The Heston Stochastic-Volatility Model

Why Model Variance as a Stochastic Process?

In a constant-volatility model, the scale of asset-price fluctuations is fixed. Market data instead show that volatility changes over time and often rises after a fall in the asset price. The Heston model represents this effect by making the asset’s instantaneous variance a random process and correlating its shocks with asset-return shocks. Its variance process is a square-root diffusion: it is pulled toward a positive long-run level, while its random fluctuations shrink as variance approaches zero.

Work on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{Q})$, where $\mathbb{Q}$ is the risk-neutral measure. Let $B^1_t$ and $B^2_t$ be Brownian motions under $\mathbb{Q}$ with $[B^1,B^2]_t=\rho t$, where $-1\le\rho\le1$. Expectations under this measure are written $\mathbb{E}^{\mathbb{Q}}$. The asset price is $S_t$, its initial value $S_0>0$, and its variance is $V_t$, with $V_0\ge0$.

Definition: The risk-neutral Heston model, with constant interest rate $r$, continuous dividend yield $q$, and parameters $\kappa,\theta,\xi>0$, is $$ \begin{aligned} dS_t&=(r-q)S_t\,dt+\sqrt{V_t}\,S_t\,dB^1_t,\\ dV_t&=\kappa(\theta-V_t)\,dt+\xi\sqrt{V_t}\,dB^2_t. \end{aligned} $$ Here $\kappa$ is the variance mean-reversion speed, $\theta$ is the long-run variance level, $\xi$ is the volatility of variance, and $\rho$ is the instantaneous correlation of the two Brownian drivers.

The coefficient of $dt$ in each equation is its drift; the coefficient of the Brownian increment is its diffusion coefficient. In particular, $\sqrt{V_t}$ is the asset’s instantaneous volatility and $V_t$ its instantaneous variance. The variance diffusion coefficient is $\xi\sqrt{V_t}$, not $\xi V_t$. The parameters $\theta$ and $V_t$ are variances, so their units are squared volatility units; $\xi$ has volatility units per square-root time.

When $|\rho|<1$, an equivalent way to construct the drivers is to take independent Brownian motions $B^1$ and $B^\perp$ under $\mathbb{Q}$, and set $B^2_t=\rho B^1_t+\sqrt{1-\rho^2}\,B^\perp_t$. At $\rho=1$ or $\rho=-1$, the second independent component disappears and $B^2=\rho B^1$. A strong solution is built using the specified Brownian drivers on the given filtered space; a weak solution allows the probability space and drivers to be chosen as part of the solution. Pathwise uniqueness means two solutions with the same drivers and initial value agree almost surely; uniqueness in law means their distributions agree. The standard CIR existence-and-uniqueness theorem gives a nonnegative strong solution for the variance equation with these parameters; its proof is beyond this tutorial.

Correlation and the Local Covariations

Quadratic covariation records the instantaneous covariance accumulated over time. The rules $dB^1_t\,dB^1_t=dt$, $dB^2_t\,dB^2_t=dt$, and $dB^1_t\,dB^2_t=\rho\,dt$, together with $dB_t\,dt=0$ and $dt\,dt=0$, give the local covariations of asset returns and variance.

Proposition: For the Heston equations, the instantaneous variance of the asset’s proportional return is $V_t\,dt$, the variance diffusion has quadratic variation $\xi^2V_t\,dt$, and the covariation of proportional return and variance is $\rho\xi V_t\,dt$.

Proof. For $S_t>0$, the random part of $dS_t/S_t$ is $\sqrt{V_t}\,dB^1_t$; the random part of $dV_t$ is $\xi\sqrt{V_t}\,dB^2_t$. Multiplying their diffusion coefficients and using the Brownian covariation rules gives $$ \begin{aligned} d[S]_t/S_t^2&=V_t\,dt,\\ d[V]_t&=\xi^2V_t\,dt,\\ d[S/S,V]_t&=\sqrt{V_t}\,\xi\sqrt{V_t}\,\rho\,dt =\rho\xi V_t\,dt. \end{aligned} $$ In the last line, $d[S/S,V]$ denotes the covariation of the proportional-return martingale part with the variance martingale part. Equivalently, applying Ito’s formula to $\log S_t$ shows that its diffusion is also $\sqrt{V_t}\,dB^1_t$, so $d[\log S,V]_t=\rho\xi V_t\,dt$. These calculations are local identities and do not assert that the asset price is a true martingale. $\square$

The sign of $\rho$ is financially important. If $\rho<0$, a negative asset-return shock tends to coincide with a positive variance shock: this is the leverage effect. The correlation is between the Brownian shocks, not a claim that the entire asset price and variance processes have a fixed ordinary correlation over a time interval.

Worked Example: Reading the Instantaneous Covariations

Take $V_t=\tfrac1{25}$, $\xi=\tfrac14$, and $\rho=-\tfrac35$ at a particular time. Compute the asset’s instantaneous volatility, the variance diffusion coefficient, and the instantaneous covariation of log price and variance.

The asset’s volatility is $\sqrt{V_t}=\sqrt{1/25}=1/5$. The variance diffusion coefficient is $\xi\sqrt{V_t}=(1/4)(1/5)=1/20$. The covariation rate is $$ \rho\xi V_t=\left(-\frac35\right)\left(\frac14\right)\left(\frac1{25}\right)=-\frac3{500}. $$ Thus over an infinitesimal time $dt$, the covariation is $-\tfrac3{500}\,dt$. Check: the product of the two diffusion coefficients is $(1/5)(1/20)=1/100$, and multiplying by the driver correlation $-3/5$ gives $-3/500$, as above.

Why the Variance Boundary Matters

The square root in the variance diffusion requires $V_t\ge0$. The nonnegative CIR solution respects this state space, but the behavior at zero depends on the parameters. When $V_t=0$, the variance diffusion coefficient vanishes, while its drift equals $\kappa\theta>0$. Thus zero is not an absorbing state under the stated positive-parameter assumptions: if it is reached, the drift pushes the process back into positive values.

Standard CIR boundary classification: For $\kappa,\theta,\xi>0$ and $V_0\ge0$, the variance process is nonnegative. If $2\kappa\theta\ge\xi^2$, zero is unattainable when $V_0>0$. If $2\kappa\theta<\xi^2$, zero is attainable; under the usual CIR solution it is instantaneously reflecting, not absorbing. This boundary classification is a standard result for the CIR diffusion; its proof is beyond this tutorial.

The inequality $2\kappa\theta\ge\xi^2$ is called the Feller condition. It is a sufficient and, for this parameterization, exact condition for zero to be inaccessible from a strictly positive starting value. It is not required for the SDE to define a nonnegative variance process. When the condition fails, the model still has a meaningful boundary behavior, but a path can reach zero. At zero, instantaneous asset volatility is also zero, so asset-price diffusion temporarily vanishes while variance is pushed upward by its positive drift.

This is a model boundary condition, not an option-pricing PDE boundary condition. In numerical or analytical work, one must respect nonnegative variance and the distinction between attainable and unattainable zero. A discretization that produces negative variance is outside the model’s state space; simply evaluating $\sqrt{V_t}$ there is not justified. The precise pricing-PDE boundary treatment is a separate question.

Worked Example: Feller Condition Holds

Let $\kappa=2$, $\theta=\tfrac1{25}$, and $\xi=\tfrac3{10}$, with $V_0>0$. Determine whether zero is attainable and compute the variance drift at zero.

The two sides of the Feller inequality are $$ 2\kappa\theta=2\cdot2\cdot\frac1{25}=\frac4{25}=\frac{16}{100}, \qquad \xi^2=\left(\frac3{10}\right)^2=\frac9{100}. $$ Since $16/100>9/100$, the Feller condition holds and zero is unattainable from a positive starting value. At zero the drift is $\kappa\theta=2/25$, while the diffusion coefficient $\xi\sqrt{0}=0$. Check: the drift points into the positive state space, and the boundary classification independently says that a path starting positive does not reach zero.

Worked Example: Feller Condition Fails

Let $\kappa=1$, $\theta=\tfrac1{25}$, and $\xi=\tfrac25$, with $V_0>0$. Decide what the failed condition means, and compute the drift and diffusion coefficients at $V_t=\tfrac1{100}$.

Here $$ 2\kappa\theta=2\cdot1\cdot\frac1{25}=\frac2{25}=\frac8{100}, \qquad \xi^2=\left(\frac25\right)^2=\frac4{25}=\frac{16}{100}. $$ Thus $2\kappa\theta<\xi^2$, so zero is attainable, but it is not absorbing. At $V_t=1/100$, the drift is $$ \kappa(\theta-V_t)=1\left(\frac1{25}-\frac1{100}\right)=\frac3{100}, $$ and the diffusion coefficient is $$ \xi\sqrt{V_t}=\frac25\sqrt{\frac1{100}}=\frac25\cdot\frac1{10}=\frac1{25}. $$ Check: at the boundary itself, the drift becomes $\kappa\theta=1/25>0$ and the diffusion becomes zero, consistent with an attainable but nonabsorbing boundary.

Log Prices and Risk-Neutral Interpretation

For $S_t>0$, Ito’s formula gives a useful representation of the asset price. On a finite horizon $[0,T]$, Ito integrals $\int_0^t H_s\,dB^i_s$ are defined for progressively measurable integrands with $\mathbb{E}^{\mathbb{Q}}\int_0^T H_s^2\,ds<\infty$; locally square-integrable integrands are handled by stopping. The variance process has finite time integral almost surely on finite intervals because it has continuous finite paths. Thus the integral with integrand $\sqrt{V_s}$ is defined locally, and is in $\mathcal{L}^2$ whenever $\mathbb{E}^{\mathbb{Q}}\int_0^T V_s\,ds<\infty$.

Proposition: A positive solution of the Heston asset equation has the representation $$ S_t=S_0\exp\left((r-q)t-\frac12\int_0^tV_s\,ds+\int_0^t\sqrt{V_s}\,dB^1_s\right). $$

Proof. Apply Ito’s formula to $f(x)=\log x$, whose derivatives are $f'(x)=1/x$ and $f''(x)=-1/x^2$. The correction term is $\tfrac12f''(S_t)(dS_t)^2$, with $(dS_t)^2=V_tS_t^2\,dt$. Therefore $$ d\log S_t=\frac{dS_t}{S_t}-\frac12\frac{(dS_t)^2}{S_t^2} =\left(r-q-\frac12V_t\right)dt+\sqrt{V_t}\,dB^1_t. $$ Integrating from $0$ to $t$ and exponentiating gives the displayed representation. Conversely, applying Ito’s formula to the exponential expression, whose first and second derivatives with respect to its log argument are both the exponential itself, restores drift $(r-q)S_t\,dt$: the $-\tfrac12V_t\,dt$ term cancels the $+\tfrac12V_t\,dt$ Ito correction. Its diffusion is $\sqrt{V_t}S_t\,dB^1_t$, so it satisfies the asset SDE and remains strictly positive. $\square$

The representation establishes positivity, but does not by itself establish that the discounted asset is a true martingale. Discounting by the risk-free rate gives $e^{-rt}S_t$, whose drift is $-q e^{-rt}S_t\,dt$; with dividends reinvested, $e^{-(r-q)t}S_t$ is a positive local martingale. A local martingale has stopped processes that are martingales along an increasing sequence of stopping times tending to the terminal horizon (or to infinity); this alone does not make the unstopped process a true martingale. A sufficient condition for the stochastic exponential in the representation to be a true martingale on $[0,T]$ is Novikov’s condition, $$ \mathbb{E}^{\mathbb{Q}}\left[\exp\left(\frac12\int_0^T V_s\,ds\right)\right]<\infty. $$ When a risk-neutral pricing argument requires a true martingale, that property must be verified or imposed; it should not be inferred solely from the local SDE.

Common Pitfall and Model Implications

Do not treat the Feller condition as necessary for nonnegative variance. For example, in the failed-condition example, $2\kappa\theta=8/100$ and $\xi^2=16/100$. The incorrect conclusion “the condition fails, so the model allows negative variance” confuses inaccessibility of zero with nonnegativity. The CIR solution remains nonnegative; the corrected conclusion is that zero can be reached and is instantaneously reflecting. Correspondingly, one should not impose a fictitious absorbing boundary at zero when interpreting that parameter regime.

The model’s practical flexibility comes from the separate roles of its parameters: $\kappa$ controls the speed of reversion, $\theta$ sets the long-run variance level, $\xi$ controls variance fluctuations, and $\rho$ controls the direction and strength of return-variance shock dependence. These jointly influence the distribution of future volatility and asset returns. The model therefore captures a source of option-price variation that a constant-volatility specification cannot represent, while retaining a precise nonnegative state process for variance.

Check Your Understanding

Use the definitions and boundary classification above. For each boundary question, assume $\kappa,\theta,\xi>0$ and $V_0>0$.

For $\kappa=1$, $\theta=1/10$, and $\xi=1/2$, compare $2\kappa\theta$ with $\xi^2$. Is zero attainable?
Write the variance diffusion coefficient and the instantaneous covariation rate between log price and variance.
Explain why $\rho<0$ represents a leverage effect in the model.
State the Ito correction term when applying the logarithm to the asset-price equation, and explain why it is negative.
What does failure of the Feller condition imply, and what does it not imply?