Quadratic Normal Volatility (QNV)

• Quadratic Normal Volatility (QNV) is defined as a one-factor local volatility diffusion where the instantaneous volatility is a quadratic function of the asset level.
• The model achieves analytic tractability by representing the process through a stopped Brownian motion, smooth measure change, and Riccati-based transformations.
• It unifies pricing methodologies by reducing to a Schrödinger-type spectral problem and clarifying volatility smile dynamics via constant-curvature geometric interpretations.

Quadratic Normal Volatility (QNV) denotes a class of one-factor local volatility diffusions in the asset level for which the instantaneous volatility is a quadratic function of the level. In the driftless formulation used for discounted prices, a QNV process $Y$ satisfies

$dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,$

with quadratic polynomial $P(z)=e_1 z^2+e_2 z+e_3$; in the risk-neutral local-volatility formulation, the asset price $S_t$ satisfies

$$dS_t = r S_t\,dt + \sigma(S_t)\,dW_t,\qquad \sigma(S)=aS^2+bS+c.$$

The class is distinguished by analytic tractability, a characterization in terms of stopped Brownian motion plus a simple change of measure, and an exact reformulation of the pricing problem as a Schrödinger-type spectral problem. More recent work has also clarified that QNV should be distinguished from “Q-variance,” which is a quadratic conditional expectation relation for realized variance rather than a structural specification of instantaneous volatility (Carr et al., 2012, Saucedo, 16 Jul 2025, Press et al., 30 May 2026).

1. Definition and canonical formulations

In the formulation centered on the discounted underlying, the basic QNV process is defined on a filtered probability space with Brownian motion $B$ by

$$dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,\qquad Y_0=y_0>0.$$

There is no drift term; under the risk-neutral measure $Q$, the discounted underlying is modeled directly as a local martingale, which may be strict in some parameter regions. The stopped version is obtained by stopping at the first time the process hits zero: $$S := \inf\{t\ge0:Y_t=0\},\qquad X_t:=Y_{t\wedge S}.$$ The polynomial $P(z)=e_1 z^2+e_2 z+e_3$ governs the qualitative behavior of the model (Carr et al., 2012).

In the generalized Black–Scholes formulation, the local volatility is written directly as a quadratic function of the asset level,

$dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,$0

and the contingent claim $dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,$1 solves

$dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,$2

The volatility is “normal” in the sense that the diffusion coefficient is $dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,$3, not $dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,$4. The local volatility is taken time-independent in this formulation (Saucedo, 16 Jul 2025).

The class contains several important special cases. When $dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,$5, the model reduces to Brownian motion with constant volatility. When $dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,$6, the process shifted by a constant is a geometric Brownian motion. When $dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,$7, the process is the reciprocal of a 3-dimensional Bessel process. These embeddings clarify that QNV interpolates between constant-volatility, lognormal-type, and reciprocal-Bessel dynamics within a single quadratic specification (Carr et al., 2012).

2. Characterization by stopped Brownian motion and simple measure change

A central structural result is that QNV models are exactly the local-martingale diffusions that can be obtained from stopped Brownian motion by a smooth monotone transformation and a change of measure whose density depends only on the terminal value of the stopped Brownian motion. Concretely, let $dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,$8 be Brownian motion, stopped at the exit time

$dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,$9

and define

$P(z)=e_1 z^2+e_2 z+e_3$0

If the Radon–Nikodym density has the form

$P(z)=e_1 z^2+e_2 z+e_3$1

and under the resulting measure $P(z)=e_1 z^2+e_2 z+e_3$2 the process $P(z)=e_1 z^2+e_2 z+e_3$3 is a local martingale, then $P(z)=e_1 z^2+e_2 z+e_3$4 necessarily satisfies the QNV SDE. Conversely, for any coefficients $P(z)=e_1 z^2+e_2 z+e_3$5, such a representation exists (Carr et al., 2012).

The analytic mechanism is Riccati. Writing

$P(z)=e_1 z^2+e_2 z+e_3$6

the relevant functions satisfy

$P(z)=e_1 z^2+e_2 z+e_3$7

while $P(z)=e_1 z^2+e_2 z+e_3$8 solves

$P(z)=e_1 z^2+e_2 z+e_3$9

and $S_t$0 solves

$S_t$1

Because Riccati equations with constant coefficients admit explicit exponential, hyperbolic, or trigonometric solutions, the transformation from Brownian motion to the QNV diffusion is explicit. This explains why pricing formulas reduce to weighted Brownian expectations and why explicit analytic formulas arise in the literature (Carr et al., 2012).

The expectation identities are correspondingly simple. For any nonnegative measurable path functional $S_t$2,

$S_t$3

For European payoffs, this becomes a one-dimensional Brownian expectation involving the event that Brownian motion stays inside an interval up to maturity. The joint law of terminal value, running minimum, and running maximum can then be used through standard reflection arguments. This is the probabilistic source of the model’s tractability (Carr et al., 2012).

3. PDE reduction, Lamperti coordinates, and Hamiltonian form

The generalized Black–Scholes PDE associated with QNV admits a canonical reduction to a self-adjoint Schrödinger operator. After reversing time with $S_t$4 and discounting via $S_t$5, one obtains

$S_t$6

Introducing the Lamperti coordinate

$S_t$7

flattens the diffusion coefficient and yields

$S_t$8

A gauge transformation $S_t$9 with $$dS_t = r S_t\,dt + \sigma(S_t)\,dW_t,\qquad \sigma(S)=aS^2+bS+c.$$0 removes the first derivative term and produces

$$dS_t = r S_t\,dt + \sigma(S_t)\,dW_t,\qquad \sigma(S)=aS^2+bS+c.$$1

The pricing problem is therefore recast as imaginary-time evolution under a self-adjoint Hamiltonian (Saucedo, 16 Jul 2025).

For the regime $$dS_t = r S_t\,dt + \sigma(S_t)\,dW_t,\qquad \sigma(S)=aS^2+bS+c.$$2, where the quadratic volatility has two real roots $$dS_t = r S_t\,dt + \sigma(S_t)\,dW_t,\qquad \sigma(S)=aS^2+bS+c.$$3, the Lamperti transform is explicit: $$dS_t = r S_t\,dt + \sigma(S_t)\,dW_t,\qquad \sigma(S)=aS^2+bS+c.$$4 which maps the finite interval $$dS_t = r S_t\,dt + \sigma(S_t)\,dW_t,\qquad \sigma(S)=aS^2+bS+c.$$5 to $$dS_t = r S_t\,dt + \sigma(S_t)\,dW_t,\qquad \sigma(S)=aS^2+bS+c.$$6. In this case, the transformed potential simplifies to the hyperbolic Pöschl–Teller form

$$dS_t = r S_t\,dt + \sigma(S_t)\,dW_t,\qquad \sigma(S)=aS^2+bS+c.$$7

The mapping between the financial parameters $$dS_t = r S_t\,dt + \sigma(S_t)\,dW_t,\qquad \sigma(S)=aS^2+bS+c.$$8 and the quantum parameters $$dS_t = r S_t\,dt + \sigma(S_t)\,dW_t,\qquad \sigma(S)=aS^2+bS+c.$$9 provides a direct dictionary, with the curvature of the volatility smile, governed by $B$0, dictating the width of the potential well, while the other parameters determine its depth and offset (Saucedo, 16 Jul 2025).

This representation gives a complete spectral machinery for semi-analytic pricing. The stationary eigenvalue problem

$B$1

has a discrete spectrum

$B$2

with eigenfunctions proportional to associated Legendre functions, and a continuous spectrum

$B$3

with hypergeometric eigenfunctions. The pricing kernel is the spectral sum-plus-integral

$B$4

Any payoff can be propagated in transformed coordinates and then mapped back to prices (Saucedo, 16 Jul 2025).

4. Geometry, discriminant regimes, and the volatility smile

A geometric interpretation follows from the metric induced by the local volatility,

$B$5

For the quadratic volatility $B$6, the sign of the discriminant

$B$7

partitions the parameter space into three distinct regimes, each corresponding to a space of constant curvature. When $B$8, the geometry is hyperbolic and therefore of constant negative curvature. When $B$9, the geometry is Euclidean and therefore of zero curvature. When $$dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,\qquad Y_0=y_0>0.$$0, the geometry is spherical and therefore of constant positive curvature (Saucedo, 16 Jul 2025).

In the hyperbolic regime, the Lamperti map is logarithmic, the transformed state space is $$dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,\qquad Y_0=y_0>0.$$1, and the effective potential is the hyperbolic Pöschl–Teller well. The stated interpretation is that the volatility smile is not treated as an ad hoc empirical observation but as the projection of a natural diffusion process from curved space into the observed price coordinate. Equal increments in the intrinsic coordinate correspond to very different increments in price space near the endpoints $$dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,\qquad Y_0=y_0>0.$$2 and near the center, and this nonlinear distortion produces option prices that deviate from lognormal pricing. The curvature parameter is essentially $$dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,\qquad Y_0=y_0>0.$$3, while the potential depth is controlled by $$dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,\qquad Y_0=y_0>0.$$4 (Saucedo, 16 Jul 2025).

This perspective also explains why QNV combines flexibility with solvability. Constant-curvature manifolds are maximally symmetric and admit separable solutions, while the corresponding financial Hamiltonian is shape-invariant in the hyperbolic Pöschl–Teller case. The resulting framework complements earlier pathwise and martingale-based tractability results by placing QNV in the setting of spectral theory, special functions, and gauge transformations (Saucedo, 16 Jul 2025).

5. Martingality, numeraire changes, and structural symmetries

The martingale property of QNV processes is determined entirely by the roots of the quadratic polynomial. For the unstopped process $$dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,\qquad Y_0=y_0>0.$$5, true martingality holds if and only if either $$dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,\qquad Y_0=y_0>0.$$6, or $$dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,\qquad Y_0=y_0>0.$$7 has two real roots $$dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,\qquad Y_0=y_0>0.$$8 and the initial value $$dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,\qquad Y_0=y_0>0.$$9 lies in $Q$0. For the stopped process $Q$1, true martingality holds if and only if either $Q$2, or $Q$3 has a root $Q$4 with $Q$5. Otherwise the process is a strict local martingale. These distinctions are relevant for put–call parity and no-arbitrage issues (Carr et al., 2012).

A particularly important practical property is stability under change of numeraire. If $Q$6 is a stopped QNV process under $Q$7 with polynomial

$Q$8

and one passes to the Föllmer measure associated with $Q$9, then the reciprocal process $$S := \inf\{t\ge0:Y_t=0\},\qquad X_t:=Y_{t\wedge S}.$$0 is again a stopped QNV process with polynomial

$$S := \inf\{t\ge0:Y_t=0\},\qquad X_t:=Y_{t\wedge S}.$$1

Thus the QNV class is closed under reciprocal transformation in the sense relevant for FX numeraires. This closedness is one of the reasons the class is attractive in multi-currency settings (Carr et al., 2012).

The literature summarized in the data also records a symmetry identity for a special polynomial $$S := \inf\{t\ge0:Y_t=0\},\qquad X_t:=Y_{t\wedge S}.$$2 with initial value $$S := \inf\{t\ge0:Y_t=0\},\qquad X_t:=Y_{t\wedge S}.$$3: $$S := \inf\{t\ge0:Y_t=0\},\qquad X_t:=Y_{t\wedge S}.$$4 for any nonnegative measurable $$S := \inf\{t\ge0:Y_t=0\},\qquad X_t:=Y_{t\wedge S}.$$5 with $$S := \inf\{t\ge0:Y_t=0\},\qquad X_t:=Y_{t\wedge S}.$$6 and finite $$S := \inf\{t\ge0:Y_t=0\},\qquad X_t:=Y_{t\wedge S}.$$7. This identity can be interpreted as a semistatic replication symmetry for certain barrier-style claims. More broadly, the structural correspondence with Brownian motion and the preservation under numeraire change make QNV unusually robust among analytically tractable local-volatility models (Carr et al., 2012).

6. Relation to Q-variance and common conceptual confusions

QNV is distinct from the “Q-variance” relation studied in recent work on realized variance and scaled returns. There, the empirical starting point is

$$S := \inf\{t\ge0:Y_t=0\},\qquad X_t:=Y_{t\wedge S}.$$8

or, in the notation $$S := \inf\{t\ge0:Y_t=0\},\qquad X_t:=Y_{t\wedge S}.$$9,

$P(z)=e_1 z^2+e_2 z+e_3$0

where $P(z)=e_1 z^2+e_2 z+e_3$1 is the return over an interval $P(z)=e_1 z^2+e_2 z+e_3$2, divided by $P(z)=e_1 z^2+e_2 z+e_3$3, so that conditioned on the latent variance over that interval one has $P(z)=e_1 z^2+e_2 z+e_3$4. In that framework, the paper shows that the Q-variance relation is exactly equivalent, in the Gaussian-mixture setting, to an Inverse Gamma distribution for $P(z)=e_1 z^2+e_2 z+e_3$5, and equivalently to a Student-t marginal for $P(z)=e_1 z^2+e_2 z+e_3$6 (Press et al., 30 May 2026).

The same work then derives a multiplicative Langevin dynamics for variance,

$P(z)=e_1 z^2+e_2 z+e_3$7

whose stationary density is exactly Inverse Gamma and whose coherence time is approximately

$P(z)=e_1 z^2+e_2 z+e_3$8

For horizons $P(z)=e_1 z^2+e_2 z+e_3$9, the variance is approximately constant over the interval, so the Q-variance relation is approximately independent of $dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,$00; for $dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,$01 comparable to or larger than $dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,$02, the relation is washed out and the coefficient $dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,$03 is driven toward zero (Press et al., 30 May 2026).

The conceptual overlap with QNV is limited but nontrivial. In a QNV model, the instantaneous variance is specified structurally as a quadratic function of the contemporaneous state,

$dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,$04

or equivalently $dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,$05 in normal coordinates. In Q-variance, by contrast, the quadratic object is the conditional expectation of realized variance given the realized return over an interval. The former is a diffusion specification inside the SDE; the latter is a statistical relation derived from a Gaussian scale mixture with Inverse Gamma variance. Both involve “variance is quadratic in something,” but they are not the same assertion. A plausible implication is that the Q-variance framework supplies a statistical target—quadratic dependence on movement size, together with Inverse Gamma/Student-t structure—that a QNV-style diffusion may be chosen to approximate in a continuous-time pricing model (Press et al., 30 May 2026).

7. Position in the literature and research directions

Within the QNV literature, the model class has been studied both for tractability and for explicit pricing. Earlier work characterizes QNV as exactly the class obtainable from stopped Brownian motion by a deterministic transformation and a simple change of measure depending only on terminal value, thereby explaining the existence of explicit analytic formulas for option prices (Carr et al., 2012). More recent work revisits the same class through a Hamiltonian and quantum-mechanical lens, identifying the generalized Black–Scholes operator with a self-adjoint Schrödinger operator and, in the $dY_t = \big(e_1 Y_t^2 + e_2 Y_t + e_3\big)\,dB_t,$06 regime, with the hyperbolic Pöschl–Teller potential (Saucedo, 16 Jul 2025).

The 2025 spectral analysis explicitly situates the model alongside the earlier QNV literature of Zülsdorff and Carr–Fisher–Ruf. Its contribution is not to replace the probabilistic viewpoint but to complement it by providing an exact spectral decomposition, a pricing kernel in terms of classical special functions, and a geometric interpretation of smiles as effects of diffusion on constant-curvature manifolds. The paper also emphasizes future research directions involving multi-factor extensions, integrable systems such as the Calogero–Sutherland model, connections to KdV via the Pöschl–Teller potential, and topological quantum field-theoretic analogies, though these directions are explicitly exploratory (Saucedo, 16 Jul 2025).

Taken together, the sources describe QNV as a rare local-volatility class in which several forms of structure coincide: Riccati solvability, explicit Brownian representation, closure under reciprocal-numeraire transformations, exact reduction of the pricing PDE to a self-adjoint operator, and a discriminant-based geometric classification of smile regimes. The recent Q-variance literature does not redefine QNV, but it sharpens the distinction between structural quadratic volatility models and statistical quadratic variance relations, while also suggesting a broader landscape in which quadraticity can enter either through the SDE itself or through conditional laws of returns and realized variance (Carr et al., 2012, Saucedo, 16 Jul 2025, Press et al., 30 May 2026).