Heston Stochastic Volatility Model

About the Heston Model

The Heston model is a mathematical model used to price options that accounts for stochastic volatility. Unlike the Black-Scholes model which assumes constant volatility, the Heston model allows volatility to fluctuate randomly.

The model is defined by these stochastic differential equations:

Where:

• Sₜ - Asset price at time t
• vₜ - Variance (volatility squared) at time t
• μ - Expected return rate
• κ - Mean reversion speed (how quickly volatility returns to θ)
• θ - Long-term variance level
• σ - Volatility of volatility
• ρ - Correlation between asset price and volatility shocks
• Wₜ¹, Wₜ² - Wiener processes (random shocks)

The model captures several observed market phenomena like volatility clustering and the leverage effect (negative correlation between asset returns and volatility).