchmarking volatility-surface models
Where the intuition breaks
What this view leaves out
- Heston can violate the strict-positivity condition for variance, has calibration degeneracies, and still misses jumps and some surface dynamics.
Common mistakes
“A stochastic-volatility model automatically fits every observed volatility surface.”
Not quite.Its parameterized dynamics restrict the shapes and term structures it can reproduce.
Where this concept sits
Follow the strongest prerequisites, applications, extensions, and contrasts.
Connections for Stochastic Volatility (Heston) and Characteristic Functions
Stochastic Volatility (Heston) and Characteristic FunctionsYOU ARE HERE
ExtendsBlack–Scholes Option PricingHeston extends the constant-volatility Black–Scholes framework by making variance stochastic.
SupportsImplied Volatility Smile and SurfaceHeston introduces stochastic variance dynamics that can generate non-flat implied-volatility smiles and surfaces.
Validated byModel Calibration and ValidationCalibration fits Heston parameters to market option data, while validation checks whether the fitted model reproduces relevant prices and dynamics.
Extends · Black–Scholes Option Pricing
Heston extends the constant-volatility Black–Scholes framework by making variance stochastic.
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