For the two-factor stochastic volatility system dFt=σtδFt dWtdF_t=\sigma_t^\delta F_t\,dW_t, dσt=ασtγ dZt+b(σt) dtd\sigma_t=\alpha\sigma_t^\gamma\,dZ_t+b(\sigma_t)\,dt with Brownian motions correlated by ρ\rho, Lions and Musiela (2007) proved that the mm-th moment of the price (m>1m>1) is finite for all maturities when ρ<ρc(m)\rho<\rho_c(m) and infinite when ρ>ρc(m)\rho>\rho_c(m), where ρc(m)=−(m−1)/m−b∞/(αm)\rho_c(m)=-\sqrt{(m-1)/m}-b_\infty/(\alpha m) and b∞≤0b_\infty\le 0 is the leading growth rate of the drift. At exactly ρ=ρc(m)\rho=\rho_c(m) their exponential-supersolution comparison method degenerates (the governing quadratic in the supersolution parameter has a double root), and the paper records only one-sided sufficient conditions, stating that the result depends on the behaviour of the drift beyond its leading term. The Heston-type critical case is in fact settled inside the paper itself (Theorem 3.2(vi)); the later literature completely classified the martingale case m=1m=1 with arbitrary correlation and the affine (Heston-type) sub-class, and proved sharp two-sided results for the log-normal volatility model with exactly quadratic drift when its subleading linear drift term is non-positive. The surviving open core is the sharp characterization, by necessary and sufficient conditions expressed through the subleading asymptotics of the drift, of LmL^m-moment finiteness at the critical correlation for the general power-law class, including concrete undecided instances such as the quadratic-drift log-normal volatility model with positive linear correction at criticality.

Background

A one-factor stochastic volatility model specifies a positive asset price FtF_t whose returns have a random instantaneous volatility σt\sigma_t, itself driven by a second Brownian motion ZtZ_t correlated with the price-driving Brownian motion WtW_t through a parameter ρ∈[−1,1]\rho\in[-1,1]. Because the two-factor stochastic differential equation (SDE) couples the price to the volatility, its coefficients typically violate the classical Lipschitz and linear-growth conditions that guarantee a well-posed equation, and two pathologies appear: the discounted price can be a strict local martingale (a nonnegative local martingale with E[Ft]<F0\mathbb{E}[F_t]<F_0, which in arbitrage theory is the signature of a price bubble), and the moments E[Ftm]\mathbb{E}[F_t^m] for m>1m>1 can be infinite even at small maturities, a phenomenon called moment explosion. Finite higher moments are needed in practice to price contracts with super-linear payoffs, to justify Monte-Carlo confidence intervals (second moment), and, through Lee's moment formula, they control the slope of the implied-volatility smile (the Black-Scholes volatility backed out of option prices as a function of strike) at extreme strikes: the largest finite moment determines the slope of the smile wings.

Lions and Musiela (2007) began the rigorous analysis of the most classical stochastic volatility families. For the log-normal volatility model dFt=σtFtdWtdF_t=\sigma_tF_tdW_t, dσt=ασtdZtd\sigma_t=\alpha\sigma_tdZ_t (the continuous part of the Scott and log-normal SABR models), they proved a sharp two-sided result including the boundary: FF is a martingale if and only if ρ≤0\rho\le 0, and for m>1m>1, sup⁡t≤TE[Ftm]<∞\sup_{t\le T}\mathbb{E}[F_t^m]<\infty if and only if ρ≤−(m−1)/m\rho\le-\sqrt{(m-1)/m} (Theorem 2.3). They then studied the power-law family

dFt=σtδFt dWt,dσt=ασtγ dZt+b(σt) dt,dF_t=\sigma_t^\delta F_t\,dW_t,\qquad d\sigma_t=\alpha\sigma_t^\gamma\,dZ_t+b(\sigma_t)\,dt,

with α>0\alpha>0, γ,δ∈(0,1]\gamma,\delta\in(0,1] (the case where the exponent on FF is below one is benign), drift bb locally Lipschitz with b(0)≥0b(0)\ge 0 and b(ξ)≤C(1+ξ)b(\xi)\le C(1+\xi), and b∞=lim⁡ξ→∞b(ξ)/ξγ+δ∈(−∞,0]b_\infty=\lim_{\xi\to\infty}b(\xi)/\xi^{\gamma+\delta}\in(-\infty,0] the leading growth rate of the drift at high volatility. The key step is a Girsanov change of measure (a reweighting of paths that tilts the drift): computing E[Ftm]\mathbb{E}[F_t^m] is equivalent to evaluating Eexp⁡(m2−m2∫0tσs2δds)\mathbb{E}\exp(\frac{m^2-m}{2}\int_0^t\sigma_s^{2\delta}ds) under a tilted volatility dynamics with drift ραmσγ+δ+b(σ)\rho\alpha m\sigma^{\gamma+\delta}+b(\sigma), and Lions and Musiela bound this exponential functional by comparing the associated linear parabolic equation with exponential supersolutions exp⁡(Aξq+Bt)\exp(A\xi^q+Bt) (functions that dominate a solution from above, via the maximum principle). The comparison is possible precisely when a quadratic in AA is nonnegative, which reduces to the correlation condition

ρ  <  ρc(m):=−m−1m−b∞αm.\rho\;<\;\rho_c(m):=-\sqrt{\tfrac{m-1}{m}}-\tfrac{b_\infty}{\alpha m}.

Their outcome (Theorems 3.2 and 3.3): E[sup⁡s≤tFsm]<∞\mathbb{E}[\sup_{s\le t}F_s^m]<\infty for all tt when ρ<ρc(m)\rho<\rho_c(m), and E[Ftm]=+∞\mathbb{E}[F_t^m]=+\infty for all t>0t>0 when ρ>ρc(m)\rho>\rho_c(m), in the cases γ+δ>1\gamma+\delta>1; for the border case γ+δ=1\gamma+\delta=1 (which contains the Heston model, γ=δ=1/2\gamma=\delta=1/2 with b(ξ)=b(σˉ−ξ)b(\xi)=b(\bar\sigma-\xi) for positive constants b,σˉb,\bar\sigma — the source's notation, with b∞=−bb_\infty=-b — where the state variable σ\sigma plays the role of the variance process) the non-strict inequality ρ≤ρc(m)\rho\le\rho_c(m) gives finiteness for all maturities, while ρ>ρc(m)\rho>\rho_c(m) makes the moment explode only after an explicit finite time T0T_0 determined by a Riccati-type ordinary differential equation. Thus the Heston critical correlation is fully characterized inside the paper.

What the paper leaves open is the borderline itself in the remaining cases. In the cases γ+δ>1\gamma+\delta>1 the quadratic above has a double root exactly at ρ=ρc(m)\rho=\rho_c(m), the exponential-supersolution construction must be retuned to the subleading behaviour of bb, and the authors state in a remark after Theorem 3.2 that the borderline is 'slightly more delicate' since 'the result depends on the behaviour of b(ξ)−b∞ξγ+δb(\xi)-b_\infty\xi^{\gamma+\delta}'; they only prove sufficient conditions, e.g. in the case γ=δ\gamma=\delta finiteness at criticality when b(ξ)−b∞ξ2γb(\xi)-b_\infty\xi^{2\gamma} is bounded above on [0,∞)[0,\infty), and a corresponding displayed condition when γ≠δ\gamma\ne\delta that also involves the Itô correction term α22(δ−γ)ξγ+δ−1\frac{\alpha^2}{2}(\delta-\gamma)\xi^{\gamma+\delta-1}. No necessary condition at the critical correlation is given. The same gap appears in their general-dispersion version, dσt=μ(σt)dZt+b(σt)dtd\sigma_t=\mu(\sigma_t)dZ_t+b(\sigma_t)dt with μ(ξ)∼μ∞ξ\mu(\xi)\sim\mu_\infty\xi, where the condition with A0=m2−m/μ∞A_0=\sqrt{m^2-m}/\mu_\infty is again only sufficient.

Later work settled the neighbouring questions but not this one. For the martingale question (m=1m=1) Mijatović and Urusov (2012) gave necessary and sufficient conditions via a boundary-classification analysis of an associated ODE, and Bernard, Cui and McLeish (Mathematical Finance 27 (2017)) extended the complete classification to arbitrary correlation −1≤ρ≤1-1\le\rho\le 1 for volatility processes that are time-homogeneous diffusions, explicitly describing the Lions–Musiela conditions as sufficient only; Akdim, Eddahbi and Haddadi applied this to the Scott model. For affine models (Heston, Bates, Barndorff-Nielsen–Shephard), Andersen and Piterbarg (2007) and Keller-Ressel (2011) determine moment explosion times and critical moments completely from generalized Riccati equations, confirming the Heston boundary behaviour. For the practically important non-affine case γ=δ=1\gamma=\delta=1 with exactly quadratic drift, dσt=(R0+R1σt)(R2−σt)dt+νσtdZtd\sigma_t=(R_0+R_1\sigma_t)(R_2-\sigma_t)dt+\nu\sigma_tdZ_t, Carr and Willems (2019) built directly on Lions and Musiela's Theorems 2.5/2.6 and proved a two-sided result, E[STm]<∞\mathbb{E}[S_T^m]<\infty for all TT if R1>ν(ρm+m2−m)R_1>\nu(\rho m+\sqrt{m^2-m}) and =∞=\infty for all TT if R1<ν(ρm+m2−m)R_1<\nu(\rho m+\sqrt{m^2-m}); at the critical value they obtain finiteness only under the additional condition R0≥R1R2R_0\ge R_1R_2 (non-positive subleading linear drift), and the complementary boundary sub-case is left undecided. Sepp and Rakhmonov (2023) develop semi-analytic moment-generating-function expansions for the same quadratic-drift log-normal family without claiming a critical-correlation characterization. The question posed below is the surviving core of the Lions–Musiela analysis: the sharp borderline.

Problem Statement

Fix α>0\alpha>0, δ,γ∈(0,1]\delta,\gamma\in(0,1] with γ+δ>1\gamma+\delta>1, and a locally Lipschitz drift bb on [0,∞)[0,\infty) with b(0)≥0b(0)\ge 0, b(ξ)≤C(1+ξ)b(\xi)\le C(1+\xi), and a limit b∞=lim⁡ξ→∞b(ξ)/ξγ+δ∈(−∞,0]b_\infty=\lim_{\xi\to\infty}b(\xi)/\xi^{\gamma+\delta}\in(-\infty,0]. Let (Ft,σt)(F_t,\sigma_t) be the nonnegative solution (in the conventions of the source) of

dFt=σtδFt dWt,dσt=ασtγ dZt+b(σt) dt,dF_t=\sigma_t^\delta F_t\,dW_t,\qquad d\sigma_t=\alpha\sigma_t^\gamma\,dZ_t+b(\sigma_t)\,dt,

where (W,Z)(W,Z) is a two-dimensional Brownian motion with correlation ρ∈[−1,1]\rho\in[-1,1] (i.e. E[WtZt]=ρt\mathbb{E}[W_tZ_t]=\rho t for all t≥0t\ge0), σ0>0\sigma_0>0 and F0>0F_0>0. For m>1m>1 set

ρc(m)=−m−1m−b∞αm.\rho_c(m)=-\sqrt{\tfrac{m-1}{m}}-\tfrac{b_\infty}{\alpha m}.

Lions and Musiela proved that E[sup⁡0≤s≤tFsm]<∞\mathbb{E}[\sup_{0\le s\le t}F_s^m]<\infty for all t≥0t\ge 0 whenever ρ<ρc(m)\rho<\rho_c(m), and that E[Ftm]=+∞\mathbb{E}[F_t^m]=+\infty for all t>0t>0 whenever ρ>ρc(m)\rho>\rho_c(m); at ρ=ρc(m)\rho=\rho_c(m) exactly, their exponential-supersolution method is inconclusive (the governing quadratic in the supersolution parameter has a double root), and they state only sufficient conditions, noting that the answer depends on the behaviour of b(ξ)b(\xi) beyond the leading term b∞ξγ+δb_\infty\xi^{\gamma+\delta}. (The source states its sufficient conditions with b∞=lim sup⁡ξ→∞b(ξ)/ξγ+δb_\infty=\limsup_{\xi\to\infty}b(\xi)/\xi^{\gamma+\delta} and its necessary conditions with the corresponding lim inf⁡\liminf; the limit assumed above makes both readings coincide, so that ρc(m)\rho_c(m) is unambiguous.)

Problem. At the critical correlation ρ=ρc(m)\rho=\rho_c(m), characterize by necessary and sufficient conditions — expressed through the asymptotic behaviour of b(ξ)b(\xi) beyond its leading term b∞ξγ+δb_\infty\xi^{\gamma+\delta}, and, in the general-dispersion analogue (the source's Section 2.5, where γ=δ=1\gamma=\delta=1) dσt=μ(σt) dZt+b(σt) dtd\sigma_t=\mu(\sigma_t)\,dZ_t+b(\sigma_t)\,dt with μ(ξ)/ξ→μ∞>0\mu(\xi)/\xi\to\mu_\infty>0, through the subleading behaviour of μ\mu as well — whether E[sup⁡0≤s≤tFsm]<∞\mathbb{E}[\sup_{0\le s\le t}F_s^m]<\infty holds for all t≥0t\ge 0. Where finiteness fails, determine whether E[Ftm]=+∞\mathbb{E}[F_t^m]=+\infty for all t>0t>0 or only beyond a critical time, and characterize that time. The characterization should: (i) recover as consequences the sufficient conditions stated in the 2007 remark (in the case γ=δ\gamma=\delta, finiteness at criticality when b(ξ)−b∞ξ2γb(\xi)-b_\infty\xi^{2\gamma} is bounded above on [0,∞)[0,\infty); in the case γ≠δ\gamma\ne\delta the corresponding displayed condition involving (b(ξ)−b∞ξγ+δ)ξδ−γ+α22(δ−γ)ξγ+δ−1(b(\xi)-b_\infty\xi^{\gamma+\delta})\xi^{\delta-\gamma}+\frac{\alpha^2}{2}(\delta-\gamma)\xi^{\gamma+\delta-1}) and identify whether they are sharp; (ii) decide the concrete instances that the known results leave open, for example b(ξ)=b∞ξγ+δb(\xi)=b_\infty\xi^{\gamma+\delta} exactly with γ<δ\gamma<\delta (where even the sufficient condition fails), b(ξ)=b∞ξγ+δ+c ξpb(\xi)=b_\infty\xi^{\gamma+\delta}+c\,\xi^p with c>0c>0 and γ+δ−1<p<γ+δ\gamma+\delta-1<p<\gamma+\delta, and the quadratic-drift log-normal volatility model dσt=(R0+R1σt)(R2−σt) dt+νσt dZtd\sigma_t=(R_0+R_1\sigma_t)(R_2-\sigma_t)\,dt+\nu\sigma_t\,dZ_t at criticality with R0<R1R2R_0<R_1R_2, which Carr and Willems leave undecided; and (iii) remain consistent with the settled boundary cases: the driftless log-normal model (γ=δ=1\gamma=\delta=1, b≡0b\equiv 0: finite at criticality), the Heston-type case γ=δ=1/2\gamma=\delta=1/2 (finite at criticality for all maturities), and the affine models.

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