--- title: Quadratic Normal Volatility (QNV) url: https://www.emergentmind.com/topics/quadratic-normal-volatility-qnv type: topic --- # Quadratic Normal Volatility (QNV) 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 [1202.6187] [2507.12501] [2606.00800]. ## 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 [1202.6187]. In the generalized Black–Scholes formulation, the local volatility is written directly as a quadratic function of the asset level, \[ \sigma(S)=aS^2+bS+c, \] and the contingent claim \(C(S,t)\) solves \[ \frac{\partial C}{\partial t}+\frac12 \sigma(S)^2\frac{\partial^2 C}{\partial S^2}+rS\frac{\partial C}{\partial S}-rC=0, \qquad C(S,T)=\Phi(S). \] The volatility is “normal” in the sense that the diffusion coefficient is \(\sigma(S)\), not \(S\sigma\). The local volatility is taken time-independent in this formulation [2507.12501]. The class contains several important special cases. When \(e_1=e_2=0\), the model reduces to Brownian motion with constant volatility. When \(e_1=e_3=0\), the process shifted by a constant is a geometric Brownian motion. When \(e_2=e_3=0\), 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 [1202.6187]. ## 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 \(W\) be Brownian motion, stopped at the exit time \[ \tau:=\inf\{t\ge0:W_t\notin(a,b)\}, \] and define \[ Y_t=f(W_t^\tau),\qquad W_t^\tau:=W_{t\wedge\tau}. \] If the Radon–Nikodym density has the form \[ Z_t=\exp\!\left(\frac{C}{2}t\right)g(W_t^\tau), \qquad C=e_1e_3-\frac{e_2^2}{4}, \] and under the resulting measure \(Q\) the process \(Y\) is a local martingale, then \(Y\) necessarily satisfies the QNV SDE. Conversely, for any coefficients \(e_1,e_2,e_3\), such a representation exists [1202.6187]. The analytic mechanism is Riccati. Writing \[ \mu(x):=\frac12\frac{f''(x)}{f'(x)}, \] the relevant functions satisfy \[ \mu'(x)-\mu(x)^2=C, \] while \(f\) solves \[ f'(x)=e_1 f(x)^2+e_2 f(x)+e_3, \] and \(g\) solves \[ -g''(x)=Cg(x). \] 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 [1202.6187]. The expectation identities are correspondingly simple. For any nonnegative measurable path functional \(h\), \[ \mathbb E\big[h(\{Y_t\}_{t\in[0,T]})\big] = \mathbb E\Big[h\big(\{f(W_t^\tau)\}_{t\in[0,T]}\big)\, 1_{\{\tau>T\}} \exp\!\Big(\frac{C}{2}T\Big)\, g(W_T^\tau)\Big]. \] 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 [1202.6187]. ## 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 \(\tau=T-t\) and discounting via \(v(S,\tau)=e^{r\tau}C(S,t)\), one obtains \[ \frac{\partial v}{\partial \tau} = \frac12 \sigma(S)^2 \frac{\partial^2 v}{\partial S^2} + rS\frac{\partial v}{\partial S}. \] Introducing the Lamperti coordinate \[ x(S)=\int^S \frac{dS'}{\sigma(S')}, \] flattens the diffusion coefficient and yields \[ \frac{\partial v}{\partial \tau} = \frac12 v_{xx}+\nu(x)v_x, \qquad \nu(x)=\frac{rS(x)}{\sigma(S(x))}-\frac12 \sigma'(S(x)). \] A gauge transformation \(v(x,\tau)=g(x)\psi(x,\tau)\) with \(g'/g=-\nu\) removes the first derivative term and produces \[ \frac{\partial \psi}{\partial \tau} = \left[-\frac12\frac{\partial^2}{\partial x^2}+\tilde V(x)\right]\psi, \qquad \tilde V(x)=-\frac12(\nu'+\nu^2). \] The pricing problem is therefore recast as imaginary-time evolution under a self-adjoint Hamiltonian [2507.12501]. For the regime \(\Delta=b^2-4ac>0\), where the quadratic volatility has two real roots \(S_l0, \] with hypergeometric eigenfunctions. The pricing kernel is the spectral sum-plus-integral \[ K(x,\tau;x',0) = \sum_{n=0}^{\lfloor\lambda\rfloor}\phi_n(x)\phi_n^*(x')e^{-E_n\tau} + \int_0^\infty \phi_k(x)\phi_k^*(x')e^{-E_k\tau}\,dk. \] Any payoff can be propagated in transformed coordinates and then mapped back to prices [2507.12501]. ## 4. Geometry, discriminant regimes, and the volatility smile A geometric interpretation follows from the metric induced by the local volatility, \[ ds^2=\frac{dx^2}{\sigma(x)^2}. \] For the quadratic volatility \(\sigma(S)=aS^2+bS+c\), the sign of the discriminant \[ \Delta=b^2-4ac \] partitions the parameter space into three distinct regimes, each corresponding to a space of constant curvature. When \(\Delta>0\), the geometry is hyperbolic and therefore of constant negative curvature. When \(\Delta=0\), the geometry is Euclidean and therefore of zero curvature. When \(\Delta<0\), the geometry is spherical and therefore of constant positive curvature [2507.12501]. In the hyperbolic regime, the Lamperti map is logarithmic, the transformed state space is \(\mathbb R\), 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 \(S_l,S_u\) and near the center, and this nonlinear distortion produces option prices that deviate from lognormal pricing. The curvature parameter is essentially \(1/\alpha^2\), while the potential depth is controlled by \(\lambda\) [2507.12501]. 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 [2507.12501]. ## 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 \(Y\), true martingality holds if and only if either \(e_1=0\), or \(P\) has two real roots \(r_1\le r_2\) and the initial value \(y_0\) lies in \([r_1,r_2]\). For the stopped process \(X\), true martingality holds if and only if either \(e_1=0\), or \(P\) has a root \(r\in\mathbb R\) with \(x_0\le r\). Otherwise the process is a strict local martingale. These distinctions are relevant for put–call parity and no-arbitrage issues [1202.6187]. A particularly important practical property is stability under change of numeraire. If \(X\) is a stopped QNV process under \(Q\) with polynomial \[ P(z)=e_1 z^2+e_2 z+e_3, \] and one passes to the Föllmer measure associated with \(X\), then the reciprocal process \(\widehat X=1/X\) is again a stopped QNV process with polynomial \[ \widehat P(z)=-z^2 P(1/z)=-e_3 z^2-e_2 z-e_1. \] 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 [1202.6187]. The literature summarized in the data also records a symmetry identity for a special polynomial \(P(z)=e z^2/L+e_2 z+eL\) with initial value \(x_0=L\): \[ \mathbb E\left[h\left(\frac{X_T}{L}\right)\right] = \mathbb E\left[h\left(\frac{L}{X_T}\right)\frac{X_T}{L}\right] \] for any nonnegative measurable \(h\) with \(h(0)=0\) and finite \(h(\infty)\). 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 [1202.6187]. ## 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 \[ \mathbb E(\sigma^2\mid z)=\sigma_0^2+k z^2, \qquad k\approx \tfrac12, \] or, in the notation \(V=\sigma^2\), \[ \mathbb E(V\mid z)=V_0+kz^2, \] where \(z\) is the return over an interval \(T\), divided by \(\sqrt{T}\), so that conditioned on the latent variance over that interval one has \(z\mid V\sim\mathcal N(0,V)\). 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 \(V\), and equivalently to a Student-t marginal for \(z\) [2606.00800]. The same work then derives a multiplicative Langevin dynamics for variance, \[ dV_t=\gamma(\bar V-V_t)\,dt+sV_t\,dW_t, \] whose stationary density is exactly Inverse Gamma and whose coherence time is approximately \[ \tau_c\approx \frac{1}{\gamma+\tfrac12 s^2}. \] For horizons \(T\ll \tau_c\), the variance is approximately constant over the interval, so the Q-variance relation is approximately independent of \(T\); for \(T\) comparable to or larger than \(\tau_c\), the relation is washed out and the coefficient \(k\) is driven toward zero [2606.00800]. 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, \[ \sigma^2(X_t)=\alpha+\beta X_t+\gamma X_t^2 \] or equivalently \(\sigma(F_t)=aF_t^2+bF_t+c\) 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 [2606.00800]. ## 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 [1202.6187]. 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 \(\Delta>0\) regime, with the hyperbolic Pöschl–Teller potential [2507.12501]. 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 [2507.12501]. 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 [1202.6187] [2507.12501] [2606.00800]. Source: https://www.emergentmind.com/topics/quadratic-normal-volatility-qnv