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 $y_t$:

$$dx_j = \big(r - \tfrac12\sigma_j^2\big)\,dt + \sigma_j\,dW_j, \qquad d\langle W_1, W_2\rangle_t = \rho_{y_t}\,dt .$$

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. 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

$$\big(S_1(T) - q\,S_2(T)\big)^+ \qquad\text{and}\qquad \big(S_1(T) - S_2(T) - K\big)^+ .$$

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

$$\bar\rho = \frac{b\,\rho_{\mathrm{calm}} + a\,\rho_{\mathrm{crisis}}}{a + b} = 0.35 .$$

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:

$$\mathcal L_y = \bar{\mathcal L} + (\rho_y - \bar\rho)\,A, \qquad A = \sigma_1\sigma_2\,\partial_{x_1}\partial_{x_2} .$$

$A$ has constant coefficients, so it commutes with $\bar{\mathcal L}$. Along any path of the chain the log prices at $T$ are Gaussian with correlation equal to the time average $\hat\rho_T = \frac1T\int_0^T \rho_{y_s}\,ds$. So the price is a mixture of constant-correlation prices, the correlation analogue of the mixing formula of Hull and White (1987):

$$C = \mathbb E\big[\,C_{\rho}\big|_{\rho = \hat\rho_T}\big].$$

For two regimes the law of $\hat\rho_T$ is explicit, and quadrature over it 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, so $\partial_\rho^2 C_\rho = T^2 A^2 C_\rho$. The rule adds the correction $K_{\rho\rho}A^2$ to the averaged operator, and because $A$ commutes with $\bar{\mathcal L}$ the result is

$$C \;\approx\; C_{\bar\rho} + \frac{K_{\rho\rho}}{T}\,\frac{\partial^2 C_\rho}{\partial\rho^2}\Big|_{\rho = \bar\rho},$$ $$\begin{aligned} K_{\rho\rho} &= \int_0^\infty \operatorname{Cov}_\pi\big(\rho_{y_0},\,\rho_{y_\tau}\big)\,d\tau = \frac{ab\,(\rho_{\mathrm{calm}} - \rho_{\mathrm{crisis}})^2}{(a + b)^3} . \end{aligned}$$

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

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 $s_y = \sigma_1^2 + \sigma_2^2 - 2\rho_y\,\sigma_1\sigma_2$. The change of measure leaves the chain alone, so the exchange option is exactly the one-dimensional problem of a switching variance, with Green–Kubo coefficient $K_{ss} = 4\,\sigma_1^2\sigma_2^2\,K_{\rho\rho}$.

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

$$C \;\approx\; S_1\,N(d_1) - q\,S_2\,N(d_2) \;+\; T\,K_{ss}\,\frac{S_1\,\varphi(d_1)\,(d_1 d_2 - 1)}{4\,\bar V^{3/2}},$$ $$\begin{aligned} d_1 &= \frac{\log\big(S_1/(qS_2)\big) + \bar V/2}{\sqrt{\bar V}}, \qquad d_2 = d_1 - \sqrt{\bar V}, \\ \bar V &= \bar s\,T, \qquad \bar s = \sigma_1^2 + \sigma_2^2 - 2\bar\rho\,\sigma_1\sigma_2 . \end{aligned}$$

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, at log-moneyness $k = \log\big(S_1/(q\,S_2)\big)$, this is

$$\rho_{\mathrm{imp}}(k) \;\approx\; \bar\rho + \frac{K_{\rho\rho}\,\sigma_1\sigma_2}{\bar V}\Big(1 + \frac{\bar V}{4} - \frac{k^2}{\bar V}\Big).$$

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:

$q$	$k$	price, numerical	price, first order	price, averaged	implied correlation, numerical	parabola
0.60	+0.511	40.4160	40.4185	40.3949	0.3351	0.3332
0.80	+0.223	23.4289	23.4238	23.4450	0.3530	0.3539
1.00	0.000	11.7258	11.7209	11.7913	0.3582	0.3588
1.25	−0.223	4.2861	4.2797	4.3062	0.3530	0.3539
1.60	−0.470	0.9319	0.9344	0.8975	0.3381	0.3372

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, and the rule is the boxed first-order formula with this $C_\rho$. At $K = 0$ the quadrature reproduces Margrabe's formula to $10^{-12}$.

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. At $K = 0$ the numerical implied correlation is $0.3582$, and at $K = 40$ it is $0.3482$.

Checks for the option prices

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

$m$	$K_{\rho\rho}$	exchange, averaged	exchange, first order	parabola, implied correlation	spread, averaged	spread, first order
1	0.012656	6.5e-02	6.4e-03	1.9e-03	6.5e-02	6.8e-03
2	0.006328	3.4e-02	1.6e-03	3.7e-04	3.4e-02	1.6e-03
4	0.003164	1.7e-02	3.9e-04	7.7e-05	1.7e-02	3.9e-04
8	0.001582	8.8e-03	9.7e-05	1.7e-05	8.8e-03	9.5e-05

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.

At $m = 1$, Lewis's Fourier formula 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, is within one standard error of the numerical prices at those strikes.

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:

$$dx_j = \mu_{j,y_t}\,dt + \sigma_{j,y_t}\,dW_j, \qquad d\langle W_1, W_2\rangle_t = \rho_{y_t}\,dt .$$

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 is the set of joint cumulants of the log returns $X = x_t - x_0$. 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.

The first-order rule

Given the regime path the returns are Gaussian, so

$$\mathbb E\,e^{\theta\cdot X} = \pi\cdot\exp\big(t\,(Q + \operatorname{diag} g(\theta))\big)\,\mathbf 1, \qquad g_y(\theta) = \theta\cdot\mu_y + \tfrac12\,\theta^{\top}c_y\,\theta .$$

This is the engine system with a constant forcing. With a stationary start the rule gives the averaged exponent plus the Green–Kubo variance of the fluctuating forcing $\tilde g$:

$$\Lambda(\theta) = \log\mathbb E\,e^{\theta_1 X_1 + \theta_2 X_2} \approx t\,\bar g(\theta) + t\,K(\tilde g, \tilde g), \qquad K(f, h) = -\pi\cdot\big(f\,Q^{\#}h\big) .$$

The part of $K(\tilde g, \tilde g)$ that is cubic in $\theta$ is $K\big(\theta\cdot\mu,\ \theta^{\top}c\,\theta\big)$, and it is the only source of third cumulants at first order. A regime whose mean is low when its covariance is high makes $K(\mu, c) < 0$ and a negative third cumulant.

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:

$$\begin{aligned} \kappa_{12} &\approx t\,\bar c_{12} + 2t\,G\,\Delta\mu_1\,\Delta\mu_2, \\ \kappa_{112} &\approx 2t\,G\,\big(2\,\Delta\mu_1\,\Delta c_{12} + \Delta\mu_2\,\Delta c_{11}\big), \\ \kappa_{122} &\approx 2t\,G\,\big(2\,\Delta\mu_2\,\Delta c_{12} + \Delta\mu_1\,\Delta c_{22}\big), \\ \kappa_{111} &\approx 6t\,G\,\Delta\mu_1\,\Delta c_{11}, \qquad \kappa_{222} \approx 6t\,G\,\Delta\mu_2\,\Delta c_{22}, \end{aligned}$$ $$G = \frac{ab}{(a + b)^3}, \qquad \Delta\mu_j = \mu_{j,\mathrm{calm}} - \mu_{j,\mathrm{crisis}}, \qquad \Delta c_{jk} = c_{jk,\mathrm{calm}} - c_{jk,\mathrm{crisis}} .$$

Here $\bar c_{12}$ is the stationary average of $c_{12}$. 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

$$\kappa_{112} \approx 2 \times 0.035156 \times \big(2 \times 0.30 \times (-0.0564) + 0.33 \times (-0.0675)\big) = -3.946\times10^{-3},$$

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.

The correlation of the two simulated 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.

Exact finite-rate cumulants for two regimes

For two states the cumulants need not be approximated. Let $A_T=\int_0^T\mathbf 1_{\{y_t=1\}}\,dt$ be the time spent in crisis over a horizon $T$, and in this section put $r=a+b$, $p_0=b/r$, $p_1=a/r$ and $x=rT$. With a stationary start its first three cumulants are

$$\kappa_1(A_T)=p_1T,\qquad \kappa_2(A_T)=\frac{2p_0p_1}{r^2}\big(rT-1+e^{-rT}\big),\qquad \kappa_3(A_T)=\frac{6p_0p_1(p_0-p_1)}{r^3}\big(x-2+(x+2)e^{-x}\big),$$

and the full note gives the fourth, together with the stationary moment generating function in closed form.

Use differences from calm to crisis, $d\mu=\mu_{\mathrm{crisis}}-\mu_{\mathrm{calm}}$ and $dc=c_{\mathrm{crisis}}-c_{\mathrm{calm}}$. Conditional on $A_T$ the returns are Gaussian with mean $\mu_{\mathrm{calm}}T+d\mu\,A_T$ and covariance $c_{\mathrm{calm}}T+dc\,A_T$.

Let $\mathcal M_n$ be the set of partitions of the labelled positions $\{1,\ldots,n\}$ into singleton and pair blocks. For $n\geq3$,

$$\kappa_{i_1\cdots i_n}(X)= \sum_{\mathcal P\in\mathcal M_n}\kappa_{|\mathcal P|}(A_T) \prod_{\{a\}\in\mathcal P}d\mu_{i_a} \prod_{\{a,b\}\in\mathcal P}dc_{i_ai_b}.$$

For $n=1$ add $T\mu_{i_1,\mathrm{calm}}$, and for $n=2$ add $Tc_{i_1i_2,\mathrm{calm}}$. The statement is exact at finite rate and horizon for every two-state initial law, with the arbitrary-start occupation cumulants given in the full note, under a chain independent of the Brownian shocks. At order three it reads

$$\kappa_{ijk}(X)=\kappa_2(A_T)\big(d\mu_i dc_{jk}+d\mu_j dc_{ik}+d\mu_k dc_{ij}\big)+\kappa_3(A_T)d\mu_i d\mu_j d\mu_k .$$

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. If only covariance switches ($d\mu=0$), every odd cumulant of order at least three vanishes exactly.

At the unscaled rates the exact formula gives $\kappa_{112}=-3.58034\times10^{-3}$: the mean–covariance term contributes $-3.20936\times10^{-3}$ and the skewed occupation time contributes $-0.37098\times10^{-3}$. Starting certainly calm gives $\kappa_{112}=-2.92476\times10^{-3}$, and starting certainly in crisis gives $-3.80148\times10^{-3}$.

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 $m=16$.

Checks for the cumulants

The numerical solution reads the cumulants off the exact generating function by Cauchy's formula, with the matrix exponential evaluated at each node. Third cumulants in units of $10^{-3}$:

$m$	$\kappa_{12}$, averaged	$\kappa_{12}$, numerical	$\kappa_{12}$, rule	$\kappa_{112}$, numerical	$\kappa_{112}$, rule	$\kappa_{122}$, numerical	$\kappa_{122}$, rule
1	0.01770	0.02336	0.02466	−3.580	−3.946	−3.362	−3.632
4	0.01770	0.01936	0.01944	−0.9734	−0.9864	−0.9020	−0.9080
16	0.01770	0.01813	0.01814	−0.2459	−0.2466	−0.2268	−0.2270

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 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 $m=16$. A further certificate checks the all-order partition identity through order six, over all 100 symmetric bivariate tensor entries of orders two through six under stationary, calm, mixed, and crisis initial laws; the maximum absolute discrepancy is $2.4\times10^{-12}$.

Prior art

Longin and Solnik (2001) find that international equity correlation rises in bear markets but not in bull markets. 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. Ang and Bekaert (2002) estimate a regime with lower means, higher volatility and higher correlation, the bear regime used here.

Stochastic correlation in option pricing is usually modelled with the Wishart process of Bru (1991). Deelstra and Simon (2017) price multi-asset options when a chain modulates the parameters, at ordinary speed.

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), and the exchange option inherits it exactly.

The all-order identity is the conditional-Gaussian form of Brillinger's conditional-cumulant formula.

Scope

With fixed volatilities the correlation operator commutes with the rest, and the price is an exact mixture over the time-averaged correlation.
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 is not worked out here.

The full note also covers the spread-option quadrature and tables, the arbitrary-start occupation cumulants, the finite-state version of the partition identity, a finite-state fast-rate theorem with its Green–Kubo and second-order tensors, the power-law maturity phase diagram, the critical boundary layer $sT=\tau$, and the Monte Carlo and three-state checks.

Full note. Certificates: verify_correlation.py and verify_fast_rate.py.