Heston Stochastic Volatility Model
Volatility smile, implied vol surface, and price path simulation
Model Parameters
Initial vol v₀
0.04
Mean reversion κ
2.0
Long-run vol θ
0.04
Vol of vol σ
0.50
Correlation ρ
-0.70
Risk-free rate r
0.02
Simulate Paths
Compute Smile
dS = rS dt + √v·S dW₁
dv = κ(θ−v)dt + σ√v dW₂
dW₁dW₂ = ρ dt
Feller: 2κθ/σ² = —
Current vol: —
Heston (1993): closed-form option prices via characteristic function. Key feature: correlation ρ<0 (leverage effect) produces skew/smile. Feller condition 2κθ>σ² ensures v stays positive. The volatility smile shows implied vol varies with strike — BS assumes flat.
Price & Volatility Paths
Implied Volatility Smile (T=1y)