CEV Dynamics Across Volatility Exponents

CEV Dynamics and the Role of the Exponent

The constant-elasticity-of-variance (CEV) model describes a positive state whose instantaneous volatility changes as a power of its current level. Its exponent determines how quickly the diffusion coefficient shrinks—or grows—near zero. Comparing exponents is useful, but the coefficient’s regularity and the actual boundary behavior are distinct questions: boundary conclusions also depend on the drift and on how the process is defined at zero.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\). For \(x\ge0\), a common CEV equation is

where \(\kappa\in\mathbb{R}\), \(\sigma>0\), and, for this discussion, \(\beta\ge0\). The drift coefficient is \(\kappa x\), while the diffusion coefficient is \(\sigma x^\beta\). An SDE solution is understood to be adapted and continuous, and its stochastic integral is an Itô integral: on every finite time interval where it is used, the integrand is progressively measurable and belongs to \(\mathcal{L}^2\), meaning \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\). For coefficients that may grow too quickly for this condition globally, the integral and equation can instead be considered up to exit times from bounded state intervals.

In applications, \(X_t\) is often restricted to nonnegative values. When discussing a process killed at zero, we mean that it follows the SDE until its first hit of zero and is then held at zero. This is one possible boundary convention, not an automatic consequence of writing down the equation. In particular, when a coefficient is not locally Lipschitz at zero, existence and uniqueness questions require care.

What Changes at Exponent One?

For \(0<\beta<1\), the coefficient tends to zero at zero, but less rapidly than a linear function. It is locally Hölder continuous with exponent \(\beta\) there, and is not locally Lipschitz at zero. At \(\beta=1\), the coefficient is linear and locally Lipschitz. For \(\beta>1\), it vanishes faster than linearly and is also locally Lipschitz at zero. These statements concern the coefficient as a function; they do not alone decide whether a sample path reaches the boundary.

Proof. If \(\beta\ge1\), then \(a\) is continuous at zero and, for \(x>0\), \(a'(x)=\sigma\beta x^{\beta-1}\). On \([0,R]\) this derivative is bounded by \(\sigma\beta R^{\beta-1}\) when \(\beta\ge1\). The mean value theorem therefore gives a finite Lipschitz bound on \([0,R]\), including pairs with one endpoint zero. Now suppose \(0<\beta<1\). For \(x>0\), the quotient \(|a(x)-a(0)|/|x-0|=\sigma x^{\beta-1}\) tends to infinity as \(x\downarrow0\), so no finite local Lipschitz constant exists. To check the Hölder claim, for \(u,v\ge0\) use \(|u^\beta-v^\beta|\le |u-v|^\beta\), which follows from \((r+s)^\beta\le r^\beta+s^\beta\) for \(r,s\ge0\). Multiplication by \(\sigma\) proves the stated bound. \(\square\)

At \(\beta=0\), the coefficient is the positive constant \(\sigma\), so it is Lipschitz but does not vanish at zero. This endpoint is therefore different from the positive exponents below one: noise does not switch off at the boundary. A nonnegative-state model needs an explicit boundary convention in this case.

The Exponent-One Case

When \(\beta=1\), the state and diffusion coefficients are linear. For positive initial state, the equation has the explicit geometric Brownian motion solution. Its pathwise expression gives a direct boundary conclusion for any finite time horizon: the process remains strictly positive. This special case is especially useful as a benchmark, but its conclusion should not simply be extended to other exponents.

Proof. Define \(Y_t=x_0\exp((\kappa-\tfrac12\sigma^2)t+\sigma B_t)\). Apply Itô’s formula to \(f(t,b)=x_0\exp((\kappa-\tfrac12\sigma^2)t+\sigma b)\). Its derivatives are \(f_t=(\kappa-\tfrac12\sigma^2)f\), \(f_b=\sigma f\), and \(f_{bb}=\sigma^2f\). The formula, including its second-derivative correction, gives \(dY_t=f_t\,dt+f_b\,dB_t+\tfrac12 f_{bb}\,dt=\kappa Y_t\,dt+\sigma Y_t\,dB_t\). Thus \(Y\) satisfies the equation. The coefficients are globally Lipschitz, so the standard existence-and-uniqueness result for Lipschitz SDEs identifies this solution with the unique solution. The exponential of a finite real number is positive, and Brownian paths are finite at finite times, so \(Y_t>0\). \(\square\)

Candidate Zero-Boundary Behavior

A clean comparison uses the driftless CEV diffusion \(dX_t=\sigma X_t^\beta\,dB_t\) on the positive half-line, considered up to its first hit of zero or an upper level. The standard one-dimensional diffusion boundary test expresses accessibility through the scale and speed densities. For a diffusion with drift \(b(x)\) and diffusion coefficient \(a(x)>0\) on \((0,\infty)\), these densities are \(s'(x)=\exp(-\int^x 2b(y)/a(y)^2\,dy)\) and \(m(x)=2/(a(x)^2s'(x))\), up to a positive constant in the scale density. The Feller boundary classification, including its accessibility criterion, is a standard result of one-dimensional diffusion theory; its proof is beyond this tutorial.

For zero drift, choose \(s'(x)=1\), so \(s(x)=x\), and \(m(x)=2/(\sigma^2x^{2\beta})\). The accessibility integral at zero is finite precisely when \(\int_0^c (s(x)-s(0))m(x)\,dx\) is finite, for any fixed \(c>0\). Here its integrand is a constant multiple of \(x^{1-2\beta}\). The integral is finite for \(\beta<1\), and infinite for \(\beta\ge1\). Under this driftless comparison, zero is accessible for exponents below one and is not accessible from the interior for exponents at least one. Accessibility means the boundary can be hit; the behavior after hitting it still requires a boundary convention.

The coefficient comparison and this boundary test tell a consistent local story, but they are not interchangeable. Local Lipschitz regularity is a property of the coefficient, while accessibility is a property of the diffusion and its boundary setup. For \(\beta>1\), a nonzero linear drift changes the scale density near zero, so the driftless test cannot be quoted as a classification for every \(\kappa\). For \(\beta<1\), the scale factor from a linear drift has a finite, nonzero limit at zero because the relevant integral has integrand proportional to \(x^{1-2\beta}\), which is integrable there; the same accessibility threshold follows in this setting. At \(\beta=1\), the explicit solution already settles the positive-initial-state case for any \(\kappa\).

Interpretation and Common Pitfalls

The exponent indicates how strongly noise acts near zero. Below one, the diffusion coefficient is not locally Lipschitz when \(0<\beta<1\), and the driftless boundary test predicts accessibility. At one, the geometric model has a positive explicit solution. Above one, the coefficient is locally Lipschitz and vanishes faster than linearly, while the driftless boundary test predicts inaccessibility. These are useful reference cases, not a substitute for stating the drift, initial condition, state space, and boundary convention.

A common error is to infer “zero cannot be reached” merely from the fact that the coefficient vanishes at zero. For example, in the driftless model with \(\beta=\tfrac12\), the coefficient \(a(x)=\sigma\sqrt{x}\) is zero at zero, yet the accessibility integral is finite. Conversely, the coefficient’s lack of local Lipschitz regularity does not by itself prove that every solution hits zero or determine what happens after a hit. Those conclusions require the diffusion boundary test and a specified solution and boundary setup.