M.2.3 Multivariable Calculus

Basic Prerequisites: 02 · Single-Variable Calculus and Linear Algebra & Matrices (vectors, matrices, quadratic forms, eigenvalues).

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1. Intuition & Practical Objective

Finance is multivariable the moment you hold more than one asset. A portfolio value depends on every price; a risk number depends on every factor loading; a model depends on every parameter. This page builds the three objects that turn “a function of many variables” into something you can reason about:

• the gradient $\nabla f$ - the vector of first partials; the direction of steepest ascent, and the multi-asset generalisation of delta;
• the Jacobian $J$ - the matrix of first partials of a vector-valued map; how a whole vector of outputs (e.g. a set of option prices) responds to a whole vector of inputs;
• the Hessian $H={\nabla }^{2}f$ - the symmetric matrix of second partials; the curvature, the multi-asset gamma, and the object that decides whether a stationary point is a minimum, maximum or saddle.

The objective is one sentence: second-order Taylor with a gradient and a Hessian is the universal local model of a multivariable quantity, and its quadratic term $\frac{1}{2}{d}^{\top }Hd$ is where all the interesting risk lives - covariance, convexity, curvature-driven hedging error.

The one-sentence essence. “In several variables the derivative is not a number but a linear map (the gradient or Jacobian), and the second derivative is not a number but a symmetric matrix (the Hessian); sign-definiteness of that matrix is the multivariable second-derivative test.”

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2. Mathematical Ground Truth & Derivations

Partial derivatives. Hold all variables but one fixed and differentiate:

$$\frac{\partial f}{\partial x_i}(x)=\underset{h\to 0}{\lim }\frac{f(x+h{e}_i)-f(x)}{h}.$$

The total derivative (the local linear map). $f:{\mathbb{R}}^n\to \mathbb{R}$ is differentiable at $x$ iff there is a vector $g$ with $f(x+d)=f(x)+g^{\top }d+o(\Vert d\Vert )$; then $g=\nabla f(x)$, the gradient, whose components are the partials. This is Bernstein’s Carathéodory idea lifted to ${\mathbb{R}}^n$ (Simon & Blume §14.4): the derivative is the best linear approximation.

Directional derivative and steepest ascent. For a unit vector $u$,

$$D_{u}f(x)=\underset{t\to 0}{\lim }\frac{f(x+tu)-f(x)}{t}=\nabla f(x{)}^{\top }u.$$

By Cauchy–Schwarz $|D_{u}f|\le \Vert \nabla f\Vert$, with equality when $u\parallel \nabla f$: the gradient points in the direction of steepest increase, and its norm is the maximum rate. This is the entire geometric content of gradient descent (page 05).

Gradient rules. Linearity $\nabla (af+bg)=a\nabla f+b\nabla g$; product $\nabla (fg)=f\nabla g+g\nabla f$; and the chain rule along a curve $x(t)$:

$$\frac{d}{dt}f(x(t))=\nabla f(x(t){)}^{\top }{x}^{\prime }(t).$$

For a composite with an inner linear map $y=Ax$, ${\nabla }_{x}f(Ax)=A^{\top }\nabla f(Ax)$ - the reason covariance matrices enter through $A^{\top }$ factors.

Jacobian (vector-valued maps). For $F:{\mathbb{R}}^n\to {\mathbb{R}}^m$, the derivative is the Jacobian matrix $J_F(x)\in {\mathbb{R}}^{m\times n}$ with $(J_F{)}_{ij}=\partial F_i/\partial x_j$, and

$$F(x+d)\approx F(x)+J_F(x)d.$$

The chain rule in matrix form is $J_{F\circ G}(x)=J_F(G(x))J_G(x)$ - matrix multiplication of local sensitivity maps. When $m=n$, $\det J_F$ is the local volume scale factor (the change-of-variables Jacobian in densities).

Hessian and the second-order Taylor model. For $f:{\mathbb{R}}^n\to \mathbb{R}$ twice differentiable, the Hessian is the symmetric matrix $H_{ij}={\partial }^{2}f/\partial x_i\partial x_j$ (symmetry is Clairaut/Schwarz’s theorem: mixed partials commute for $C^2$ functions). Then

$$f(x_0+d)=f(x_0)+\nabla f(x_0{)}^{\top }d+\frac{1}{2}{d}^{\top }H(x_0)d+o(\Vert d{\Vert }^2).$$

The multivariable second-derivative test (Simon & Blume Ch 16, §17.3): at a stationary point $\nabla f(x^{\ast })=0$,

• $H\succ 0$ (positive definite, all eigenvalues $>0$) $\Rightarrow$ strict local minimum;
• $H\prec 0$ (negative definite) $\Rightarrow$ strict local maximum;
• $H$ indefinite (mixed-sign eigenvalues) $\Rightarrow$ saddle point;
• $H$ semidefinite (a zero eigenvalue) $\Rightarrow$ test inconclusive (need higher order).

For a $2\times 2$ Hessian this is the familiar test: $H_{11}>0$ and $\det H>0$ $\Rightarrow$ positive definite.

Finance reading - the Greeks as a gradient/Hessian. With $V(S_1,\dots ,S_n,t,\sigma )$:

$$dV\approx \sum _i{\Delta }_{i}d{S}_i+\Theta dt+\mathcal{V}d\sigma +\frac{1}{2}\sum _{i,j}{\Gamma }_{ij}d{S}_{i}d{S}_j+\cdots ,{\Gamma }_{ij}=\frac{{\partial }^{2}V}{\partial S_i\partial S_j}.$$

The gamma matrix $\Gamma$ is exactly the Hessian’s asset block; its off-diagonals are cross-gammas. And the covariance matrix of a linear portfolio $w^{\top }R$ is itself a quadratic form $w^{\top }\Sigma w$ - the same object appearing as curvature.

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3. Computational Implementation - gradient, directional derivative, Hessian, Jacobian

Stdlib only. We take $f(x,y)=x^2+y^2+xy$ (a convex quadratic), compute the analytic gradient and Hessian, cross-check the gradient by central differences, verify the directional derivative, then build the Jacobian of a nonlinear map and check the first-order Taylor model.

Read the results. The analytic gradient matches the finite-difference gradient to six decimals; the directional derivative along $(1,1)/\sqrt{2}$ equals $\Vert \nabla f\Vert$ because that direction is exactly the gradient direction ($\nabla f=(3,3)$) - so the directional derivative is maximal there, which is the steepest-ascent fact in numbers. The Hessian eigenvalues $(3,1)$ are both positive: $f$ is strictly convex and its single stationary point is the global minimum. The Jacobian’s determinant is $0$ - the map fails to be locally invertible at $(1,2)$, a genuine degeneracy (its two rows are identical). Finally the first-order Taylor model reproduces $F(1.001,2.001)$ to $10^{-6}$, and the error scales like $\Vert d{\Vert }^2=10^{-6}$ - the quadratic remainder.

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4. Failure Modes & First-Principles Breakdowns

1. A zero gradient is not an answer; the Hessian decides its nature. The same condition $\nabla f=0$ holds at a min, a max and a saddle. Report the eigenvalues of $H$ (or at least its definiteness) or you have not characterised the point. In $n$ variables a saddle is the generic outcome, not the exception.
2. A singular or nearly-singular Jacobian means the map is locally non-invertible. Here $\det J=0$; nearby, the implicit-function theorem fails and the inverse mapping (e.g. recovering implied vols from prices) is unstable or undefined. Near-singular $\det J$ (small but nonzero) is worse in practice: the inverse exists but amplifies noise by $1/\det J$.
3. The Hessian’s conditioning, not its definiteness, sets numerical robustness. Positive definiteness guarantees a minimiser exists; a huge condition number $\kappa (H)={\lambda }_{\max }/{\lambda }_{\min }$ means the level sets are thin ellipsoids and any gradient-based solver crawls (page 05). Asset covariance matrices are routinely ill-conditioned.
4. Discontinuous or non-differentiable objectives break everything. A cardinality constraint (“at most 10 names”) makes the feasible set non-convex and the objective non-smooth; $\nabla f$ and $H$ do not exist at the kinks, so gradient and Newton methods are invalid there (page 06).
5. Partial derivatives are local, and cross-derivatives can be non-commuting if $C^2$ fails. Clairaut’s theorem ($H$ symmetric) needs the mixed partials continuous; for pathological functions $H$ is asymmetric and the quadratic-form picture collapses. In practice, always symmetrise an empirical Hessian: $H\leftarrow \frac{1}{2}(H+H^{\top })$.
6. The gradient direction is not scale-invariant. With variables of wildly different units (a price in dollars, a vol in percent), the gradient’s largest component is chosen by units, not importance; this is why one standardises inputs before gradient-based fitting.

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5. References

• Simon & Blume: Mathematics for Economists
• Stewart, Clegg & Watson: Calculus: Early Transcendentals (9th ed.)
• Hubbard, John H. & Hubbard, Barbara Burke: Vector Calculus, Linear Algebra, and Differential Forms (5th ed.)
• Spivak, Michael: Calculus

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6. Connected Graph Bridges

• Back: 02 · Single-Variable Calculus · Index Hub
• Next: 04 · Constrained Optimization (add equality/inequality borders to $\nabla f=0$) → 05 · Gradient & Newton (follow $-\nabla f$, utilise $H$)
• Base: Linear Algebra & Matrices (quadratic forms, eigenvalues, definiteness)
• Applied: Greeks & Hedging (delta vector, gamma matrix) · Mean–Variance Optimization ($w^{\top }\Sigma w$)