stochastic-volatility and interest-rate models

The Black-Scholes model assumes volatility is a known constant and interest rates are fixed — two assumptions the market visibly violates. Stochastic-volatility and interest-rate models are the next layer of realism: they let the volatility, and the interest rate itself, be random processes with their own dynamics. They are what practitioners actually use to price and hedge in markets where option prices reveal a volatility smile (implied volatility varying by strike) and where bonds of different maturities move together but not identically (a term structure that wiggles).

Stochastic volatility makes the variance a second SDE. The Heston model is the canonical example: the asset follows dS_t = mu S_t dt + sqrt(v_t) S_t dB^1_t while the variance v_t follows a mean-reverting square-root (CIR) process dv_t = kappa (theta - v_t) dt + xi sqrt(v_t) dB^2_t, with the two Brownian motions correlated (correlation rho, typically negative — the leverage effect, where falling prices come with rising volatility). Because there are now TWO sources of risk (price and volatility) but volatility is not directly tradable, the market is INCOMPLETE: there is no unique risk-neutral measure, only a family indexed by the market price of volatility risk, which must be calibrated to observed option prices rather than derived. Interest-rate models make the short rate r_t a process. Short-rate models include Vasicek, dr_t = a(b - r_t) dt + sigma dB_t (mean-reverting, Gaussian, but allows negative rates), and Cox-Ingersoll-Ross, dr_t = a(b - r_t) dt + sigma sqrt(r_t) dB_t (square-root, keeps rates nonnegative when the Feller condition 2ab >= sigma^2 holds). The whole yield curve is then P(t,T) = E^Q[ exp(- integral_t^T r_s ds) given F_t ], a Feynman-Kac expression; the Heath-Jarrow-Morton framework instead models the entire forward-rate curve directly, with a no-arbitrage drift condition that ties the drift of forward rates to their volatilities.

These models matter because they fit market data that Black-Scholes cannot: the smile, the term structure of volatility, the correlation between rates and their volatility. Heston is prized because its characteristic function is known in closed form, so option prices follow by Fourier inversion. Honest caveats that practitioners must respect: incompleteness is the defining feature, not a technicality — with volatility (or jumps, or rate) risk untradeable, perfect replication fails, pricing requires a chosen martingale measure (a calibrated risk premium), and hedges are only approximate. The CIR rate process needs the Feller condition 2ab >= sigma^2 to stay strictly positive; Vasicek can go negative (sometimes a feature, sometimes a bug). And calibration is an inverse problem that can be ill-posed — many parameter sets fit the same option prices, so model risk (the price changes with the model you chose) is real and must be managed, not assumed away.