Cumulants of integrated variance under a fast regime

The fourth cumulant of a return whose CIR variance has a fast switching level, and what its Green–Kubo form leaves out at every maturity.

The model

A stationary finite-state chain $Y_t$ switches the long-run level of a square-root (CIR) variance:

The Brownian motion $B$ is independent of $Y$ and of the price Brownian motion. We keep $\kappa$ and $\xi$ fixed and assume $2\kappa\min_i\theta_i\ge\xi^2$, so the variance stays strictly positive.

The integrated variance is $V_T=\int_0^T v_s\,ds$. With $Z$ a standard normal independent of it, the Brownian return and the risk-neutral log return are

Leverage, a correlation between the price and variance Brownian motions, is taken up below.

The quantity

The cumulants of $V_T$, $M_T$ and $X_T$. Conditional normality gives the cumulant-generating functions

and comparing coefficients proves the exact identities

The last two formulas are not interchangeable. The expression $3\operatorname{Var}(V_T)$ is exact for the martingale return $M_T$, but the drift $-V_T/2$ of the log return is random too, and its third and fourth cumulants enter.

The exact variance of integrated variance

Write

Integrating the CIR equation and then integrating over time splits $V_T$ into a regime integral and a martingale, and the two are uncorrelated. Itô isometry gives the exact decomposition

The first line is the integrated-variance variance of the averaged CIR model, $\operatorname{Var}_{\rm av}(V_T)$. The second is exactly the extra variance created by the hidden regimes. This separation does not require an asymptotic expansion.

The Green–Kubo correction and a uniform bound

Put $I_2(T)=\int_0^T B_\kappa(s)^2\,ds$. Replacing the narrow correlation kernel by its mass $K_m$ gives

and the remainder can be bounded without fixing the maturity:

If the generator is $mQ_0$ and $|C_m(u)|\le A e^{-m\rho u}$, the right side is at most $4A/(m^2\rho^2)$. Consequently

This is a finite-rate statement, uniform over all maturities.

A two-state check

The certificate uses a symmetric chain with flip rate $m$, $\theta=(0.08,0.02)$, $\kappa=2$, $\xi=0.2$, $v_0=0.04$, and $T=1.5$. Here $C_m(u)=0.03^2e^{-2mu}$ and $K_m=0.03^2/(2m)$. An independent matrix exponential for every mixed moment of total degree at most four agrees with the covariance decomposition to better than $2\times10^{-13}$.

The scaled residual settles, confirming the $m^{-2}$ remainder.

Leverage

Now allow the price and variance Brownian motions to have correlation $\rho$. Conditional on the variance path, $M_T$ is no longer an independent centered normal with variance $V_T$. Write $\kappa_{r,s}$ for the joint cumulant with $r$ copies of $M_T$ and $s$ copies of $V_T$; multilinearity gives

For the certificate parameters at switching speed $m=8$, the independent shortcut is exact at $\rho=0$ but materially wrong under leverage:

Positive and negative leverage do not give the same log-return kurtosis. At $\rho=-0.7$, the independence formula misses about 60% of the exact fourth cumulant.

Further results in the full note

Option prices. For a symmetric two-state chain with $\delta=(\theta_+-\theta_-)/2$ and any fixed correlation $\rho$, the Heston Riccati coefficient $B_s$ does not depend on the regime. At fixed maturity, uniformly for $s$ in any compact subset of the MGF strip,

Carr–Madan inversion turns this into a call correction $C_m(k)=C_{\rm av}(k)+c_1(k)/m+O(m^{-2})$. On compact strike sets where Black–Scholes Vega $\mathcal V_{\rm av}$ has a positive lower bound, the implied volatility is corrected by $b_1(k)/m$ with $b_1(k)=c_1(k)/\mathcal V_{\rm av}(k)$.

Positivity. Let the drift $c_i=\kappa_i\theta_i$, the mean reversion $\kappa_i$ and the vol-of-vol $\xi_i$ all switch. Zero is inaccessible from every positive initial value and every initial regime if

and for an irreducible finite chain this condition is also necessary. The Feller test cannot be applied only to the averaged model: a two-state example passes on average with $2\bar c/\overline{\xi^2}=1.6$, yet from $v_0=0.04$ in the deficient regime the exact probability of hitting zero before the next switch is $0.2341927683$.

Simultaneous switching. When $\kappa$ switches too, the averaged drift is $\bar c-\bar\kappa v$, whose effective mean level is $\bar c/\bar\kappa$, generally not $\bar\theta$. The first-order correction to the moments of $V_T$ through order four, and so to its cumulants, then holds at fixed maturity with an $O_T(m^{-2})$ remainder, and the dependence on $T$ is necessary. In a two-state example the long-run mean is

and $\mathbb E[V_T]=\mu_mT+O_m(1)$ as $T\to\infty$, so the residual left after the averaged and first-order slopes grows linearly in maturity for every fixed $m$. Retaining the exact slope $\mu_m$ and one slow exponential mode restores an $O(m^{-2})$ approximation to $\mathbb E[V_T]$ uniform over $T\ge0$.

Invariant moments. With $a_i=\xi_i^2$, the row vectors $r_i^{(p)}=\mathbb E_{\Pi_m}[v^p\mathbf1_{\{Y=i\}}]$ of the invariant law satisfy an exact triangular recursion:

All positive integer invariant moments are finite. This needs mean reversion, not the statewise Feller inequality.

Long maturities. Started in the joint invariant law, the variance of $V_T$ has an exact slope and intercept, each from finite linear solves, with an exponentially decaying remainder:

For every fixed integer $k\ge2$ and every fixed switching rate $m>0$ the limit $\lim_{T\to\infty}\kappa_k(V_T)/T$ exists and is given by a triangular recursion of Poisson equations.

The full note also covers explicit Green–Kubo forms of the slow mode, truncated slopes on growing maturity windows, uniform Gamma and cumulant-disk bounds for the invariant law, inverse-speed expansions of the invariant moments and long-run rates, arbitrary-start boundary constants, the transfer to independent and leveraged return cumulants, and standardized cumulants.

Scope

The chain starts in stationarity; a known or filtered starting regime adds an initial-memory term. The exact decomposition, the maturity-uniform bound and the call correction assume fixed $\kappa$ and switch only the CIR mean level. The maturity-uniform bound is proved for the independent-price benchmark; no maturity-uniform leverage bound is claimed.

The conditional integrated-variance viewpoint goes back to Hull and White (1987), and the finite-dimensional moment construction follows the polynomial-process framework of Cuchiero, Keller-Ressel and Teichmann (2012).

Proofs, further tables and references: the full note. Certificates: verify_cumulants.py, verify_switching_kappa.py, verify_switching_feller.py.