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Finance & economics · Topic
Stochastic Volatility
Stochastic volatility models extend Black-Scholes by allowing volatility to vary randomly. They explain the implied volatility smile observed in option markets — a pattern Black-Scholes cannot reproduce. The Heston model is the most widely used.
The Heston model
Asset price and variance $(S_t, v_t)$ follow coupled SDEs:
\(dS_t = \mu S_t\,dt + \sqrt{v_t}\,S_t\,dW_t^S\) \(dv_t = \kappa(\theta - v_t)\,dt + \xi\sqrt{v_t}\,dW_t^v\)
where $\text{Corr}(dW^S, dW^v) = \rho\,dt$ (typically $\rho < 0$: falling prices → rising vol).
Pricing via characteristic functions
The Heston model is affine in $(S, v)$, so the characteristic function of $\log S_T$ has a closed form:
\[\varphi(u) = \mathbb{E}^\mathbb{Q}[e^{iu\log S_T}] = e^{C(T,u) + D(T,u)v_0 + iu\log S_0}\]where $C, D$ satisfy Riccati ODEs. Option prices follow by Fourier inversion.
The volatility smile
Black-Scholes implies a flat implied-volatility surface. In practice, implied vol is:
- Higher for OTM puts (skew/smirk): investors pay for crash protection
- Higher for short maturities at extreme strikes (smile)
The Heston model reproduces the skew via correlation $\rho$ and the smile curvature via vol-of-vol $\xi$.