Heston Model: Stochastic Variance, Skew, and Calibration Risk

For educational purposes only; not investment advice.

Direct answer

The Heston model replaces Black–Scholes’ constant variance with a random, mean-reverting variance process correlated with the asset price. Under a common risk-neutral specification:

dS = (r − q)S dt + sqrt(v)S dW_S

dv = κ(θ − v)dt + ξsqrt(v)dW_v, with dW_S dW_v = ρdt.

The core parameters are initial variance v0, long-run variance θ, mean-reversion speed κ, volatility of variance ξ, and spot–variance correlation ρ. This compact structure can generate option skew and changing variance over time, but a calibrated parameter set is not a unique truth about the market.

How the five parameters shape the surface

v0 anchors short-horizon variance. θ is the level toward which expected variance reverts, and κ controls the speed. ξ widens the distribution of future variance and generally adds smile curvature. Negative ρ, common in equity calibrations, links price declines with variance increases and helps create richer downside Put prices.

Expected variance at horizon t is E[v(t)] = θ + (v0 − θ)e^(−κt). A useful mean-reversion half-life is ln(2)/κ. The sufficient Feller condition 2κθ ≥ ξ² keeps the square-root variance process strictly positive. A calibration that violates it is not automatically unusable, but boundary behavior, simulation scheme, and numerical stability require more care.

Heston is still a continuous diffusion. It does not automatically reproduce price jumps, event-specific discontinuities, every maturity and strike simultaneously, transaction costs, or liquidity. Parameters can compensate for one another, so two sets may fit today’s prices similarly but imply different future dynamics and hedges.

Reading one parameter set

Suppose v0 = 0.09, θ = 0.04, κ = 2, ξ = 0.30, and ρ = −0.70.

Initial volatility: sqrt(0.09) = 30%.
Long-run volatility level: sqrt(0.04) = 20%.
Variance half-life: ln(2)/2 = 0.347 year, about 4.16 months.
Six-month expected variance: 0.04 + (0.09 − 0.04)e^(−2×0.5) = 0.0584.
Square root of that expected variance: sqrt(0.0584) = 24.16%.
Feller check: 2κθ = 0.16, while ξ² = 0.09, so the sufficient condition holds.

24.16% is the square root of expected variance, not expected volatility; in general sqrt(E[v]) ≠ E[sqrt(v)]. The negative correlation means downward price shocks tend to coincide with upward variance shocks, steepening downside skew. Changing only ρ toward zero can flatten asymmetry; lowering ξ can reduce smile curvature.

Calibration and validation checklist

Use synchronized, executable option Bid/Ask quotes and consistent forwards, discount factors, dividends, and timestamps.
Remove stale, zero-Bid, crossed, arbitrage-violating, and corporate-action-distorted observations under documented rules.
Constrain v0, θ, κ, and ξ to sensible positive ranges and ρ to [−1, 1].
Fit prices with weights reflecting Vega or Bid/Ask reliability; inspect errors by maturity and moneyness, not only one RMSE.
Run optimization from multiple starting points and report parameter instability and alternative near-optimal sets.
Compare model errors with market spreads; a small dollar error can still be economically large in a tight market.
Check characteristic-function implementation, integration convergence, branch handling, and simulation bias.
Validate out of sample and compare hedge P/L, not merely same-day surface fit.
Stress jumps, more negative ρ, larger ξ, term-structure breaks, and parameter recalibration independently.

Common misconceptions

“Heston predicts volatility.” It specifies conditional stochastic dynamics after parameters are chosen.
“A better fit means a better hedge.” Overfitting and unstable parameters can worsen out-of-sample behavior.
“Each parameter has one observable market meaning.” Parameters interact and can offset each other.
“Violating Feller invalidates the model.” It changes boundary and numerical considerations, not automatically every price.
“Stochastic volatility captures crashes.” Continuous paths still omit discontinuous price jumps.
“One calibration works for all products and dates.” Surface regimes, events, and liquidity change, and path-dependent products expose different model weaknesses.