Stochastic Volatility: Heston

In the last chapter you watched Black-Scholes break. You fed it a single number for volatility and it handed you back a flat surface — one price of risk, the same at every strike, the same at every expiry. The market never agrees. The smile, the skew, the term structure: every one of them is the market telling you that volatility is not a constant. So the honest next move is to stop pretending it is, and to let volatility itself become a random thing that moves over time. That is the whole idea behind stochastic volatility, and the Heston model is its most important working example.

Let us build the intuition before a single symbol.

Imagine you are not modeling one moving quantity but two. The first is the price of the asset, which you already know wanders. The second is the asset's variance — the square of its volatility — which also wanders, but not freely. Variance does not run off to infinity and it does not collapse to zero and stay there. It paces. When fear spikes, variance jumps high; then, over days and weeks, it drifts back down toward some normal, long-run level. When the market is calm, variance sits low, but the smallest shock nudges it back up. Volatility, in other words, is mean-reverting. It is tethered. That tether is the heart of Heston, and it is why the model feels so much more like a real market than a flat line ever could.

Now make it precise. Heston writes the asset price with the familiar diffusion, but it replaces the constant volatility with the square root of a variance that has its own life. The variance, which we call little-v, follows this equation, and it is the one you came here for:

dv equals kappa times the quantity theta minus v, dt, plus xi times the square root of v, times dW.

Read it slowly, because every piece earns its place. The term kappa times theta-minus-v is the leash. Theta is the long-run average variance — the level volatility wants to come home to. Kappa is how hard the leash pulls: the speed of mean reversion. If variance is currently above theta, then theta-minus-v is negative and the drift pulls it down; if it is below theta, the drift pushes it up. A large kappa snaps variance back fast, so shocks are short-lived; a small kappa lets a vol spike linger for weeks. Then comes the second term. Xi, the volatility of volatility, controls how violently variance itself shakes. And critically, that randomness is scaled by the square root of v. When variance is near zero, the square root is near zero, so the random kicks shrink and the deterministic pull upward dominates — that square-root floor is precisely what keeps variance from going negative, which would be nonsense. This square-root structure is the famous Cox-Ingersoll-Ross process, borrowed from interest-rate modeling, and it is why Heston is sometimes called a square-root stochastic-volatility model.

There is one more ingredient, and it is the one that makes equity surfaces look the way they do. The random shock driving the price and the random shock driving the variance are correlated, with correlation rho. In equity markets rho is strongly negative — typically somewhere around minus-point-seven. That negative correlation says: when price falls, variance tends to rise. Crashes are violent; rallies are calm. This single number is what bends the implied-volatility surface into the downward skew you see on every index — low strikes priced at higher implied vol, because the model knows a drop and a vol-spike arrive together.

So Heston has five parameters, and each one maps onto something you can feel. Theta sets where vol lives. Kappa sets how fast it returns there, which shapes the term structure — how the smile flattens as you look further out. Xi sets how pronounced the smile's curvature is — more vol-of-vol, fatter tails, steeper smile. Rho tilts the smile into a skew. And v-naught, the variance you start from today, sets the near-term level. Five dials, and between them they can reproduce the shape of a real surface far better than any constant ever could.

What makes Heston not just intuitive but usable is that it has a semi-closed-form solution for European option prices. Through its characteristic function and a single numerical integral, you can price the whole surface in milliseconds and then calibrate — turn the dials until the model's surface matches the market's. That tractability is why Heston, published in nineteen-ninety-three, is still the workhorse it is today.