Volatility in real markets behaves like weather: it clusters, it drifts back toward a typical climate, and it stays stubbornly random from day to day. The Heston model turns these observations into a compact, intuitive story for option pricing: the asset price diffuses as usual, but the instantaneous variance becomes a state variable that wanders randomly, tends to come back to a long-run level, and carries memory. In practice, this is enough to generate an upward or downward sloping term structure of at-the-money implied volatility, realistic skew, and volatility clustering, without resorting to a zoo of ad-hoc parameters.

The Heston Model

The model starts by letting the asset price grow at the risk-free rate while its diffusion scale is the square root of a stochastic variance. The variance itself follows a square-root diffusion (CIR Process). Under the risk-neutral measure, the core dynamics are:

The terms in the above dynamic equations are interpreted as such:

The variance is mean-reverting with speed kappa toward a long-run level theta, it is buffeted by its own randomness with intensity sigma (vol-of-vol), and its shocks can be correlated with price shocks through rho.

Two immediate practical features fall out. First, the square-root term keeps variance nonnegative and, under a familiar condition below, away from zero. Second, by allowing correlation between price and variance shocks, the model can create realistic downside skew in equities.

Mean reversion is the backbone of the model’s intuition. If today’s variance starts below its long-run level, the drift term pushes it upward; if it starts above, the drift pulls it down. That story is made precise by the conditional expectation of the variance as time moves forward. This single formula already tells you how quickly the market “forgets” today’s variance regime and drifts toward the background climate:

Autocorrelation—volatility clustering—drops out immediately from the same mean-reverting structure. Two nearby points in time share much of the same history, and that correlation decays exponentially as you separate them. This is both a realistic statistical description and a key pricing ingredient, because it tells you how much today’s volatility regime bleeds into tomorrow’s:

Options, however, do not care about the instantaneous variance in isolation. What they “see” is the time-average of variance over their life. At a high level, the at-the-money implied variance for maturity T is well-approximated by the risk-neutral expectation of that time-average, with small corrections we will discuss shortly. This makes the integral of variance—the realized variance over the option’s life—the central object. The model gives a clean, closed-form expression for the expected integrated variance and, dividing by maturity, for the expected average variance:

This compact formula is the engine behind the term structure. The factor multiplying the initial deviation from the long-run level is a decreasing function

When today’s variance is below its long-run level, the initial deviation is negative. Multiplying a negative number by a function that decreases with maturity makes the overall correction term rise toward zero as maturity grows, which means the expected average variance rises toward the long-run level.

Interpretation of Parameters

What we described above is precisely an upward-sloping at-the-money variance term structure. Mirror the argument when today’s variance is above its long-run level and you get a downward-sloping term structure. If today equals the long-run level, the term structure is flat—nothing in the model needs to work harder than this basic mean-reversion arithmetic.

A useful first-order bridge between the variance path and option prices is the approximation that at-the-money implied variance equals the expected average variance, up to small corrections that vanish as vol-of-vol goes to zero.

However, refinements matter for practitioners. First, option prices are convex in variance. Randomizing variance while holding its mean fixed pushes option prices up compared with a constant-variance world at that mean; in implied-variance terms, this adds a small upward “convexity premium” proportional to the variance of the time-average of variance. In Heston, that variance is governed mainly by vol-of-vol and mean-reversion speed.

Second, the correlation between price and variance shocks—the leverage effect—tilts the surface, especially at short maturities, by making downside price moves coincide with increases in variance. These refinements shift levels and near-term curvature, but they do not overturn the central message: the sign of the at-the-money slope is determined by where today’s variance sits relative to the long-run level and by the monotone shape of the factor shown above.

You can summarize the “why upward?” logic in a crisp checklist when the market opens in a calm regime (today’s variance below its long-run level). One, the conditional mean of variance increases over time, so the path you expect to average lies above today’s level and points toward the long-run anchor. Two, the expected average variance is a decreasing transform of that initial gap, so it moves upward with maturity. Three, short maturities remain anchored to today’s regime due to autocorrelation, leaving room for longer maturities to blend in more future days closer to the long-run climate. Four, convexity and leverage sprinkle adjustments on top—raising levels slightly and shaping skew—but the basic slope logic holds. The mirror image applies in stressed regimes: a high starting variance leads to a downward slope that relaxes toward calmer long-run conditions.

Parameter roles fall neatly into place once you view the surface through this lens. The long-run level pins the far-end at-the-money variance (plus a modest convexity cushion), the mean-reversion speed controls how quickly the curve flattens and how fast volatility memory fades, the vol-of-vol deepens smiles and increases the convexity uplift, and the correlation sets the sign and intensity of skew.

The initial variance sets the short end and, relative to the long-run level, determines the sign of the slope. This parameter map is not just academically tidy; it is operational. For example, if you observe an unusually steep upward at-the-money term structure on a quiet morning, the model’s story would be: initial variance is below its long-run level, the gap is sizeable, and mean reversion is not so fast that the surface is already flat—hence the slope.

Analytic Solution

For completeness, we must mention that the model remains analytically tractable because the log price has an exponential-affine characteristic function, which lets you price vanillas via Fourier inversion. The details are standard, but the critical point for intuition is that tractability comes from the variance process being affine in its own state; all of the term-structure logic above survives regardless of whether you compute prices by quadrature or by an FFT.

Over very short maturities, a small-time expansion further clarifies the initial slope. The leading-order term is simply today’s variance; the next term contains the mean-reversion pull toward the long-run level and a leverage tweak. When today’s variance is below the long-run level, that next term is positive, which is exactly the upward initial slope you see on quiet days.

Taken together, the Heston model’s appeal is its minimalism with punch. By turning variance into a mean-reverting, autocorrelated, stochastic state, it reproduces the key term-structure patterns you see on trading screens. Today’s variance sets the short end; mean reversion and averaging set the long end; autocorrelation, convexity, and leverage draw the curve between them. And because the model is both interpretable and tractable, it gives you levers - long-run level, mean-reversion speed, vol-of-vol, and correlation - that map cleanly to surface features you care about in practice.

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