Fast regimes in any pricing equation

One first-order rule for every model whose parameters are switched by a fast hidden chain.

The examples switch a volatility, a mean level, a hazard rate or a jump intensity, and they all follow one rule. To first order, a fast chain adds a second-order term to the pricing operator, weighted by a Green–Kubo matrix of the switched parameters.

The rule has a symmetric part and an antisymmetric part. It is checked below on option smiles, on the joint survival of many names, and on a model where the order of the regimes matters.

The setting

A price $u(t, x)$ depends on time to maturity $t$ and on a slow state $x$, such as a log stock price or a short rate. With the parameters held fixed, it solves a pricing equation

where $\mathcal L$ is the generator of the slow process, including any discounting.

Now let some of the parameters depend on a hidden state $y_t$. The hidden state is a Markov chain with generator $Q$ and stationary law $\pi$, and it switches much faster than $x$ moves. Write the operator in regime $y$ as its average plus fluctuations:

Here $\phi_k(y)$ is the deviation of the $k$-th switched parameter from its $\pi$-average, and $A_k$ is the operator that the parameter multiplies. Most models in finance fit this form:

A parameter that multiplies another is not a coefficient on its own. When $\kappa$ and $\theta$ switch together, the drift is $\kappa_y\theta_y - \kappa_y x$, so the coefficients are $\kappa\theta$ and $\kappa$ with the fixed operators $\partial_x$ and $-x\,\partial_x$, and the averaged drift is $\mathbb E_\pi[\kappa\theta] - \mathbb E_\pi[\kappa]\,x$.

Averaging $\kappa$ and $\theta$ separately misses $\operatorname{Cov}_\pi(\kappa, \theta)$ at leading order. With a symmetric two-state chain, $\kappa = (1, 3)$ and $\theta = (0.01, 0.09)$, the effective level is $\mathbb E_\pi[\kappa\theta]/\mathbb E_\pi[\kappa] = 0.07$, not $\mathbb E_\pi[\theta] = 0.05$. The same product $\kappa\theta$ sits in the Heston variance drift.

The regime-dependent prices $u_y$ are coupled by the chain:

We want the price seen by someone who cannot observe the regime, so the regime is drawn from $\pi$ and the payoff does not depend on it.

Deriving the rule

Scale the chain as $Q = Q_0/\varepsilon$ and expand $u = u_0 + \varepsilon u_1 + \varepsilon^2 u_2 + \cdots$. Each order is handled with the two steps of the solvability page: impose the condition that makes the equation solvable, then solve it.

Order $\varepsilon^{-1}$. Only the chain appears:

So $u_0$ is constant across regimes. The leading price does not depend on the hidden state.

Order $1$. The equation for the first corrector is

Averaging under $\pi$ kills the left side and the fluctuations $\phi_k$. So a solution exists only if

which is the averaged model. The equation is then solved with the group inverse:

The second term remembers the regime. It is large in a regime whose parameters sit above average and persist.

Order $\varepsilon$. The solvability condition at the next order fixes the average $\bar u_1$. Averaging the equation under $\pi$ gives

Substituting $u_1$ from the previous step turns the last term into

Putting $\varepsilon$ back into the chain, the coefficient is the Green–Kubo matrix

The rule

Here $f$ is the payoff. The averaged operator gains the correction

which should be applied as a first-order perturbation, as in the formula above. Exponentiating the corrected operator is not always safe, because it can contain higher derivatives with the wrong sign for a diffusion.

Covariance and order

Split the matrix into its symmetric and antisymmetric parts, $K = K^{\mathrm{sym}} + K^{\mathrm{anti}}$. The correction splits the same way:

• The symmetric part is a covariance rate. For a stationary chain, write $C(s)$ for the covariance matrix of $\phi(y_0)$ and $\phi(y_s)$, and $C^{\mathrm{sym}}$ for its symmetric part. The integrated parameters have covariance $$\operatorname{Cov}\Big(\int_0^t \phi(y_s)\,ds\Big) = 2\int_0^t (t - s)\,C^{\mathrm{sym}}(s)\,ds,$$ while $K^{\mathrm{sym}} = \int_0^\infty C^{\mathrm{sym}}(s)\,ds$. With the chain sped up to $Q_0/\varepsilon$, $2t\,K^{\mathrm{sym}}$ is the leading term, of order $\varepsilon$, and the remainder is of order $\varepsilon^2$ at fixed $t$. Dividing by $2t$ and letting $t$ grow shows that $K^{\mathrm{sym}}$ is positive This is the term the averaging page explains as half the variance of a fluctuating integral.
• The antisymmetric part is about order. $K^{\mathrm{anti}}_{jk}$ compares parameter $j$ followed by parameter $k$ with the reverse order. It vanishes for a reversible chain, which looks the same run backwards in time. It survives when the chain cycles in a preferred direction.

With a payoff that does not depend on the regime, the antisymmetric part only acts through the commutator $[A_j, A_k] = A_j A_k - A_k A_j$. So the direction of the cycle matters only when the operators do not commute: when it matters whether one parameter acts before the other.

A payoff or a running source that depends on the regime opens a second route. The race studied on the kinetics site is the example: each channel's own rate is the source of its win probability, and the asymmetry of $K$ is visible from the order of arrivals even though every operator there is a multiplication. The symmetry of $K$ under detailed balance, and its use as a witness of a hidden irreversible driver, are developed there (Cotton, working draft).

Constant-coefficient operators commute. In Black–Scholes, and whenever the level and variance of Vasicek are switched, only the covariance enters. That is why the yield curve sees a symmetric $2\times2$ matrix.

The smile from switching volatility

Switch the variance in Black–Scholes. With $x = \log S$ there is one operator,

It is the derivative with respect to total variance $V = \sigma^2 t$: applied to a Black–Scholes price, it gives $\partial_V$. So the correction $K\,A^2$ is $K\,\partial_V^2$, and the rule gives a closed form:

This is a volga term, the convexity of the price in variance. It is positive in the wings and negative near the money, so no single volatility can absorb it: it is a smile.

It is the first-order form of the mixing formula of Hull and White, with the variance of integrated variance replaced by its leading term $2t\,K$. The two differ by a term of order $\varepsilon^2$, bounded uniformly in maturity; the cumulants page gives the bound, the exact cumulants of the return, and the CIR case.

This smile is known. Sircar and Papanicolaou (1999) derive it for a fast ergodic volatility driver whose integrated fluctuations obey a central limit theorem, as a finite-state chain does. Stationarity and ergodicity alone are not enough: the covariance tail sets the correction scale. Fouque, Lorig and Sircar (2016) give the correlated fast-factor expansion to second order.

The check below confirms the rule reproduces it. A Heston example shows the antisymmetric part moving the smile.

The check uses three regimes with volatilities of 10%, 30% and 20%. The chain is twice the irreversible three-state generator

so that regimes last about two to three months.

$S = 100$, maturity 1 year, $r = 2\%$, averaged volatility 23.80%.

The table shows implied volatilities. The rule reproduces the smile to within a few hundredths of a volatility point. With the chain four times faster, the largest price error falls from $1.2\times10^{-3}$ to $7\times10^{-5}$, sixteen times smaller, as a second-order remainder should.

Many names from one matrix

Give each of several firms a hazard rate $h_m(y)$ that depends on the common regime, and let the defaults be independent given the regime path. The survival of a set $S$ of names is

Every operator is $-1$, so all of them commute, and the rule gives

One matrix gives the joint survival of every subset. For a pair of names the dependence is

Two consequences follow.

• The sign of the dependence between two names is the sign of $K^{\mathrm{sym}}_{12}$, the integrated lagged cross-covariance of their hazards, and it can be negative. It is not fixed by the regimes in which each hazard is high, nor by the sign of the contemporaneous covariance $\operatorname{Cov}_\pi(h_1, h_2)$: with several relaxation times the lagged terms can outweigh it. For the reversible chain with rates $1$ between states 1 and 2 and $10$ between states 2 and 3, and hazards $h_1 = (0.03, 0.05, 0.04)$ and $h_2 = (0.03, 0.02, 0.07)$, the contemporaneous covariance is $-10^{-4}/3$ while $K^{\mathrm{sym}}_{12}$ is $+10^{-4}/(3m)$ at speed $m$, and the numerical log ratio at $t = 5$ is positive at every speed.
• The matrix is positive semidefinite. With constant hazards its rank is at most one less than the number of regimes, so a three-regime economy supports at most two independent patterns of default dependence, however many names there are. When hazards load on the regime with different maturity profiles, the integrated matrix can have higher rank, as the positive CIR construction shows. The same rank property appears for races on the kinetics site.

Joint survival sees only the symmetric part of $K$. The order of defaults sees more.

Which name defaults first is a race under a shared driver, and two names already suffice for its law to reveal the asymmetry of $K$, provided the chain has at least three states and is not reversible. Races of this kind are studied on the kinetics site.

For two names, starting from the stationary law, the probability that name 1 defaults first is

Expanding the resolvent for a fast chain, with $\bar h = \bar h_1 + \bar h_2$,

The entry $K_{21}$ appears without $K_{12}$. Reversing the direction of the cycle transposes $K$, and moves $p_1$ by

An example: a three-state cycle with rate 2 one way round and 0.1 the other, hazards $h_1 = (0.07, 0.01, 0.01)$ and $h_2 = (0.01, 0.07, 0.01)$, and the generator multiplied by 10. Name 1 defaults first with probability 0.500150 with the cycle one way and 0.499850 the other way. Every single-name and joint survival probability is the same for both directions.

With three or more causes, however, one unweighted race sees only the row sums of the antisymmetric matrix and can miss a nonzero cyclic current completely. The joint-default note proves that varying positive cause exposures recovers the full antisymmetric matrix and gives an exact three-cause counterexample to identification from one unweighted race.

The check uses four names and the three-state chain $Q_A$ from the smile example. The pairwise dependence over five years, in units of $10^{-4}$, is:

Its two smallest eigenvalues are zero. With the chain ten times faster, the rule matches the numerical log survival of all fifteen subsets to within $3\times10^{-6}$. At forty times faster it matches within $2\times10^{-7}$.

The order of the regimes

Now switch the reversion speed and the variance of a Vasicek short rate together. The two operators are

and they do not commute. A short calculation gives

So the antisymmetric part of $K$ acts as extra variance. The effective variance becomes

The intuition is about sequence. If high volatility tends to be followed by fast reversion, shocks are forgotten quickly and the rate is effectively calmer. If high volatility is followed by slow reversion, shocks linger.

The averages and the covariances are the same in both cases. Only the direction of the cycle differs.

The check uses a chain that cycles almost only one way, $1 \to 2 \to 3 \to 1$, at rate 2 per year, with a reverse rate of 0.1:

speeds $\kappa = (0.3,\ 1.5,\ 3.0)$, variances $\sigma^2 = (0.0004,\ 0.0036,\ 0.0016)$, $\theta = 4\%$.

For this chain the order term changes the effective variance by $-19\%$, from $0.00187$ to $0.00151$. Running the cycle the other way changes it by $+19\%$. The table shows errors in a two-year bond price, on a finite-difference grid, as the chain is sped up by a factor $m$:

The full rule is second order. Keeping only the covariance does worse than no correction at all, because here the order term is the larger of the two.

Higher orders and special cases

• The rule is first order for any pricing equation. When the model has an exponential-affine factor with a regime-free exponent, the engine gives every order.
• The yield-curve page is the rule for Vasicek with switching level and variance, where the operators commute.
• When the payoff depends on the regime at expiry, an initial layer enters at order zero, as on the bond options page.

Certificate: verify_general.py.