Stochastic Volatility and the Heston Model

The Heston model, published by Steven Heston in 1993, is the best known stochastic volatility model. Instead of assuming volatility is constant (Black Scholes) or a fixed function of price (local volatility), Heston lets variance move randomly over time, pulled back towards a long run average and correlated with the underlying's price. These features let the model produce volatility smiles and skews similar to those seen in real markets, while still allowing fast pricing of European options through a semi closed form solution.

The model#

The underlying price S and its variance v follow:

dS = r × S dt + √v × S dW1
dv = κ × (θ - v) dt + ξ × √v dW2
corr(dW1, dW2) = ρ

The five parameters#

Volatility is the square root of variance, so v0 = 0.04 means 20% volatility.

How Heston creates smiles and skews#

• Negative correlation (ρ < 0): when prices fall, volatility tends to rise. That fattens the left tail of returns, making out of the money puts more valuable, which creates the downward sloping skew typical of equity indices. Equity calibrations often find ρ between about minus 0.5 and minus 0.9.
• Volatility of volatility (ξ): larger ξ makes extreme moves more likely in both directions, raising implied volatility for far out of the money options on both sides and curving the smile. See Volatility Smile and Skew.
• Mean reversion (κ and θ): when current volatility is below its long run level, the term structure slopes upward; when above, it slopes downward, as after a market shock. See Volatility Term Structure.

The Feller condition#

If 2κθ > ξ², the variance process stays strictly positive. Calibrations to equity markets often violate this condition, which means variance can touch zero in simulation. This needs careful handling in numerical methods.

Pricing and calibration#

Heston derived a characteristic function for log prices, which allows European option prices to be computed with numerical integration or Fourier methods quickly. To calibrate:

1. Collect market prices or implied volatilities across strikes and expiries.
2. Choose the five parameters that minimise the difference between model and market prices.
3. Check the fit and stability over time.

Exotic options are then priced using the calibrated parameters, usually by Monte Carlo or finite differences. See Monte Carlo Option Pricing.

Strengths and weaknesses#

Short dated skews in equity markets are often steeper than Heston can produce, which has led to models adding jumps (such as the Bates model) or rough volatility models, a more recent research direction in which volatility paths are much rougher than standard diffusion.

Heston vs local volatility#

Local volatility fits today's prices exactly but has unrealistic dynamics; Heston has more realistic dynamics but fits only approximately. Local stochastic volatility models combine them. See Local Volatility.

Frequently asked questions#

What is the Heston model?#

A stochastic volatility model in which variance follows a random, mean reverting process correlated with the underlying price, used to price options consistently with volatility smiles.

Why is correlation negative in the Heston model for stocks?#

Because stock prices and volatility tend to move in opposite directions: when markets fall, volatility usually rises. Negative correlation creates the observed skew.

Can the Heston model be priced quickly?#

Yes, European options have a semi closed form solution using numerical integration, which makes calibration practical.

Next, learn the industry standard model for interest rate smiles in SABR Model.

Sources#

• Wikipedia, Heston model