Subordinator-tracking portfolios#
In a generalized hyperbolic (GH) model the latent subordinator \(Y\) is the volatility clock: conditionally on \(Y=y\), returns are Gaussian with covariance \(y\Sigma\) and mean \(\mu + \gamma y\). The Bayes estimate \(E[Y\mid X]\) is already computed in the EM E-step. This note asks a different question: can a portfolio \(w^\top X\) be dominated by \(Y\), so that the portfolio itself reveals the clock?
The answer in the model is yes, and the analysis is closed-form. The optimal direction is \(w \propto \Sigma^{-1}\gamma\). The achievable signal-to-noise is the scalar \(\tilde q = \gamma^\top\Sigma^{-1}\gamma\) that already appears in the GH marginal density and the conditional GIG posterior. “Dominated by \(Y\)” can be made precise in three inequivalent ways. We did not find this tracking reading in the literature, but each ingredient exists separately.
Background: The Generalized Hyperbolic Distribution, The Generalized Inverse Gaussian Distribution, EM Algorithm for Generalized Hyperbolic Distributions, Mean-Risk Optimization for Normal Mixture Distributions. The S&P 500 measurement is Subordinator tracking on the S&P 500.
Setup#
Let \(X\in\mathbb{R}^d\) be a normal mean–variance mixture BarndorffNielsen1977 McNeil2010
with \(\mu,\gamma\in\mathbb{R}^d\) and \(\Sigma\succ 0\). The mixing law of \(Y\) is GIG\((p,a,b)\) for GH, Gamma for variance-gamma (VG) MadanSeneta1990, InverseGaussian for normal-inverse-Gaussian (NIG) BarndorffNielsen1997, or InverseGamma for normal-inverse-gamma (NInvG). Write
for the Mahalanobis radius, the \(\gamma\)-energy in the \(\Sigma\) metric, and the third central moment of \(Y\). The model has the scaling gauge \((\gamma,\Sigma,Y)\mapsto(\gamma/c,\,\Sigma/c,\,cY)\); quantities below are flagged when they are invariant.
Three standing facts.
Projection. For any \(w\), the portfolio \(P_w = w^\top X\) is a univariate mixture with the same \(Y\) and parameters \((w^\top\mu,\, w^\top\gamma,\, w^\top\Sigma w)\) Blaesild1981. This is (2) in Mean-Risk Optimization for Normal Mixture Distributions, implemented as
project().Posterior. \(Y\mid X=x\) is GIG with updated parameters \((p-d/2,\, a+\tilde q,\, b+q(x))\) (EM Algorithm for Generalized Hyperbolic Distributions). The posterior depends on \(x\) only through \(q(x)\). The linear statistic \(s(x)=\gamma^\top\Sigma^{-1}(x-\mu)\) enters the joint density as a \(y\)-free tilt \(e^{s(x)}\) and cancels from the conditional.
Bayes optimality. \(E[Y\mid X]\) minimises mean-squared error among measurable functions of \(X\). Normix computes it via
conditional_expectations().
The tracking portfolio#
Assume \(\gamma\neq 0\) and take any \(w\) with \(m=w^\top\gamma\neq 0\). Define the tracker
Conditionally on \(Y\), \(\hat Y_w\sim N(Y,\, t_w Y)\): an unbiased observation of \(Y\) whose noise variance is proportional to the signal level. The mean-squared error is therefore
Proposition 1 (optimal direction). \(t_w \ge 1/\tilde q\) for all \(w\), with equality if and only if \(w\propto\Sigma^{-1}\gamma\).
Proof. Cauchy–Schwarz in the \(\Sigma\) inner product:
with equality iff \(\Sigma^{1/2}w\) is parallel to \(\Sigma^{-1/2}\gamma\). \(\square\)
Two remarks on the informal recipe “maximise \(w^\top\gamma\), minimise \(w^\top\Sigma w\)”.
The Pareto frontier of that bi-objective problem is exactly the ray \(\{c\,\Sigma^{-1}\gamma : c>0\}\). No other direction improves one objective without hurting the other.
The degree-1 ratio \(w^\top\Sigma w/w^\top\gamma\) is not a dominance measure: scaling \(w\) down sends it to \(0\). The scale-invariant objective is \(t_w\). The degree-1 ratio does have an exact meaning — it is four times the expected pathwise floor of the portfolio below its location (Proposition 2).
Fix the normalisation \(m=1\):
Computing \(w^\star\) from a fitted model is two triangular solves
against the stored Cholesky factor L_Sigma.
Two invariant signal-to-noise ratios#
Under the gauge \(c\): \(\tilde q\mapsto\tilde q/c\), \(E[Y]\mapsto c\,E[Y]\), \(\mathrm{Var}(Y)\mapsto c^2\mathrm{Var}(Y)\). Two dimensionless, gauge-invariant ratios control everything below:
Their ratio \(\kappa/\kappa_{\mathrm{lev}} = \mathrm{Var}(Y)/E[Y]^2\) is the squared coefficient of variation of \(Y\). Then
\(\kappa_{\mathrm{lev}}\gg 1\) means the portfolio level is dominated by the subordinator term. \(\kappa\gg 1\) means the tracker resolves \(Y\)’s fluctuations, which is the estimation-relevant statement. In the identifiability gauge \(E[Y]=1\) they collapse to \(\kappa_{\mathrm{lev}}=\tilde q\) and \(\kappa=\tilde q\,\mathrm{Var}(Y)\).
The correlation of the tracker with \(Y\) follows at once from the fluctuation SNR:
At \(\kappa=0.05\) this is about \(0.22\); at \(\kappa=1\) it is \(1/\sqrt{2}\approx 0.71\). Daily-equity \(\kappa\) in the \(10^{-2}\) range is a prior, not a theorem — see the empirics.
The tracker as one synthetic asset#
\(P^\star=(w^\star)^\top X\) is the univariate mixture with parameters \(((w^\star)^\top\mu,\, 1,\, 1/\tilde q)\). The \(d=1\) posterior on the tracker alone is
to be compared with the full-information posterior \(\mathrm{GIG}(p-d/2,\, a+\tilde q,\, b+q(x))\). The \(a\)-updates agree. The information lost by compressing \(X\) into the scalar \(\hat Y\) sits in the \(b\)-slot and the GIG order.
Decompose \(x-\mu\) into its \(\gamma\)-component and its \(\Sigma\)-orthogonal complement:
The full posterior sees the squared tracker plus the orthogonal Mahalanobis radius \(q_\perp\) — a realized-dispersion statistic that is informative about \(Y\) even when \(\gamma=0\). Two channels:
Linear (drift). The tracker, strength \(\tilde q\). Requires \(\gamma\neq 0\). Tradable: it is a portfolio.
Quadratic (dispersion). The radius, strength of order \(d\) (one \(\chi^2_1\) observation of scale \(Y\) per orthogonal direction, visible in the posterior order \(p-d/2\)). Alive even for elliptical models. Not a linear payoff — in market terms it is a variance-swap-like object.
Three estimators of \(Y\)#
Estimator |
Form |
Needs |
MSE / \(\mathrm{Var}(Y)\) |
|---|---|---|---|
Tracker |
\(\hat Y = s(X)/\tilde q\) |
direction \(\Sigma^{-1}\gamma\), location \(\mu\) |
\(1/\kappa\) |
Linear Bayes |
\(E[Y] + \frac{\kappa}{1+\kappa}(\hat Y-E[Y])\) |
also \(E[Y]\), \(\mathrm{Var}(Y)\) |
\(1/(1+\kappa)\) |
Posterior mean |
\(E[Y\mid X]\) |
full model |
\(\le 1/(1+\kappa)\) |
The linear-Bayes row is the best affine function of \(X\). With \(\mathrm{Cov}(X)=E[Y]\,\Sigma + \mathrm{Var}(Y)\,\gamma\gamma^\top\) and \(\mathrm{Cov}(X,Y)=\mathrm{Var}(Y)\,\gamma\), the Sherman–Morrison formula gives
— the same direction again — and the usual credibility shrinkage \(\kappa/(1+\kappa)\) toward \(E[Y]\), with MSE \(\mathrm{Var}(Y)/(1+\kappa)\). The posterior mean improves on every affine rule because it also uses \(q_\perp\). It has no closed form (Bessel ratios), but \(E[\mathrm{Var}(Y\mid X)]\le \mathrm{Var}(Y)/(1+\kappa)\) is a usable upper bound. Even at \(\kappa=0\) the posterior remains consistent for \(Y\) as \(d\to\infty\) under the model.
The tracker therefore costs a factor \((1+\kappa)/\kappa\) in MSE relative to linear Bayes. That factor is negligible when \(\kappa\gg 1\) and ruinous when \(\kappa\ll 1\): unbiasedness is expensive when the signal is weak. The tracker’s compensations: it is a portfolio (tradable, holdable, a return series), it is linear (no Bessel evaluations), and it needs only the direction \(\Sigma^{-1}\gamma\) — not the mixing law, and not even \(\tilde q\) if only relative movements matter.
Three meanings of “dominated by \(Y\)”#
(i) Observation-level. In \(\hat Y_w = Y + \sqrt{Y}\,\varepsilon_w\), the signal exceeds the noise on the event \(Y\gg t_w\). For the optimal tracker, \(t_w=1/\tilde q\). Typical observations are subordinator-dominated iff \(\kappa_{\mathrm{lev}}\gg 1\).
(ii) \(L^2\) / distributional. Along any sequence of models with \(\tilde q\to\infty\) and the mixing law of \(Y\) held fixed, \(\hat Y\to Y\) in \(L^2\), so the law of the portfolio converges to the mixing law shifted by the location:
VG: \(P^\star - (w^\star)^\top\mu \Rightarrow \mathrm{Gamma}\);
NIG: \(\Rightarrow\) InverseGaussian;
GH: \(\Rightarrow\) GIG.
The portfolio becomes its mixing distribution. At finite \(\tilde q\) the tracker is the univariate mixture with residual variance \(1/\tilde q\): the minimal Gaussian blur of \(Y\) achievable by any portfolio.
(iii) Pathwise floor.
Proposition 2. For \(m=w^\top\gamma>0\) and \(\varepsilon\sim N(0,t_w)\),
pathwise, because \(y+\sqrt{y}\,\varepsilon \ge -\varepsilon^2/4\) for all \(y\ge 0\) (minimise the quadratic in \(\sqrt{y}\)). Hence the expected worst-case shortfall below location is at most \(m\,t_w/4\), which is the degree-1 ratio \(w^\top\Sigma w/(4\,w^\top\gamma)\). For the optimal tracker the left tail below location is Gaussian-thin:
since the event requires \(\varepsilon^2\ge 4c\). The tracker is the model’s maximally asymmetric portfolio: GIG (semi-heavy, \(e^{-ay/2}\)) upside inherited from \(Y\), downside below location bounded by \(\chi^2_1/(4\tilde q)\). Shorting it is the model’s linear “short-volatility” trade.
No-near-arbitrage. As \(\tilde q\to\infty\) the tracker converges to a non-negative payoff above location — a limiting free lunch unless \((w^\star)^\top\mu\) is pushed negative. In an economy where such portfolios are priced, either \(\tilde q\) stays moderate or the location term compensates. Empirically relevant \(\kappa\) should therefore be \(O(1)\) or smaller. That is a prior about markets, not a theorem.
The same direction from four angles#
The ray \(\Sigma^{-1}\gamma\) is canonical for the family, not just for tracking.
Max-SNR (Proposition 1).
Best linear predictor of \(Y\) given \(X\) (Sherman–Morrison above).
Maximal skewness. Portfolio skewness depends on \(w\) only through \(t=w^\top\Sigma w/(w^\top\gamma)^2\). For \(m>0\),
\[ \mathrm{skew}(P_w) = \frac{\mu_3 + 3t\,\mathrm{Var}(Y)} {\bigl(\mathrm{Var}(Y) + t\,E[Y]\bigr)^{3/2}}. \]Differentiating in \(t\),
\[ \frac{d}{dt}\mathrm{skew} \gtrless 0 \iff t \lessgtr t^\dagger := \frac{2\,\mathrm{Var}(Y)}{E[Y]} - \frac{\mu_3}{\mathrm{Var}(Y)}. \]Skewness is unimodal on the feasible interval \([1/\tilde q,\infty)\). The maximiser is \(w^\star\) if and only if \(t^\dagger\le 1/\tilde q\); in particular for all \(\tilde q\) whenever
\[ E[Y]\,\mu_3 \ge 2\,\mathrm{Var}(Y)^2 \](equivalently \(\kappa_1\kappa_3\ge 2\kappa_2^2\) in cumulants of \(Y\)). This holds with equality for Gamma mixing (VG) — both sides equal \(2\alpha^2/\beta^4\) — and strictly for InverseGaussian (\(E[Y]\,\mu_3 = 3\,\mathrm{Var}(Y)^2\)) and InverseGamma (\(E[Y]\,\mu_3 = \frac{4(\alpha-2)}{\alpha-3}\mathrm{Var}(Y)^2 > 2\,\mathrm{Var}(Y)^2\) for \(\alpha>3\)). For general GIG we use the inequality as a per-fit check; Cauchy–Schwarz on the Lévy measure gives only the weaker \(\kappa_1\kappa_3\ge\kappa_2^2\). When \(t^\dagger>1/\tilde q\), the maximiser sits at the interior point \(t=t^\dagger\) and is a whole cone of weights, so max-skewness and max-SNR decouple.
Markowitz tangency under a pure volatility premium. In the model, \(E[X]-\mu = \gamma\,E[Y]\): all expected return beyond the location is compensation for subordinator exposure. If the location is common and riskless, \(\mu=r\mathbf{1}\), the tangency weights Markowitz1952 are \(\mathrm{Cov}(X)^{-1}(E[X]-r\mathbf{1})\propto\Sigma^{-1}\gamma\). In general this is the skewness–variance fund of MenciaSentana2009: riskless + mean–variance fund + a third fund that loads on the skewness vector. That third fund is the minimum-dispersion portfolio per unit of \(w^\top\gamma\) — the tracker, subject to their budget constraint.
Item 4 is already in the efficient-surface reduction of Mean-Risk Optimization for Normal Mixture Distributions: portfolios are parametrised by \((\tilde\mu,\tilde\gamma)\) with minimal dispersion \(g(\tilde\mu,\tilde\gamma)\). The constrained tracker is the \(\tilde\gamma\)-extreme of that surface.
When can \(\tilde q\) be large?#
Cross-sectional growth of \(\tilde q_d=\gamma_d^\top\Sigma_d^{-1}\gamma_d\) depends on how skewness aligns with the covariance factors.
Equicorrelation. Let \(\Sigma=\sigma^2\bigl[(1-\rho)I+\rho\mathbf{1}\mathbf{1}^\top\bigr]\) with \(0<\rho<1\), and split \(\gamma=g\mathbf{1}+\delta\) with \(\mathbf{1}^\top\delta=0\). Then
If skewness loads only on the market (\(\delta=0\)) — the usual first-order description of equities, where the index carries the leverage asymmetry — \(\tilde q_d\) saturates at \(g^2/(\sigma^2\rho)\). Diversification cannot separate \(Y\) from market noise, because the market factor is exactly the direction where the Gaussian component refuses to average out. The \(d\to\infty\) tracker is the index itself, and the ceiling is the index’s own \(\kappa\).
Unbounded recovery needs idiosyncratic skewness dispersion: \(\lVert\delta\rVert^2\asymp d\). Then the tracker is a long–short in the \(\delta\) direction, and recovery of \(Y\) is a law-of-large-numbers effect across names.
Contrast with the Bayes channel. Conditionally on \(Y=y\), the whitened residuals \(\Sigma^{-1/2}(X-\mu-\gamma y)/\sqrt{y}\) are i.i.d. standard normal in every direction. The Mahalanobis radius per dimension therefore concentrates at \(Y\) as \(d\to\infty\), even for bounded \(\tilde q\) and even for \(\gamma=0\). The quadratic channel does not saturate. It does need a well-estimated \(\Sigma^{-1}\) (serious at large \(d\); see Shrinkage with Penalized Likelihood and Factor Analysis for Generalized Hyperbolic Distributions), it is quadratic in returns rather than a position, and it leans on exact conditional Gaussianity. The tracker degrades more gracefully on all three counts — when the linear signal is actually there.
Open problems#
(a) Time series. Per-period \(Y_t\) with dependence (subordinated Lévy or stochastic-volatility dynamics). Does the tracker plus online EM (Online EM Algorithm, Cappe2009) yield a real-time activity index comparable to realized variance? The empirics answer this in the negative on daily US large caps: the object that tracks volatility is \(q_\perp\), not \(\hat Y\).
(b) Nonlinear tradable payoffs. Among payoffs \(\phi(w^\top X)\) (options on the tracker), what closes the gap to the quadratic channel?
(c) Several subordinators. Common plus idiosyncratic clocks Semeraro2008. Which linear combinations track the common clock, and is the answer again a generalised eigenvector problem?
A GIG cumulant sweep for \(\kappa_1\kappa_3\ge 2\kappa_2^2\) is also open. The Gamma boundary is tight; the four GH fits in the empirics all satisfy \(t^\dagger\le 1/\tilde q\).
Literature#
We did not find the statement “the minimum-variance-per-unit-skewness portfolio of a normal mean–variance mixture is an optimal linear estimator of the latent subordinator, with conditional law \(N(Y,\,Y/\tilde q)\)”. The surrounding pieces sit in five strands.
GH structure and the posterior. BarndorffNielsen1977 introduced GH; Blaesild1981 proved closure under affine maps; MadanSeneta1990 (VG) and BarndorffNielsen1997 (NIG) are the workhorse special cases; McNeil2010 Ch. 6 is the standard mixture treatment. The GIG posterior and its Bessel-ratio moments drive the EM algorithms of Protassov2004 and Hu2005 — normix’s E-step. What EM uses as a latent-variable imputation, this note reads as a volatility measurement.
Portfolio selection in mixtures. MenciaSentana2009 is the closest work: for location-scale mixtures of normals, any portfolio law is determined by (mean, variance, skewness), the mean–variance–skewness frontier is closed-form, and its efficient part is spanned by three funds, the third loading on the skewness vector. Section 5(4) above is the observation that their skewness fund is a mimicking portfolio for the latent mixing variable — an interpretation they do not pursue (their \(\xi\) is integrated out, not estimated). The repo’s own mean-risk reduction (Mean-Risk Optimization for Normal Mixture Distributions, after Shi2016) contains the same \((\tilde\mu,\tilde\gamma,\tilde\sigma)\) geometry.
Skewness-maximising projections. Loperfido2010 posed projection pursuit by skewness for skew-normal families; ArevalilloNavarro2020 ArevalilloNavarro2021 prove that for scale mixtures of skew-normal vectors the max-skewness direction is proportional to the shape vector (scaled by the scatter inverse). Structurally identical to the \(t^\dagger\) calculation, with a truncated-normal latent instead of a subordinator. The feasibility boundary \(t^\dagger\le 1/\tilde q\) and the Gamma case sitting exactly on it appear to be new for GH-type mixtures. The canonical-form literature for skew-elliptical families AzzaliniCapitanio2014 makes the same point that one linear combination carries all the asymmetry.
Mimicking / tracking portfolios. HubermanKandelStambaugh1987 characterise portfolios that can replace factors in pricing relations; BreedenGibbonsLitzenberger1989 build the consumption-mimicking maximum-correlation portfolio; Lamont2001 tracks macro variables. The subordinator tracker is a maximum-correlation mimicking portfolio where the “factor” is the model’s own latent activity. The mixture structure supplies in closed form what that literature estimates by regression.
Time-change recovery and traded volatility. Clark1973 began the subordinated-returns program; AneGeman2000 claimed recovery of the transaction clock’s moments (contested: MurphyIzzeldin2010; see also RichardsonSmith1994). Realized variance estimates the integrated clock from high-frequency data BarndorffNielsenShephard2002 — the quadratic channel, done across time instead of across assets. Variance swaps CarrWu2009 are the traded quadratic instrument. Multivariate subordination with common plus idiosyncratic clocks Semeraro2008 LucianoSemeraro2010 LucianoMarenaSemeraro2016 BallottaBonfiglioli2016 is the natural setting for open problem (c).