Stochastic Volatility and the Heston Model: When Volatility Has a Life of Its Own

Quant ResearchBy QuantDaddyOctober 14, 20268 min readSeries: From Black-Scholes to the Vol Surface · Part 4

From Black-Scholes to the Vol Surface

A six-part series on how option markets price volatility, from the textbook model to the arbitrage-free surfaces used by professional desks.

Black-Scholes and the Greeks
Implied Volatility and the Smile
Local Volatility and Dupire's Formula
Stochastic Volatility: the Heston Model (you are here)
The SABR Model
Arbitrage-Free Surfaces: SVI and Mixtures of Lognormals

Local volatility, the subject of Part 3, fits every option price but makes volatility a fixed function of the stock price, which leads to unrealistic dynamics. Stochastic volatility models take the opposite approach: they give volatility its own random process. The most widely used of these is the model Steven Heston published in 1993, valued for combining realistic behavior with prices that can be computed almost in closed form.

The model

The Heston model describes the stock price S and its instantaneous variance v with two coupled equations:

dS = μ S dt + √v · S dW₁

dv = κ (θ − v) dt + ξ √v dW₂ with corr(dW₁, dW₂) = ρ

The variance follows a mean-reverting process, often called a square-root or CIR process after Cox, Ingersoll and Ross, who used it for interest rates. Its √v term makes the noise shrink as variance approaches zero, which helps keep variance positive. If 2κθ ≥ ξ², known as the Feller condition, variance never reaches zero at all. Calibrated parameters often violate this condition in practice, which is harmless for pricing but requires care in simulation.

What each parameter does

Parameter	Meaning	Effect on the surface
v₀	Current variance	Sets the level of short-dated implied volatility
θ	Long-run variance	Sets the level of long-dated implied volatility
κ	Speed of mean reversion	Controls how quickly the term structure moves from √v₀ toward √θ
ξ	Volatility of variance	Adds curvature: both wings of the smile rise, and tails get fatter
ρ	Correlation between price and variance	Tilts the smile; negative ρ produces the equity index skew

The correlation ρ is the key to the skew. For equity indices it is strongly negative, often around −0.7. When the price falls, variance tends to rise, so the distribution of future prices develops a fat left tail and a thin right tail, exactly the shape Part 2 inferred from the smile. This is the model's version of the leverage effect.

The term structure of volatility

Illustration only. Approximate at-the-money volatility implied by the expected average variance in the Heston model with θ = 0.04 (20% volatility) and κ = 2. Volatility for each maturity is pulled toward the long-run level at a speed set by κ.

Because variance mean-reverts, the expected average variance over the life of an option has a simple formula:

E[ average variance over 0..T ] = θ + (v₀ − θ) · (1 − e^−κT) / (κT)

Its square root is a good first approximation to at-the-money implied volatility at each maturity, and it is also the fair strike of a variance swap in the model. For example, with θ = 0.04 (a long-run volatility of 20%), κ = 2 and current variance v₀ = 0.09 (30% volatility), the formula gives about 28.2% for three months, 24.8% for one year and 22.9% for two years. When current variance is above its long-run level, as after a market shock, the term structure slopes downward: near-term options are expensive and longer ones less so. In calm markets, when v₀ is below θ, it slopes upward. These are the same two regimes seen in the VIX term structure, backwardation and contango, and Heston reproduces both with one parameter set.

How options are priced

Heston's key contribution was showing that the model has a closed-form characteristic function, the Fourier transform of the log-price distribution. Option prices can then be computed by a one-dimensional numerical integral rather than a slow simulation. The call price has a form that echoes Black-Scholes:

C = S · P₁ − K e^−rT · P₂

Here P₁ and P₂ play the roles of N(d₁) and N(d₂), but each is computed by integrating the characteristic function. In 1999 Peter Carr and Dilip Madan showed how to price a whole strip of strikes at once with the fast Fourier transform, which made calibration to an entire surface fast enough for daily use. Later work fixed a numerical issue in the original formula, sometimes called the "little Heston trap," where the complex logarithm jumps between branches for long maturities.

Calibrating to the market

Calibration means choosing the five parameters so that model prices match market prices as closely as possible, usually by minimizing squared differences in implied volatility weighted by vega or by bid-ask width. A few practical lessons are widely shared:

Parameters are partly interchangeable. A higher ξ with a lower κ can produce similar prices to the opposite combination, so the best fit may be unstable from day to day.
Fixing or constraining some parameters, for example κ, usually gives more stable results than fitting all five freely.
The quality of fit across maturities is good for medium and long expirations but noticeably weaker for very short ones.

Strengths and limitations

Heston's strengths are exactly where local volatility was weak. Volatility moves on its own, so the model produces realistic forward smiles and sensible dynamics for products that depend on future volatility, such as forward-start options and options on variance. Its prices are fast to compute, and its parameters have clear economic meaning.

Its main weakness is at the short end. A diffusion cannot move far in a short time, so the model struggles to produce the very steep skews seen in options expiring within weeks. Fitting those skews requires an unrealistically large volatility of variance. The common fix is to add jumps to the price process, as in the 1996 Bates model, which keeps the Fourier pricing machinery while fattening the short-dated left tail. A second limitation is that, unlike local volatility, Heston cannot reprice every vanilla option exactly. Desks often combine the two in a stochastic local volatility model, using Heston-style dynamics with a local volatility correction that restores an exact fit.

Key takeaways

Heston models variance as a mean-reverting square-root process correlated with the price.
Negative correlation ρ creates the equity skew, volatility of variance ξ creates curvature, and κ and θ shape the term structure.
Expected average variance has a closed form, which gives the at-the-money term structure and variance swap prices.
Options are priced quickly through the characteristic function and Fourier methods.
Dynamics are realistic, but very short-dated skews need jumps, and exact fits need a local volatility correction.