Heston model
Quantitative Finance glossary · Derivatives Pricing & Volatility Surfaces
Definition
Stochastic-volatility model: dS_t = μ S_t,dt + √(v_t),S_t,dW_t^S, dv_t = κ(θ - v_t),dt + xi√(v_t),dW_t^v, with dlangle W^S, W^vrangle_t = ρ,dt. Closed-form characteristic function admits semi-analytical option pricing via Carr-Madan FFT or Lewis-Lipton contour integration. Captures volatility clustering and (via ρ < 0) the equity-index skew; struggles with very short-dated steep skew, motivating rough-volatility extensions.
Heston model in plain English
The famous Black-Scholes option model pretends a stock's 'wobbliness' (its volatility) is a fixed number, but in reality volatility itself jumps around — calm weeks, then panicky ones. The Heston model fixes this by letting volatility be its own randomly moving quantity that drifts back toward a long-run average, like a thermostat always pulling temperature back to a set point but getting knocked around along the way. Crucially it lets the stock and its volatility be negatively linked (ρ < 0): when prices crash, volatility spikes — exactly what real markets do, which is why Heston reproduces the 'skew' (crash insurance being pricier than you'd naively expect). It's a workhorse because, despite the added realism, options can still be priced quickly with special math rather than slow simulation.
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