The direction of a regime cycle in the Heston smile
A chain or continuous periodic factor that cycles in one direction contributes a commutator correction invisible to averages and symmetric covariances.
The smile from fast volatility
When volatility fluctuates fast and independently of the stock, the option price is Black–Scholes averaged over integrated variance (Hull and White 1987). Expanding that average around the mean variance gives a smile at first order in the fluctuation time scale.
Sircar and Papanicolaou (1999), section 3.3, derive it for a stationary ergodic volatility driver whose integrated fluctuations obey a central limit theorem, which a finite-state Markov chain does. Ergodicity alone does not fix the rate: long memory changes the power of the small parameter.
The implied volatility is quadratic in log-moneyness, with a curvature set by the variance of integrated variance. Fouque, Lorig and Sircar (2016) carry the fast-factor expansion to second order with correlation and a terminal layer.
In that work the fast driver is a one-dimensional diffusion or a generic ergodic process whose correlation function enters through a variance. A stationary scalar diffusion on an interval or the line is reversible under the usual zero-current boundary conditions. Dimension alone is not enough: a diffusion on a periodic domain can carry stationary current, break detailed balance, and have an antisymmetric Green–Kubo matrix.
The central limit condition sets the order of the correction. A stationary ergodic driver with long memory, whose correlation decays like $t^{-\alpha}$ with $0 < \alpha < 1$, has no finite Green–Kubo integral.
With $\varepsilon$ the fluctuation time scale, the variance of its integrated variance scales like $\varepsilon^\alpha$ in place of $\varepsilon$, and so does the smile correction. The chain below mixes exponentially fast and has the order-$\varepsilon$ correction.
The model
The log price $X_t$ and the variance $v_t$ follow Heston's model with zero correlation. The variance level and the vol-of-vol depend on a hidden regime $y_t$:
$$dX_t = -\tfrac12 v_t\,dt + \sqrt{v_t}\,dW_t, \qquad dv_t = \kappa\,\big(\theta_{y_t} - v_t\big)\,dt + \xi_{y_t}\sqrt{v_t}\,dZ_t .$$The regime is a three-state chain that cycles mostly one way, $1 \to 2 \to 3 \to 1$. The reverse chain has the same stationary law and the same symmetric Green–Kubo matrix. The variance reverts over $1/\kappa$, half a year, while the regimes last a few weeks.
Parameters: $\kappa = 2$; levels $\theta = (0.02,\ 0.10,\ 0.05)$; vol-of-vol $\xi = (0.2,\ 0.4,\ 1.0)$; $v_0 = 0.04$; $S_0 = 100$; zero interest; one year to expiry. Forward rate 2 and reverse rate 0.1 per year along the cycle, multiplied by a speed factor $m$.
The quantity
European call prices and their Black–Scholes implied volatilities, for a regime drawn from its stationary law. The prices come from the characteristic function of the return,
$$\phi(u) = \mathbb{E}\big[e^{iu(X_T - X_0)} \mid v_0\big],$$through Lewis's formula:
$$C = S_0 - \frac{\sqrt{S_0K}}{\pi}\int_0^\infty \operatorname{Re}\big[e^{iu\log(S_0/K)}\,\phi(u - \tfrac i2)\big]\,\frac{du}{u^2 + 1/4}.$$Averaging
When the chain switches infinitely fast, the limit is Heston's model with the stationary averages $\bar\theta$ and $\bar\xi^2$.
The first-order rule
After the Fourier transform in $x$, the switched parameters $\theta$ and $\xi^2$ multiply two operators in the variance:
$$A_\theta = \kappa\,\partial_v, \qquad A_{\xi^2} = \tfrac12\,v\,\partial_{vv} .$$They do not commute:
$$\big[A_\theta,\,A_{\xi^2}\big] f = \tfrac{\kappa}{2}\big(\partial_v(v f'') - v f'''\big) = \tfrac{\kappa}{2}\,f'' .$$By the first-order rule, the averaged operator gains
$$\sum_{j,k} K^{\mathrm{sym}}_{jk}\,A_j A_k \;+\; K^{\mathrm{anti}}_{\theta\xi^2}\,\tfrac{\kappa}{2}\,\partial_{vv}, \qquad K^{\mathrm{anti}}_{\theta\xi^2} = \tfrac12\big(K_{\theta\xi^2} - K_{\xi^2\theta}\big).$$The last term adds a constant diffusion to the variance. Its sign flips when the cycle is reversed, and the symmetric terms do not change.
The intuition is a lag. The variance follows its level with a delay of about $1/\kappa$.
When high vol-of-vol tends to follow a high level, it arrives while the variance is high and shakes it harder. When high vol-of-vol comes first, it acts on a low variance.
Method
For each Fourier frequency the coupled equations for the three regimes are solved on one finite-difference grid in $v$, which gives the numerical solution. The rule is applied on the same grid, as a first-order Duhamel correction to the averaged operator, so both share the discretisation error.
Results
Implied volatilities in percent, with the chain four times faster than the base rates. Each regime then lasts about six weeks.
| strike | forward, numerical | forward, rule | reverse, numerical | reverse, rule | averaged Heston |
|---|---|---|---|---|---|
| 70 | 23.98 | 23.96 | 23.88 | 23.78 | 24.02 |
| 85 | 21.53 | 21.51 | 21.56 | 21.54 | 21.55 |
| 100 | 20.62 | 20.65 | 20.74 | 20.88 | 20.62 |
| 115 | 21.32 | 21.30 | 21.36 | 21.37 | 21.33 |
| 130 | 22.70 | 22.67 | 22.65 | 22.57 | 22.74 |
The forward cycle has the deeper smile: lower at the money, higher in the wings. The difference is about a tenth of a volatility point. The rule gets its sign and shape, and overstates it at this speed.
As the chain speeds up, the largest implied-volatility error across strikes, in volatility points, is:
| SPEED $m$ | averaged | symmetric part only | full rule | forward minus reverse at the money |
|---|---|---|---|---|
| 4 | 0.037 | 0.138 | 0.031 | −0.116 |
| 8 | 0.024 | 0.065 | 0.009 | −0.078 |
| 16 | 0.014 | 0.031 | 0.003 | −0.047 |
| 32 | 0.008 | 0.015 | 0.0007 | −0.026 |
The full rule's error falls by about four for each doubling, as a second-order remainder should. The effect of the direction falls by about two, as a first-order effect should, and at the fastest speed the rule reproduces it within about 20% at every strike.
Keeping only the symmetric part is worse than no correction. In the forward direction the symmetric and antisymmetric parts of the correction largely cancel, so either one alone misses by more than the average does.
Beyond finite chains
The reversal property of the Green–Kubo matrix does not need a finite chain. Let $P_t$ be a stationary Markov semigroup on the real space $L^2(\pi)$, and let $P_t^{\leftarrow}=P_t^\dagger$ be its stationary time reversal. For centered observables $f,g$ whose correlations are integrable, define
$$K^{\rightarrow}(f,g)=\int_0^\infty\langle f,P_tg\rangle_\pi\,dt.$$Semigroup adjointness gives, without a diffusion or finite-state assumption,
Thus stationary time reversal preserves the symmetric Green–Kubo form and flips the antisymmetric form. This is the integrated-correlation consequence of the general Markov time-reversal framework of Nagasawa (1964).
For a scalar diffusion on the circle the two parts are explicit. Let
$$dY_t=b(Y_t)\,dt+\sqrt{2a(Y_t)}\,dW_t\pmod {2\pi},\qquad a>0,$$have a smooth positive stationary density $p$. Its stationary probability current $J=b(y)p(y)-(a(y)p(y))'$ is constant around the circle, and its generator has the exact decomposition
In $L^2(p)$, $S$ is self-adjoint and $A$ is skew-adjoint. Consequently the stationary time-reversed diffusion has generator $L^\dagger=S-A$, current $-J$, and drift $b^\dagger=b-2J/p$. In particular, a regular scalar diffusion on the circle is reversible if and only if $J=0$ (Haussmann and Pardoux 1986).
Drifted Brownian motion on the circle is the closed-form example:
$$dY_t=c\,dt+\sqrt{2D}\,dW_t\pmod {2\pi},\qquad D>0.$$Its stationary law is uniform, but its stationary current is $c/(2\pi)$ and time reversal changes $c$ to $-c$. For the centered feature pair $\cos(ny)$, $\sin(ny)$ of Fourier mode $n\ge1$, the Green–Kubo block is exactly
If $\cos Y$ and $\sin Y$ multiply two fixed operators $A_c,A_s$, their first correction is
$$\frac{D}{2(D^2+c^2)}(A_c^2+A_s^2) +\frac{c}{2(D^2+c^2)}[A_c,A_s].$$For $D=1,c=2$, the block is $K_1=\left(\begin{smallmatrix}0.1&0.2\\-0.2&0.1\end{smallmatrix}\right)$. A periodic CTMC discretization converges to this block at rate 2.001. Keeping only the symmetric block leaves a first-order error; the forward-minus-reverse residual after the commutator correction converges at rate 2.000.
The full note also covers exact occupation-law blindness at every switching rate, random-probe detection of finite-state irreversibility with finite-probe certificates, deterministic stability under generator estimation, a generator radius from a fully observed path, and a sharp quantitative irreversibility bound.
Full note: cycle-smile-full.html. Certificates: verify_cycle_smile.py, verify_circle_factor.py.