Heston stochastic volatility

Definition Fix a filtered probability space with a pricing measure $\mathbb{Q}$, equivalent to the physical measure $\mathbb{P}$ and in general not unique, and constants $r$ and $q$. The Heston model specifies the ex-dividend stock price $S_t > 0$ and its instantaneous variance $v_t \ge 0$ by

where $W^S$ and $W^v$ are $\mathbb{Q}$-Brownian motions with $d[W^S, W^v]_t = \rho \, dt$. The parameters satisfy $S_0 > 0$, $v_0 > 0$, $\kappa > 0$, $\theta > 0$, $\xi \ge 0$ and $-1 \le \rho \le 1$. Here $\kappa$ is the speed of mean reversion per year, $\theta$ the long-run level of the variance and $\xi$ the volatility of the variance, and all of them are parameters under $\mathbb{Q}$. The variance equation has a unique solution, and it is nonnegative, a standard result for square-root diffusions that the course states without proof. The stock is then

The volatility is $\sqrt{v_t}$, not $v_t$. The discounted stock with dividends reinvested, $e^{-(r - q)t} S_t$, is a positive local martingale by construction, and that it is a true martingale is a known result for this model that the lesson does not prove. With $\xi = 0$ the variance path is deterministic. With $\xi > 0$ and $\lvert \rho \rvert < 1$ the variance is driven in part by a Brownian motion independent of $W^S$.

Definition The variance of the Heston model is the square-root diffusion of Cox-Ingersoll-Ross,

with $\kappa > 0$, $\theta > 0$, $\xi \ge 0$ and $v_0 \ge 0$. Its drift is positive below $\theta$ and negative above it. The moment bounds of this process, stated here without proof, give $\mathbb{E}^{\mathbb{Q}}\bigl[\int_0^T v_t \, dt\bigr] < \infty$, so the stochastic integral term is a martingale with mean zero. Taking expectations then gives the linear equation $\frac{d}{dt}\mathbb{E}^{\mathbb{Q}}[v_t] = \kappa\bigl(\theta - \mathbb{E}^{\mathbb{Q}}[v_t]\bigr)$, whose solution is

Since $v \ge 0$, the time integral and the expectation can be exchanged, and the expected integrated variance is

Neither formula involves $\xi$. The integrated variance $\int_0^T v_t \, dt$ is the quadratic variation of the log price over $[0, T]$.

Definition The leverage correlation is the parameter $\rho$ in $d[W^S, W^v]_t = \rho \, dt$, the instantaneous correlation between the shocks of the stock and of its variance. For the log price $x_t = \ln S_t$, Itô’s lemma gives

so the quadratic variation and covariation rates are

The generator of the pair $(x, v)$, acting on twice continuously differentiable functions $f(x, v)$, is

The mixed term collects the two equal cross terms $\tfrac12 f_{xv} \, d[x, v]$ and $\tfrac12 f_{vx} \, d[v, x]$ of Itô’s formula. A negative $\rho$ makes a negative stock shock come with a positive variance shock on average, which gives the model a leverage effect. Quadratic covariation does not change under an equivalent change of measure, so if the physical dynamics have the same diffusion form, they have the same $\rho$ and $\xi$, while the drift of the variance under $\mathbb{Q}$ is a modeling choice. When $\xi > 0$ and $\lvert \rho \rvert < 1$, the variance shock has a component independent of the stock, so the stock and the money-market account cannot replicate every claim, and replication does not identify the $\mathbb{Q}$ drift of the variance.

Definition For the square-root diffusion with $\kappa > 0$, $\theta > 0$, $\xi > 0$ and $v_0 > 0$, the Feller condition is

When it holds, the boundary point $0$ is inaccessible, so $v_t > 0$ for every $t$ with probability one. When $2\kappa\theta < \xi^2$, zero is accessible and the variance reaches it in finite time with positive probability, but the solution still never becomes negative. It is reflected at zero, where the drift equals $\kappa\theta > 0$. These boundary classification results for the Cox-Ingersoll-Ross process are stated here without proof. They concern the continuous-time solution. The Euler step

with $Z_{n+1} \sim N(0, 1)$, produces a negative value with positive probability whenever $\xi > 0$ and $v_n > 0$, whether or not the condition holds, and a simulation scheme has to treat that case, for example by truncating at zero.

Definition For a real number $u$, the conditional characteristic function of the log price $x_T = \ln S_T$ is $\varphi_t(u) = \mathbb{E}^{\mathbb{Q}}[e^{i u x_T} \mid \mathcal{F}_t]$. In the Heston model it has the exponential-affine form

with $\tau = T - t$, where the complex-valued coefficients $A$ and $B$ solve the Riccati equations of the next card with $A(0, u) = B(0, u) = 0$. By the Markov property, $\varphi_t(u) = f(t, x_t, v_t)$ for the function $f(t, x, v) = \mathbb{E}^{\mathbb{Q}}[e^{iux_T} \mid x_t = x, v_t = v]$, which satisfies the backward equation $\partial_t f + \mathcal{L} f = 0$ with $f(T, x, v) = e^{iux}$ when it is smooth enough. Substituting the exponential-affine guess turns this equation into ordinary differential equations, which the next card derives. That for every real $u$ these equations have solutions on the whole horizon, and that the resulting exponential is the characteristic function, is a standard result for affine models, stated here without proof. Since $\lvert e^{iux_T} \rvert = 1$, every value satisfies $\lvert \varphi_t(u) \rvert \le 1$, and $\varphi_t(0) = 1$.

Definition Write $\tau = T - t$ and substitute the guess

into the backward equation in the form $\partial_\tau f = \mathcal{L} f$. Each derivative is $f$ times a simple factor, namely $iu$ for $f_x$, $-u^2$ for $f_{xx}$, $B$ for $f_v$, $B^2$ for $f_{vv}$ and $iuB$ for $f_{xv}$, while $\partial_\tau f$ is $f$ times $iu(r - q) + \partial_\tau A + v \, \partial_\tau B$. Dividing by the nonzero $f$ and matching the terms without $v$ and the coefficients of $v$ gives the Riccati system

with $A(0, u) = B(0, u) = 0$. The equation for $B$ is quadratic in $B$, and $A(\tau, u) = \kappa\theta \int_0^\tau B(s, u)\, ds$ follows by integration. In the constant term, $-\tfrac12 u^2$ comes from the curvature term $\tfrac12 v f_{xx}$ and $-\tfrac12 iu$ from the drift correction $-\tfrac{v}{2}$ of the log price. That constant term vanishes at $u = 0$, where the solution is $A = B = 0$ and $\varphi = 1$. It also vanishes at $u = -i$, where the solution is again zero and the formal value $e^{x_T} = S_T$ gives $\mathbb{E}^{\mathbb{Q}}[S_T \mid \mathcal{F}_t] = S_t e^{(r - q)\tau}$, in line with the carry. Making that substitution rigorous needs the martingale property of the discounted stock. The system also has a closed-form solution in terms of complex square roots and logarithms, whose evaluation needs a consistent branch of the complex logarithm. That implementation, the conditions under which exponential moments of the log price are finite, and calibration to option prices lie beyond this lesson.