Quadrature scenarios for mixture CVaR#
We derive deterministic scenarios for the normal variance-mean mixtures of The Generalized Hyperbolic Distribution. The portfolio notation and loss convention follow Mean-Risk Optimization for Normal Mixture Distributions; the derivative formulas are developed in CVaR Derivatives for Normal Mixture Distributions. A worked comparison is in Quadrature scenarios for portfolio CVaR.
Portfolio returns and losses#
Let \(\mathbf X=\mu+\gamma Y+\sqrt{Y}Z\), where \(Y>0\), \(Z\sim N(0,\Sigma)\) is independent of \(Y\), and \(\Sigma\succ0\). For portfolio weights \(w\ne0\), define
where \(Z_0\sim N(0,1)\) is independent of \(Y\). The portfolio loss is \(L=-X\). For tail probability \(\alpha\in(0,1)\), write \(x_\alpha=F_X^{-1}(\alpha)\). Then \(\operatorname{VaR}_\alpha=-x_\alpha\) and \(\operatorname{CVaR}_\alpha=-E[X\mid X\le x_\alpha]\). Positive risk denotes a loss.
Given \(Y=y\), put \(m_y=\tilde\mu+\tilde\gamma y\), \(s_y=\tilde\sigma\sqrt y\), and \(z_y(x)=(x-m_y)/s_y\). With \(\Phi\) and \(\varphi\) denoting the standard normal CDF and density,
The CDF is continuous and strictly increasing by dominated convergence and strict increase of every conditional CDF. Its quantile is unique. Density and derivative formulas below assume that the displayed expectations are finite at the quantile. The density can be unbounded at the location for some Gamma mixing laws; continuity of the CDF does not require a bounded density.
Rockafellar–Uryasev and conditioning#
Proposition. Suppose \(E|L|<\infty\) and \(F_L\) is continuous. The function
is convex, has derivative \(G'(t)=1-\alpha^{-1}\mathbb P(L>t)\), and attains its minimum at the \((1-\alpha)\)-quantiles of \(L\). Its minimum is CVaR (RockafellarUryasev2000).
Proof. Each function \(t\mapsto(\ell-t)^+\) is convex. Also, \(E[(L-t)^+]=\int_t^\infty\mathbb P(L>u)\,du\), so differentiation gives the stated derivative. At a minimizer \(t_*\), \(\mathbb P(L>t_*)=\alpha\); expanding the positive part gives \(G(t_*)=\alpha^{-1}E[L\mathbf1_{\{L>t_*\}}]\). \(\square\)
For loss distributions with atoms, the same minimum defines expected shortfall with the appropriate fraction of mass at the quantile (RockafellarUryasev2002). Here only \(Y\) will be discretized: the normal component keeps the return distribution continuous.
Substituting \(t=-x\) gives the return form
Lemma. For \(V\sim N(m,s^2)\), \(s>0\), and \(z=(x-m)/s\),
Proof. Write \(V=m+sZ_0\) and integrate \(\int_{-\infty}^z(z-u)\varphi(u)\,du\). Since \(\varphi'(u)=-u\varphi(u)\), the integral equals \(z\Phi(z)+\varphi(z)\). The second identity follows by the same integration. \(\square\)
Thus conditioning on \(Y\) integrates the normal variable exactly. Define
The tower property gives \(\operatorname{RU}(x)=-x+\alpha^{-1}E[h_x(Y)]\). At \(F_X(x_\alpha)=\alpha\) this becomes
Whenever second moments exist, conditioning also reduces Monte Carlo variance: \(\operatorname{Var}(E[(x-X)^+\mid Y])\le\operatorname{Var}((x-X)^+)\). The conditional integrand is smooth, even though the positive part has a kink.
The envelope and parameter derivatives#
Let \(\theta\) be a model parameter and assume differentiation under the expectation is valid. At the minimizing root, \(\partial_x\operatorname{RU}=0\); hence
The root can therefore be held fixed for a first derivative. The expression \(C(x)=-\alpha^{-1}E[m_Y\Phi(z_Y)-s_Y\varphi(z_Y)]\) instead satisfies \(C(x)=\operatorname{RU}(x)-x(F_X(x)-\alpha)/\alpha\). Its derivative at a frozen root contains the extra term \(-x\partial_\theta F_X/\alpha\). Equality of values at a root does not imply equality of their frozen-root derivatives.
Differentiating the conditional positive part gives
The analytic Hessian and the chain rule to \(w\) are in CVaR Derivatives for Normal Mixture Distributions, following Tasche1999 and RauHasanov2004. A frozen-root second derivative omits the motion of the minimizer. It must not replace that Hessian.
Scenarios as a probability measure#
Definition. A scenario measure is \(Q=\sum_{j=1}^N p_j\delta_{Y_j}\), with \(Y_j>0\), \(p_j\ge0\), and \(\sum_jp_j=1\). Write \(E_Q[g]=\sum_jp_jg(Y_j)\) and let \(X_Q=\tilde\mu+\tilde\gamma Y_Q+\tilde\sigma\sqrt{Y_Q}Z_0\), where \(Y_Q\sim Q\). An i.i.d. sample has \(p_j=1/N\). Quadrature supplies deterministic nodes and nonuniform probabilities. Weighted historical scenarios are another instance.
Proposition. Every conditional CDF, CVaR, gradient, and Hessian identity above holds exactly for \(X_Q\) with \(E\) replaced by \(E_Q\). In particular, \(F_Q\) is continuous and strictly increasing. If \(q_j=m_{Y_j}+s_{Y_j}\Phi^{-1}(\alpha)\) for the positive-mass atoms, then
Proof. \(X_Q\) is a probability mixture of normals. At the minimum component quantile every component CDF is at most \(\alpha\), and at the maximum every component CDF is at least \(\alpha\). Taking a convex combination preserves both inequalities. All conditional identities follow from the same tower property. \(\square\)
CVaR remains coherent and convex in \(w\) for this fixed law (Artzner1999). The efficient-surface reduction of Mean-Risk Optimization for Normal Mixture Distributions therefore holds for \(X_Q\) as well. Negative weights need not define a CDF and destroy this argument.
Proposition (error transfer). If both risks are finite, then
Proof. Evaluate \(\operatorname{RU}_Q\) at the minimizer of \(\operatorname{RU}\) to bound one difference; interchange the two functions for the reverse inequality. \(\square\)
To first order the quantile error is \(x_{\alpha,Q}-x_\alpha\simeq-(F_Q-F_X)(x_\alpha)/f_X(x_\alpha)\). CVaR stationarity removes this root error to first order, whereas derivative errors also depend on the root shift.
For i.i.d. nodes, \(\operatorname{Var}(E_Q[g])=\operatorname{Var}(g(Y))/N\) when that variance exists. The usual \(N^{-1/2}\) rate does not use smoothness. For inverse-gamma shape at most two and negative skew, the second moment of the CVaR integrand diverges; this rate is then unjustified.
Gauss–Legendre quadrature#
The Legendre polynomial \(P_M(t)=(2^M M!)^{-1}(d/dt)^M(t^2-1)^M\) is orthogonal to every polynomial of degree below \(M\) on \([-1,1]\). Its \(M\) roots \(t_j\) are distinct and interior (DLMF, §18.3). Let \(\ell_j(t)=\prod_{k\ne j}(t-t_k)/(t_j-t_k)\) and \(w_j=\int_{-1}^1\ell_j(t)\,dt\).
Proposition. The rule \(\sum_jw_jh(t_j)\) is exact for polynomials of degree at most \(2M-1\). Its weights satisfy
Proof. Divide a polynomial \(q\) of degree at most \(2M-1\) as \(q=P_Mr+s\), with \(\deg r,\deg s<M\). Orthogonality gives \(\int q=\int s\), and \(q=s\) at the nodes. Interpolation integrates \(s\) exactly. Applying this result to \(\ell_j^2\) proves positivity; applying it to 1 gives the sum. The closed form follows from the Legendre Christoffel–Darboux identity and \((1-t^2)P_M'(t)=M(P_{M-1}(t)-tP_M(t))\) (DLMF, §18.2 and §18.9). \(\square\)
No \(M\)-node rule can integrate every degree-\(2M\) polynomial: the nonnegative polynomial \(\prod_j(t-t_j)^2\) vanishes at all nodes and has positive integral. For \(M=2\), \(t_j=\pm1/\sqrt3\) and \(w_j=1\). Cubics integrate exactly; the error on \(t^4\) is \(2/5-2/9=8/45\).
For \(h\in C^{2M}[-1,1]\), the remainder is
for some \(\xi\in(-1,1)\) (DavisRabinowitz1984). To obtain the constant, subtract the Hermite interpolant matching \(h,h'\) at the roots. Bound its remainder above and below by the extreme values of \(h^{(2M)}\) times \(\prod_j(t-t_j)^2/(2M)!\), then use continuity. The leading coefficient of \(P_M\) is \((2M)!/(2^M(M!)^2)\) and \(\int_{-1}^1P_M^2=2/(2M+1)\).
More generally, if \(E_n(h)\) is the best uniform polynomial approximation error of degree \(n\), positivity and exactness imply \(|\int h-\sum_jw_jh(t_j)|\le4E_{2M-1}(h)\). Analyticity in a Bernstein ellipse with parameter \(\rho>1\) gives geometric convergence in \(M\); the bound also depends on the maximum magnitude in that ellipse (Trefethen2013). Entire integrands can still require many nodes when they vary rapidly near the real axis.
Log coordinates, windows, and heavy tails#
Set \(s=\log y\). Then \(f_s(s)=f_Y(e^s)e^s\) and all four mixing laws have
Here \(A,B\ge0\). We use \(a_Y\) for Gamma or inverse-gamma shape to distinguish it from the tail probability \(\alpha\).
Law |
\(\nu\) |
\(A\) |
\(B\) |
Positive moments |
|---|---|---|---|---|
Gamma\((a_Y,\beta)\) |
\(a_Y\) |
\(\beta\) |
0 |
all |
InverseGamma\((a_Y,\beta)\) |
\(-a_Y\) |
0 |
\(\beta\) |
\(k<a_Y\) |
InverseGaussian\((\mu_Y,\lambda)\) |
\(-1/2\) |
\(\lambda/(2\mu_Y^2)\) |
\(\lambda/2\) |
all |
GIG\((p,a,b)\), \(a,b>0\) |
\(p\) |
\(a/2\) |
\(b/2\) |
all |
The log density is strictly concave. At its mode \(s_0\), let \(A_c=Ae^{s_0}\), \(B_c=Be^{-s_0}\), and \(u=s-s_0\). Then
A quadrature rule on a finite window, including its change-of-variable Jacobian in \(\omega_j>0\), yields \(Y_j=e^{s_0+u_j}\) and \(p_j\propto\omega_j e^{g_c(u_j)}\). Normalization removes the unknown density constant, so no Bessel normalizer is needed. The sinh coordinate \(u=c\sinh t\), with \(c=(A_c+B_c)^{-1/2}\), concentrates nodes near the mode.
Truncation with moment tilts#
For an integrand growing as \(y^k\), the relevant exponent is \(g_c(u)+ku\). A right endpoint where this exponent has dropped by \(P\) controls that moment’s tail, provided the moment exists. Density truncation alone does not control \(Y\)-weighted tail integrals.
Lemma. Let \(\psi(s)=\log f_s(s)\). If \(\lambda_b=\psi'(b)+k<0\), then
Proof. Concavity places \(\psi(s)+ks\) below its tangent at \(b\). Exponentiate that tangent and integrate. \(\square\)
For inverse-gamma shape \(a_Y\), the right tail in \(s\) decays like \(e^{-a_Ys}\); a \(k\)-tilted window has length roughly \(P/(a_Y-k)\). The \(k=1\) tilt covers CVaR and its skew derivative for negative skew. At zero skew the skew Hessian requires \(E[Y^{3/2}]\). Gamma with small shape has a long left exponential tail. Interior GIG and inverse Gaussian have double-exponential walls in \(s\); near the inverse-gamma boundary a long exponential stretch precedes the right wall.
Writing \(k_+=\sup\{k\ge0:E[Y^k]<\infty\}\), the inverse-gamma domain conditions give:
Quantity |
Divergence condition |
Value |
|---|---|---|
CVaR |
\(\tilde\gamma<0\), \(k_+\le1\); or \(\tilde\gamma=0\), \(k_+\le1/2\) |
\(+\infty\) |
\(\partial_{\tilde\gamma}\operatorname{CVaR}\) |
\(\tilde\gamma\le0\), \(k_+\le1\) |
\(-\infty\) |
\(\partial^2_{\tilde\gamma}\operatorname{CVaR}\) |
\(\tilde\gamma=0\), \(k_+\le3/2\) |
\(+\infty\) |
Every finite rule has finite moments. Increasing its node count cannot establish existence of moments of the original law. These domain conditions must accompany any truncation-based computation.
The near-step regime#
If \(y_c=(x_\alpha-\tilde\mu)/\tilde\gamma>0\), put \(s_c=\log y_c\). Differentiation gives
Thus the conditional CDF changes over log width \(\delta_s\). The density factor in the Hessian is a narrow peak on the same scale. A fixed rule can resolve the CVaR value while missing most of this peak. For inverse-gamma mixing and negative skew, small tail probabilities move \(y_c\) outward; in the drift-dominated limit \(\rho_{\mathrm{loc}}\asymp\alpha^{-1/(2a_Y)}\).
The ratio \(R=\delta_s/h(s_c)\), where \(h(s_c)\) is the local log-node spacing, measures resolution. It is a heuristic, not a bound. Agreement of two node counts can also miss an unresolved peak. Increasing the node count or placing panel boundaries near \(s_c\) improves resolution. Boundaries that move with \(x\) or \(w\) no longer define one fixed discrete law over an optimization surface; convexity then holds only up to integration error.
Finally, the component-quantile bracket can be enormous when a log window contains very large \(Y_j\). A monotone coordinate \(x=c+h\sinh t\), \(h>0\), compresses the bracket before bisection. This changes neither the root nor the bracketing proof. The center and scale must themselves remain near the relevant probability mass: a truncated mean of an infinite-mean law does not supply such a center.