Local & Stochastic Volatility Models

Black–Scholes is a useful baseline. In its standard form the underlying follows a continuous diffusion and volatility is constant over the option’s life. A simplified risk-neutral process is dS = (r − q) S dt + σ S dW. One σ is used for every strike and maturity.

Real option markets commonly show different implied volatilities by strike and expiry. That map is the surface you already walked: smile, skew, and term structure.

One volatility may fit one option and misprice the rest. Matching today’s surface and modeling how that surface moves are different problems. The formula stays useful for benchmarks, intuition, many vanilla calculations, and quoting implied volatility.

Local volatility lets the instantaneous width depend on the price level and on time: σ_local = σ_local(S, t). As the price moves or the clock advances, the model may read a different width. There is no separate random factor for volatility.

Strength: a well-specified, arbitrage-consistent vanilla surface can be reproduced closely. Limitation: future smile dynamics can be unrealistic even when today’s fit looks tight. Noisy or sparse quotes, and extrapolation past liquid strikes, can swing the risk numbers.

Dupire asks: if European call prices C(K, T) are known across strikes and maturities, can we infer a local-volatility function consistent with those prices under a diffusion? Under standard assumptions the local variance is a ratio of a calendar derivative and a strike convexity term. Exact forms depend on rate and carry conventions.

In practice the derivatives are taken from a smooth, arbitrage-consistent call surface — not raw noisy quotes. Second derivatives amplify noise. An ALM warning: a surface can look smooth and still produce unstable local volatility. Production use needs diagnostics, bounds, and an explicit rule for illiquid wings.

Stochastic-volatility models treat variance itself as a random process. Price has a Brownian shock. Variance has another. Those shocks may be correlated. Volatility can cluster, mean-revert, and stay uncertain even when today’s implied vol is known.

The trade-off is richer future-vol behavior against more parameters, harder calibration, and heavier numerics.

Heston is a widely used stochastic-volatility model. Variance is pulled toward a long-run level theta at speed kappa, with shock size xi (vol-of-vol). rho is the correlation between the price shock and the variance shock. v0 is where variance starts.

A commonly cited condition for variance to stay strictly positive is 2 κ θ ≥ ξ². Implementations still have to choose a numerical scheme. Heston can describe mean reversion, vol-of-vol, and price/vol correlation. It is not a universal fit.

The rates cousin on this site is SABR. That lesson stays under Risk & ALM. Do not treat Heston as a swaption smile.

Volatility is uncertainty in the underlying price. Vol-of-vol is uncertainty in the volatility level itself. Two names can share a similar implied vol today and still differ in how wildly that vol is expected to move. That difference shows up on long-dated and vol-sensitive claims, and it can be hard to pin down from vanilla quotes alone.

In many equity markets rho is negative: a down price shock often arrives with rising implied volatility. That pattern sits next to downside skew. Other markets and regimes can differ. Correlation is not causation and is not a fixed constant.

Choose the model from the risk question, not from a headline error. Then calibrate and validate, and read hedge and surface risk. Educational only.