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Correlation that switches with the market

Two assets whose correlation jumps between a calm regime and a crisis regime: the correlation smile it puts into exchange and spread options, and the co-skewness it puts into returns when correlation rises in down markets.

Correlations are low in calm markets and high in crises. When the regimes switch fast compared with the horizon, the first-order rule turns this into two closed forms: a correlation volga in option prices, and a third cross-cumulant in the joint law of returns.

The model

Two log prices $x_1, x_2$ follow Black–Scholes dynamics under the pricing measure, with fixed volatilities and a correlation set by a hidden regime $y_t$:

The regime is calm or crisis. It moves from calm to crisis at rate $a$ and back at rate $b$, and starts from its stationary law because it cannot be observed.

There are three time scales. A crisis lasts $1/b$ on average, a calm spell lasts $1/a$, and the options run for time $T$. The expansion is in the ratio of the first two to the third, and the checks speed the chain up by a factor $m$.

Parameters: $\rho = (0.2,\ 0.8)$ in (calm, crisis); $a = 4/3$ and $b = 4$ per year, so crises last three months and take a quarter of the time; $\sigma_1 = 0.30$, $\sigma_2 = 0.20$, $S_1 = S_2 = 100$, $r = 2\%$, $T = 1$.

The quantity

Two payoffs that depend on the correlation. The exchange option and the spread option pay

Each price is quoted as an implied correlation: the constant $\rho$ that reproduces it in the two-asset Black–Scholes model. A model with one correlation gives the same implied correlation at every strike.

Averaging

The stationary law puts weight $b/(a + b) = 0.75$ on calm and $a/(a + b) = 0.25$ on crisis. When the chain switches infinitely fast the model is two-asset Black–Scholes with the averaged correlation

The averaged model has a flat implied correlation at $0.35$.

Reduction to a mixture over correlation

The pricing operator in regime $y$ is the averaged operator plus the correlation fluctuation times one operator:

$A$ has constant coefficients, so it commutes with $\bar{\mathcal L}$. Along any path of the chain the operators can be collected, and the log prices at $T$ are Gaussian with correlation equal to the time average

So the price is a mixture of constant-correlation prices, the correlation analogue of the mixing formula of Hull and White (1987):

For two regimes the law of $\hat\rho_T$ is explicit. With $v$ the time spent in crisis and $u = T - v$, a path that starts calm has density

on $0 < v < T$, plus an atom $e^{-aT}$ at $v = 0$. The two terms count paths that end calm and end in crisis.

A path that starts in crisis has the same form with $a \leftrightarrow b$ and $u \leftrightarrow v$. Quadrature over this law gives the numerical solution used in the checks.

The first-order rule

Differentiating a constant-correlation price in $\rho$ brings down the operator that $\rho$ multiplies, for the whole time $T$:

The rule adds the correction $K_{\rho\rho}A^2$ to the averaged operator. Because $A$ commutes with $\bar{\mathcal L}$, the Duhamel integral is $T$ times $K_{\rho\rho}A^2$ applied to the averaged price, and $A^2$ is $T^{-2}\partial_\rho^2$:

The correction is half the variance of $\hat\rho_T$, which is $2K_{\rho\rho}/T$ to first order, times the convexity of the price in correlation. This convexity is a correlation volga. It changes sign across strikes, so no single correlation absorbs it, and the implied correlation varies with the strike.

Here $K_{\rho\rho} = 0.01266$. Over one year $\hat\rho_T$ has a standard deviation of $0.14$, against $0.16$ from $2K_{\rho\rho}/T$.

The exchange option written out

Take $S_2$ as numeraire. The payoff becomes $S_2(T)\,(Z_T - q)^+$ with $Z = S_1/S_2$, and under the new measure $Z$ is a martingale with variance rate

The change of measure involves only $W_2$ and the fixed $\sigma_2$, so it leaves the chain alone. The exchange option is therefore exactly the one-dimensional problem of a switching variance, the smile of the general rule, with switched variance $s$ and Green–Kubo coefficient

The price is Margrabe's formula at the averaged exchange variance plus a volga term:

The interest rate drops out, as it does in Margrabe's formula.

The correlation smile in closed form

To first order the implied correlation moves from $\bar\rho$ by the correction divided by the correlation vega:

For the exchange option $\partial_\rho = -2\sigma_1\sigma_2 T\,\partial_V$. The ratio of volga to vega in total variance is $(d_1 d_2 - 1)/(2\bar V)$, and $d_1 d_2 = k^2/\bar V - \bar V/4$ at log-moneyness $k$. Together:

The implied correlation is a downward parabola in log-moneyness. It peaks at the money, above the average correlation, and falls below $\bar\rho$ once $k^2 > \bar V(1 + \bar V/4)$.

The shape is the volatility smile turned over. A higher correlation lowers the exchange variance, so the variance smile, which rises in the wings, becomes a correlation frown.

A worked example

With the parameters above, $\bar s = 0.088$, $\bar V = 0.088$, $K_{\rho\rho} = 0.012656$ and $K_{ss} = 1.8225\times10^{-4}$. The parabola peaks at $0.35 + 0.00882 = 0.3588$ and crosses $\bar\rho$ at $|k| = 0.300$, that is at $q = 0.74$ and $q = 1.35$.

Exchange option prices and implied correlations, one year, crises lasting three months:

The averaged model quotes $0.35$ everywhere. The numerical smile spans two correlation points between the money and the wings, and the parabola reproduces it to within $0.002$ with regimes that last a quarter of the maturity. The at-the-money price is $0.6\%$ below the averaged price.

The spread option

The spread option has no closed form at a fixed correlation, so $C_\rho$ comes from one-dimensional quadrature. Conditioning on the second asset leaves a Black–Scholes price in the first:

with the bracket replaced by $F(z) - S_2(z) - K$ where $S_2(z) + K \le 0$. The integral is evaluated with 120 Gauss–Hermite nodes, and $\partial_\rho^2 C$ by a five-point difference in $\rho$. At $K = 0$ it reproduces Margrabe's formula to $10^{-12}$.

The rule is the boxed formula of the previous section with this $C_\rho$. One year, crises lasting three months:

The correlation volga is negative near the money and turns positive far out of the money, so the implied correlation of the spread option also peaks near $K = 0$ and falls below $\bar\rho$ in the wing.

Checks for the option prices

The largest price error over the strikes in the two tables, as the chain is sped up by $m$ with the maturity fixed:

The averaged error halves with each doubling and the first-order error falls by four, as a second-order remainder should. The measured rates over the last doubling are 2.02 for the exchange option and 2.04 for the spread option.

The numerical solution is checked three ways at $m = 1$:

• Lewis's Fourier formula, with the characteristic function of $\log Z_T$ from the matrix exponential of $T(Q + \operatorname{diag} g)$, agrees with the quadrature over the occupation law to $2\times10^{-10}$ at $q = 0.7$, $1$ and $1.4$.
• Monte Carlo over one million exactly simulated chain paths, with the conditional Margrabe price, gives 31.3893, 11.7257 and 2.2485 at those strikes, each within one standard error of the numerical prices.
• A Monte Carlo of the terminal prices themselves, two million paths with no time stepping, gives $7.814 \pm 0.012$ for the spread option at $K = 10$, against $7.815$.

Correlation that rises in down markets

Under the physical measure the crisis regime also has lower drifts and higher volatilities. Let every entry of the drift and of the covariance switch:

Write the covariance entries as $c_{11} = \sigma_1^2$, $c_{22} = \sigma_2^2$ and $c_{12} = \rho\,\sigma_1\sigma_2$, each a function of the regime. Here $\mu_j$ is the drift of the log price.

Parameters, in (calm, crisis): $\mu_1 = (0.10,\ -0.20)$, $\mu_2 = (0.08,\ -0.25)$, $\sigma_1 = (0.15,\ 0.30)$, $\sigma_2 = (0.12,\ 0.25)$, $\rho = (0.2,\ 0.8)$; the same chain; horizon $t = 1$.

The quantity

The joint cumulants of the log returns $X = x_t - x_0$, read from the cumulant generating function

with the index written out, so that $\kappa_{21}$ is $\kappa_{112}$ and $\kappa_{11}$ is $\kappa_{12}$. The third cumulant $\kappa_{112} = \mathbb E\big[(X_1 - \mathbb E X_1)^2(X_2 - \mathbb E X_2)\big]$ is the co-skewness. It is negative when the second asset does badly at times when the first moves a lot.

Reduction

Given the regime path the returns are Gaussian, so

This is the engine system with a constant forcing. All operators are multiplications, so they commute and only the symmetric part of the Green–Kubo matrix enters.

The first-order rule

With a stationary start the rule gives the averaged exponent plus the Green–Kubo variance of the fluctuating forcing $\tilde g$:

Expanding $K(\tilde g, \tilde g)$ in $\theta$ gives three pieces:

The first is quadratic in $\theta$ and adds to the covariance. The second is cubic, and it is the only source of third cumulants at first order. The third is quartic and fattens the tails.

The coefficient of $\theta_1^2\theta_2$ in the cubic piece is $2K(\mu_1, c_{12}) + K(\mu_2, c_{11})$, so

For a portfolio with weights $w$ the same step gives the third cumulant

A regime whose mean is low when its covariance is high makes $K(\mu, c) < 0$ and a negative third cumulant. The term $2w_1w_2\,c_{12}$ inside $w^{\top}c\,w$ is the part carried by the correlation.

Exact finite-rate cumulants for two regimes

For two states the covariance, co-skewness and co-kurtosis need not be approximated. Let $A_T=\int_0^T\mathbf 1_{\{y_t=1\}}\,dt$ be the time spent in crisis, put $r=a+b$, and write $p_0=b/r$, $p_1=a/r$, $u=p_0p_1$ and $x=rT$. With a stationary start its first four cumulants are

For completeness, put $D(z)=\sqrt{1+2(p_1-p_0)z+z^2}$. Diagonalizing the dimensionless tilted generator gives the stationary moment generating function in closed form:

Four derivatives of its logarithm at zero give the displayed stationary cumulants, including the fourth; this also shows that the formula is finite-horizon exact rather than a large-$r$ expansion.

Indeed, for $Z_t=\mathbf 1_{\{y_t=1\}}-p_1$, stationarity and the two-state semigroup give $\mathbb E[Z_0Z_t]=p_0p_1e^{-rt}$. Since $Z_t^2=(p_0-p_1)Z_t+p_0p_1$, for $s\leq t\leq u$ they also give $\mathbb E[Z_sZ_tZ_u]=p_0p_1(p_0-p_1)e^{-r(u-s)}$. Integrating over the square and the six orderings of the cube proves the first three formulas.

The fourth follows by expanding the exact tilted two-state generator. The modified-Bessel occupation density used above is a special case of the finite-state construction of Kovchegov, Meredith and Nir (2010).

The stationary-start restriction can also be removed exactly. Let $q=\mathbb P(y_0=1)$, $\delta=q-p_1$ and $x=rT$. Define

Then the arbitrary-start occupation cumulants are

For the proof, let $M_i(z)=\mathbb E_i e^{zA_T}$ and $M_\pi=(1-p_1)M_0+p_1M_1$. Linearity in the initial law gives $M_q=M_\pi\{1+\delta R(z)\}$ with $R=(M_1-M_0)/M_\pi$ and $R(0)=0$. Direct integration of the two-state transition probabilities gives $R^{(j)}(0)=H_j/r^j$ for $j=1,\ldots,4$.

Differentiating $\log M_q=\log M_\pi+\log(1+\delta R)$ gives the displayed corrections. This conditional-cumulant organization is the finite-mixture specialization of Brillinger (1969) and exposes the corrections as a finite-horizon initial layer rather than part of the stationary Green–Kubo coefficient.

Now use differences from calm to crisis, $d\mu=\mu_{\mathrm{crisis}}-\mu_{\mathrm{calm}}$ and $dc=c_{\mathrm{crisis}}-c_{\mathrm{calm}}$. Conditional on $A_T$,

so its cumulant generating function is the calm Gaussian exponent plus $K_{A_T}(\theta\cdot d\mu+\tfrac12\theta^\top dc\,\theta)$. Differentiating gives the exact finite-rate identities

The result is exact for every $T,a,b>0$ and every initial probability $q\in[0,1]$ under a two-state chain independent of the Brownian shocks. For a stationary start, the Green–Kubo rule is recovered by replacing $\kappa_2(A_T)$ with $2p_0p_1T/r$ and omitting $\kappa_3(A_T)$, which is one order smaller in the fast rate. For a symmetric chain $p_0=p_1$, the occupation-skew term vanishes exactly.

All orders: singleton–pair partitions

The preceding tensors are the first cases of one exact identity. Let $\mathcal M_n$ be the set of partitions of the labelled positions $\{1,\ldots,n\}$ into singleton and pair blocks. For $n\geq3$,

For $n=1$ add $T\mu_{i_1,\mathrm{calm}}$, and for $n=2$ add $Tc_{i_1i_2,\mathrm{calm}}$. This is the singleton–pair specialization of the multivariate Faà di Bruno formula applied to $K_{A_T}(\theta\cdot d\mu+\tfrac12\theta^\top dc\,\theta)$; it is also the conditional-Gaussian form of Brillinger's conditional-cumulant formula. The statement is exact at finite rate and horizon for every two-state initial law.

There is an exact finite-state version, but it is genuinely vector-valued. Choose state zero as reference, write $A_T^a$ for time in state $a=1,\ldots,m-1$, and let $d\mu^a=\mu^a-\mu^0$ and $dc^a=c^a-c^0$.

Color every block $B$ of $\mathcal P$ by a state contrast $z(B)$. Then, for $n\geq3$,

The order-one and order-two baseline additions are unchanged. This formula follows by applying the same multivariate chain rule to the joint occupation CGF.

It is invariant to the chosen reference state, although its contrast coordinates are not. Unlike the two-state case, its inputs include mixed occupation cumulants; replacing them by separate marginal cumulants is generally wrong.

Finite-state fast-rate theorem

The rate law is not special to two states. Let $Q$ be any fixed irreducible finite-state generator, let the accelerated chain have generator $sQ$, and fix $T>0$ and an arbitrary initial law $\nu$. For bounded state observables $f_a$, put $A_T^a=\int_0^T f_a(Y_t)\,dt$.

Define $\lambda(z)$ to be the simple eigenvalue continuing zero from $Q+\operatorname{diag}(\sum_a z_af_a)$, $P(z)$ its analytic spectral projection, and $b_\nu(z)=\log[\nu P(z){\bf1}]$. Then every mixed occupation cumulant of order $k\geq2$ has the expansion

The bulk coefficient is independent of the initial law; the first initial-law contribution is one inverse-rate order smaller. To prove the display, write the occupation CGF as

Analytic perturbation of the isolated simple eigenvalue zero and the spectral gap give, uniformly on a fixed complex polydisc in $z$ for all sufficiently large $s$,

Cauchy estimates permit termwise mixed differentiation and give the theorem. This is a finite-dimensional use of Kato's analytic eigenvalue and projection perturbation theory; the broader singular-perturbation and occupation-measure setting is treated by Yin and Zhang.

The statement is a fixed-horizon fast-rate result. It is also uniform on compact maturity intervals bounded away from zero, but it does not cover a joint limit with $sT=O(1)$.

The first nontrivial eigenvalue coefficient is explicit. For centered state indicators $f_a$, the group inverse $Q^{\#}$ gives the symmetric Green–Kubo rate matrix

Therefore the leading finite-state co-skewness and co-kurtosis tensors are

These are fixed-$Q$, fixed-maturity contractions of the Green–Kubo matrix. They do not assert the same rank after combining different maturity-dependent loadings.

Power-law maturity phase diagram

The gap between fixed maturity and the critical layer has a complete scaling description. Put $T_s=\tau s^{-\beta}$, $L_s=sT_s$, and center and normalize the return as

After the time change $u=st$, conditional means carry the factor $\sqrt{T_s}/L_s$ and conditional covariances the factor $1/L_s$. A singleton–pair partition with $q$ blocks and $r$ singleton blocks therefore contributes

If $0\le\beta<1$, then $L_s\to\infty$ and every order-$q\ge2$ occupation cumulant is $O(L_s)$. The partitions with the fewest blocks give, for $p\ge1$,

Thus $Y_s$ converges to the Gaussian law with stationary averaged covariance. These are upper orders: tensor contractions or symmetry may cancel the leading coefficient.

At $\beta=1$, $L_s=\tau$ and the exact finite-switch covariance mixture below remains. If $\beta>1$, $L_s\to0$: with probability tending to one the chain never leaves its initial regime, and

Hence a mixed initial law generally leaves persistent even cumulants at both sides of the critical point for entirely different reasons: finite occupation randomness at $\beta=1$, and a frozen initial-regime mixture at $\beta>1$. A deterministic initial regime makes the latter limit Gaussian.

The certificate checks the interpolation at $\beta=1/2$. The measured speed exponents for one-asset cumulants of orders three through six are $0.744523$, $0.495374$, $1.239972$, and $0.990538$, against $0.75$, $0.50$, $1.25$, and $1.00$. At $\beta=3/2$ and $s=16384$, the order-two, order-four, and order-six cumulants differ from their frozen-mixture limits by $6.0\times10^{-5}$, $1.6\times10^{-5}$, and $1.4\times10^{-6}$; the odd cumulants are already below $3.9\times10^{-6}$ and $3.2\times10^{-7}$.

The critical boundary layer $sT=\tau$

The excluded near end has a different, exact scaling law. Put $T_s=\tau/s$ and $Z_s=\sqrt{s}\,X_{T_s}$.

Under the time change $u=st$, the regime process has generator $Q$ on $[0,\tau]$, while Brownian scaling removes $s$ from the conditional covariance. If $A_\tau^a$ are the non-reference occupation times of that unit-speed chain, then exactly

Consequently, if $r(\mathcal P)$ is the number of singleton blocks in a singleton–pair partition, the all-order tensor has the exact finite-$s$ identity

For even $n$, perfect pairings have no singleton, so covariance switching can leave a nonzero limiting cumulant of every even order. For odd $n$, every partition has a singleton, giving $\kappa_n(Z_s)=O(s^{-1/2})$; the leading term uses one drift contrast and $(n-1)/2$ covariance contrasts. Thus

which is generally a non-Gaussian covariance mixture. It becomes Gaussian when the integrated covariance is deterministic in every observed direction; in particular, pure mean switching disappears at this scale.

This proves that the fixed-maturity fast-rate hierarchy above cannot be uniform down to $T=O(s^{-1})$. The process is the finite-state Markov-modulated Brownian motion studied by Breuer; the occupation vector, rather than a new limiting object, carries the entire crossover.

For the nonreversible three-state certificate with $\tau=0.75$, the limiting asset-one variance is $0.03133818$, the fourth and sixth cumulants are $3.20891983\times10^{-4}$ and $1.30228960\times10^{-5}$, and the limiting excess kurtosis is $0.326747$. The exact coefficient polynomial agrees with independent tilted-matrix extraction through order six within $6.8\times10^{-12}$.

Under each quadrupling of $s$, the residuals after the odd $s^{-1/2}$ limits and the persistent even limits fall by factors $4.000$, $4.010$, $3.999$, and $4.038$ at orders three through six. A separate calculation at $s=16$ compares the original horizon $\tau/s$ directly with the rescaled unit-speed system through order four and agrees within $2.8\times10^{-10}$.

Arbitrary-start covariance layer

The abstract boundary coefficient above has a simple first-order form. For any state field $f$, write $\widetilde f=f-\pi f$. The semigroup identity is exact:

Thus the initial-layer coefficient is $-\nu Q^{\#}\widetilde f$. In the row-generator convention used here, $-Q^{\#}$ is the continuous-time deviation matrix $\int_0^\infty(e^{tQ}-P)\,dt$ studied by Coolen-Schrijner and van Doorn. Applying the identity to the drift and covariance fields gives

Here $\Gamma(f,g)=-\pi[fQ^{\#}g]-\pi[gQ^{\#}f]$. The covariance-field transient is therefore at the same inverse-rate order as the stationary Green–Kubo drift term; it vanishes for $\nu=\pi$, but not for a known starting regime. This is a fixed-$T$ expansion, uniform for maturities bounded away from zero, and does not cover $sT=O(1)$.

The next boundary coefficient is explicit as well. Put $R=Q^{\#}$ and let products of state fields denote pointwise multiplication. For centered $f,g$, define

For the proof, normalize the left eigenvector by $\ell(z){\bf1}=1$ and the right one by $\ell(z)r(z)=1$, so that $P(z){\bf1}=r(z)$. The differentiated eigenvector equations give

Since $b_\nu(z)=\log[\nu r(z)]$, its Hessian is exactly $\mathcal B_\nu$. Hence the integrated-field covariance has the sharper expansion

Even when $\nu=\pi$, the constant boundary term need not vanish: it records the two finite-horizon endpoints, not merely failure to start in equilibrium. This refines the classical linear variance-growth description for continuous-time Markov reward chains studied by van Dijk and Sladký.

The next spectral coefficient is also explicit. Write $(u_1,u_2,u_3)=(a,b,c)$. Differentiating the normalized right eigenvector of the tilted generator twice gives the symmetric third occupation-rate tensor

Products here mean multiplication by the indicated state function. Centering makes every first derivative of $\lambda$ vanish, so no lower-order correction terms appear. More explicitly, if $r(z)$ is the right eigenvector normalized by $\pi r(z)=1$, the first perturbation equation gives $\partial_a r(0)=-Q^{\#}f_a$; projecting the first and second differentiated eigenvector equations onto $\pi$ yields the displayed six-permutation sum.

Together, $\mathcal B_\nu$ and $\Theta$ give the complete second-order co-skewness and co-kurtosis correction. Let $\mathcal M_n^{(q)}$ denote singleton–pair partitions with $q$ blocks, and put

Then, for $n=3,4$,

For order three these are the singleton–pair and all-singleton partitions. For order four they are the two-pair and pair–singleton–singleton partitions. The initial law enters only through $\mathcal B_\nu$ at this order.

Let $\mathcal M_n^{(3)}$ be the singleton–pair partitions of $\{1,\ldots,n\}$ having exactly three blocks. The leading fifth- and sixth-order return tensors are therefore, for $n=5,6$,

For order five the 15 partitions have one singleton and two pairs; for order six they are the 15 perfect pairings. Thus leading fifth-order non-Gaussianity requires both mean and covariance switching, whereas leading sixth order can be generated by covariance switching alone. This is again a fixed-generator, fixed-maturity contraction, not a rank statement after mixing maturity-dependent loadings.

Choosing the $m-1$ non-reference state indicators as the $f_a$ proves that every mixed finite-state occupation cumulant is $O(s^{1-k})$. Every singleton–pair partition has at least $\lceil n/2\rceil$ blocks, so the exact colored formula above yields

The bound is generically attained, although symmetry or parameter contractions can cancel its leading coefficient. Thus orders three and four are $O(s^{-1})$, orders five and six are $O(s^{-2})$, and so on.

If only covariance switches ($d\mu=0$), every odd cumulant of order at least three vanishes exactly and the even formula reduces to pairings. If only the mean switches ($dc=0$), only the all-singleton partition remains, giving $\kappa_n(A_T)\prod_a d\mu_{i_a}=O(s^{1-n})$.

For example, the exact mixed fourth cumulant is

At fast switching, its leading term is $(2p_0p_1T/r)(dc_{11}dc_{22}+2dc_{12}^2)$; the occupation-skew and occupation-kurtosis pieces begin at orders $r^{-2}$ and $r^{-3}$. Thus covariance switching alone creates an order-$r^{-1}$ excess co-kurtosis even when the regime drifts coincide.

For a nonstationary start, the leading covariance correction has an additional initial-layer term:

whereas the leading third-cumulant formula is unchanged, with the initial-law dependence entering at $O(r^{-2})$. Thus one must not apply the stationary covariance rule to a known starting regime.

Written out for two regimes

For the two-state chain $K(f, h) = ab\,\Delta f\,\Delta h/(a + b)^3$, with $\Delta$ the calm value minus the crisis value. Every cumulant through third order is then:

The variances $\kappa_{11}$ and $\kappa_{22}$ follow the pattern of $\kappa_{12}$. With a bear regime, $\Delta\mu_j > 0$ and $\Delta c < 0$, and all third cumulants are negative.

A worked example

With the parameters above, $G = 0.035156$, $\Delta\mu = (0.30,\ 0.33)$, $\Delta c_{11} = -0.0675$, $\Delta c_{22} = -0.0481$ and $\Delta c_{12} = 0.0036 - 0.0600 = -0.0564$. So

and $\kappa_{122} \approx -3.632\times10^{-3}$. Of $\kappa_{112}$, the correlation switch through $\Delta c_{12}$ supplies $60\%$ and the volatility of the second asset the rest.

For the equal-weight portfolio, $\kappa_3 \approx -3.79\times10^{-3}$, half of it from the correlation switch. Its skewness over one year is $-0.64$ in the numerical solution. The averaged model is Gaussian, with no skewness at all.

At the unscaled rates the exact formula instead gives $\kappa_{112}=-3.58034\times10^{-3}$: the mean–covariance term contributes $-3.20936\times10^{-3}$ and the skewed occupation time contributes $-0.37098\times10^{-3}$. Thus the finite-rate discrepancy is not numerical noise; it is the explicit boundary and occupation-skew correction omitted by the first-order rule.

The initial regime is visible at this horizon. Starting certainly calm gives $\kappa_{112}=-2.92476\times10^{-3}$; starting certainly in crisis gives $-3.80148\times10^{-3}$. Both values converge to the same stationary fast-rate law as the initial layer decays.

The same simulated returns show the asymmetry that the empirical literature measures.

The correlation of the two returns, given that both lie below their means, is $0.563$. Given that both lie above, it is $0.254$. A Gaussian with the same overall correlation, $0.612$, gives $0.367$ on both sides.

Checks for the cumulants

The closed finite-rate tensor agrees with the independent numerical solution to at most $8.6\times10^{-12}$ over stationary and nonstationary starts and rate multipliers through $m=16$. Occupation-density quadrature independently reproduces all four arbitrary-start cumulants to $1.2\times10^{-16}$, and relabelling the regimes gives a residual below $2\times10^{-19}$. The numerical solution reads the cumulants off the exact generating function by Cauchy's formula: a two-dimensional FFT of $\Lambda$ on polydiscs of radii $0.05$ through third order and $0.15$ at fourth order, with the matrix exponential evaluated at each node.

At one point it agrees with the engine's high-precision ODE solver to $10^{-14}$. Third cumulants in units of $10^{-3}$:

The relative error of the rule falls from $10\%$ with three-month crises to $1.3\%$ at $m = 4$ and $0.3\%$ at $m = 16$. Over the last doubling the absolute errors fall at rate 2.06 for the third cumulants and 2.00 for the second, as a second-order remainder should.

The fourth-order tensor has the same independent check. At $m=1$, the exact $\kappa_{1122}=9.78461\times10^{-4}$, while its leading variance-switching term is $6.75610\times10^{-4}$.

By $m=16$ these are $4.43009\times10^{-5}$ and $4.22256\times10^{-5}$. Across all five tensor entries and rate multipliers the closed formula agrees with the matrix-exponential extraction within $8\times10^{-12}$; the error of the leading term falls at rate $1.96$ over the last doubling.

An additional certificate checks the all-order partition identity against an independent matrix-exponential extraction through order six. It covers all 100 symmetric bivariate tensor entries of orders two through six under stationary, calm, mixed, and crisis initial laws (including 52 fifth- and sixth-order entries); the maximum absolute discrepancy is $2.4\times10^{-12}$. For the displayed parameters, two stationary higher-order examples are $\kappa_{11112}=-2.29443\times10^{-4}$ and $\kappa_{111222}=2.10610\times10^{-5}$.

A separate nonreversible three-state certificate computes the mixed cumulants of two reference-state occupation coordinates by a two-dimensional Cauchy transform of the tilted generator. Across stationary, deterministic, and mixed initial laws, all 54 symmetric bivariate return-tensor entries of orders two through five agree with a direct return-CGF extraction within $2.8\times10^{-13}$; recomputing all stationary entries with each of the other two reference states leaves at most a $2.4\times10^{-13}$ residual.

Its stationary example gives $\kappa_{11122}=-6.09744\times10^{-6}$. This verifies the vector formula, not a closed-form law for general finite-state occupation times.

A focused three-state certificate checks the fast-rate theorem independently. For mixed occupation orders two through six, the observed decay exponents at the last rate doubling are $0.983$, $1.968$, $2.961$, $3.939$, and $4.937$; after subtracting $Ts^{1-k}\partial^k\lambda(0)$, the residual exponents are $2.000$, $3.000$, $4.000$, $4.998$, and $5.886$. The corresponding maximum return-tensor magnitudes have exponents $0.987$, $1.005$, $1.954$, and $1.902$ at orders three through six, matching $s^{1-\lceil n/2\rceil}$.

The principal-eigenvalue derivatives are obtained by an independent two-dimensional Cauchy transform, and their first derivatives recover the stationary probabilities to numerical precision. The group-inverse $\Gamma$ agrees with the second eigenvalue derivatives within $2.0\times10^{-15}$; the errors of the explicit third- and fourth-order Green–Kubo tensors decay with exponents $2.014$ and $1.978$, respectively. The group-inverse $\Theta$ agrees with the independently extracted third eigenvalue derivatives within $2.8\times10^{-15}$; the errors of the explicit fifth- and sixth-order leading tensors decay with exponents $2.981$ and $2.989$, matching the $O(s^{-3})$ remainder.

For initial laws $(1,0,0)$ and $(0.2,0.3,0.5)$, the exact occupation means also agree with the displayed semigroup identity within $1.3\times10^{-15}$, and the covariance errors after subtracting the arbitrary-start first correction decay with exponent $2.000$. For the state-zero start, the asset-two covariance-field boundary coefficient is $-9.18107\times10^{-3}$, which overwhelms the stationary Green–Kubo coefficient $+6.40041\times10^{-3}$ and reverses the total first correction to $-2.78066\times10^{-3}$.

The same certificate checks the explicit $\mathcal B_\nu$ matrix. At rate multiplier four, the residual after extracting it from the exact occupation covariance is $3.4\times10^{-10}$ for the deterministic start and $6.5\times10^{-11}$ for the mixed start.

After subtracting both the $s^{-1}$ and complete $s^{-2}$ terms, the co-skewness errors decay with exponent $3.000$ for both starts; the co-kurtosis exponents are $2.981$ and $2.996$. These are direct return-CGF checks of the full tensor correction, not only of the occupation inputs.

Monte Carlo with no discretization: given the time spent in crisis the returns are exactly Gaussian with the integrated mean and covariance, so only the chain path and one Gaussian pair are drawn, four million times.

The third cumulant of the equal-weight portfolio, rule against numerical, in units of $10^{-3}$: $-3.79$ against $-3.48$ at $m = 1$, $-0.949$ against $-0.939$ at $m = 4$, and $-0.2371$ against $-0.2367$ at $m = 16$.

The same exact tensor gives excess kurtosis $1.030$ for the equal-weight one-year portfolio at $m=1$, falling to $0.278$ at $m=4$ and $0.068$ at $m=16$. This is excess over the Gaussian value, not raw kurtosis.

Prior art

That correlations rise in falling markets is well documented.

Longin and Solnik (2001) find, with extreme-value methods, that international equity correlation rises in bear markets but not in bull markets, and that it is tied to the trend rather than to volatility. Ang and Chen (2002) find much larger downside than upside exceedance correlations for US stocks, and that regime-switching models capture the asymmetry better than the alternatives they try. Hong, Tu and Zhou (2007) give model-free tests for asymmetric correlations and betas.

Ang and Chen also report that their exceedance correlations differ from co-skewness as a measure of asymmetry. In the fast-switching model the co-skewness above is the first-order fingerprint in the cumulants, and the simulated exceedance correlations show the asymmetry directly. Ang and Bekaert (2002) estimate a regime with lower means, higher volatility and higher correlation, the bear regime used here.

Correlation risk is priced and hedged. Driessen, Maenhout and Vilkov (2009) find a price of correlation risk in index options against options on the components. Buraschi, Porchia and Trojani (2010) solve portfolio choice with a stochastic covariance matrix and find a separate hedging demand against correlation.

Stochastic correlation in option pricing is usually modelled with the Wishart process of Bru (1991), as in Da Fonseca, Grasselli and Tebaldi (2007) and Gourieroux and Sufana (2010). Deelstra and Simon (2017) price multi-asset options when a chain modulates the parameters, at ordinary speed. Spread options are surveyed by Carmona and Durrleman (2003).

No fast-switching or fast-factor asymptotics for correlation, in option pricing or in the joint law of returns, appear in the literature reviewed. The nearest is Fouque, Pun and Wong (2016), who treat an ambiguous, worst-case correlation between two assets whose volatilities are fast mean-reverting.

The smile for a switching volatility is known (Sircar and Papanicolaou 1999). The exchange option inherits it exactly, and the parabola above is its correlation form.

Scope

• With fixed volatilities the correlation operator commutes with the rest, and the price is an exact mixture over the time-averaged correlation. When the volatilities also switch, the operators still have constant coefficients, and the correction is $T\sum K_{jk}\,\partial^2 C/\partial\Sigma_j\partial\Sigma_k$ over the entries of the integrated covariance matrix.
• Under the pricing measure the drift is $r$ in every regime, so the co-skewness of the second part matters for allocation, risk and density forecasts, not for prices, unless the regime carries a risk premium.
• The exact finite-rate cumulant tensor allows any specified initial probability for a two-state chain independent of the Brownian shocks. The displayed Green–Kubo formulas remain fast-rate approximations outside that exact two-state calculation.
• A chain independent of the price shocks gives an exchange-option smile that is even in log-moneyness. A fast diffusive correlation factor whose shocks are correlated with the prices would add a term of order $\sqrt\varepsilon$, odd in moneyness, that skews the implied correlation, in the manner of the Fouque, Papanicolaou and Sircar volatility skew. That direction is not worked out here.

Certificate: verify_correlation.py.