Free lesson · Partial derivatives

Hessians, mixed derivatives and second-order scenarios

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Start with the idea

A Hessian gathers second derivatives.
Its diagonal entries describe curvature in one coordinate; its off-diagonal entries describe how one sensitivity changes as another input changes.
Those interactions matter when inputs move together.

Symbols, units & horizon

H: Hessian matrix, not a candle high here
fₓₓ,fᵧᵧ: pure second partial derivatives
fₓᵧ,fᵧₓ: mixed second partials
h=(Δx,Δy): input-change vector
∇f: gradient at the starting point
hᵀHh: quadratic form
Δf: output change
Each Hessian entry: output units divided by the two associated input units

When and why to use this

Use Hessians to inspect local curvature, compare first- and second-order scenario estimates, and understand optimisation conditioning.

For a function with continuous second partials in a neighbourhood, mixed partial derivatives agree.
Without the regularity assumptions that symmetry cannot simply be assumed.
A Hessian is local, just like a gradient.
The second-order multivariable Taylor term is one half of a quadratic form in the input changes.
Off-diagonal terms appear twice in the full sum, so the two-variable cross contribution has no remaining one-half factor when the Hessian is symmetric.

H = (\begin{matrix} f_{x x} & f_{x y} \\ f_{y x} & f_{y y} \end{matrix}), Δ f \approx \nabla f^{𝖳} h + \frac{1}{2} h^{𝖳} H h

Calculus: derivation and arithmetic

Hessians, mixed derivatives and second-order scenarios

For f=x²+xy+2y², compute fₓ=2x+y and fᵧ=x+4y. Differentiate again to obtain H=[[2,1],[1,4]].
Expand hᵀHh=2(Δx)²+2ΔxΔy+4(Δy)². Multiplying by one half gives the curvature correction.
At (1,1), gradient=(3,5). With h=(.1,−.2), the linear term is −.7. Curvature is .01−.02+.08=.07, giving total −.63.

Work it by hand

Original value f(1,1)=4. New value f(1.1,.8)=3.37. The change is exactly −.63 because this function is quadratic and has no higher-order remainder.

An analogy to remember

Bending a sheet in two directions can create an interaction that neither one-direction bend captures. Mixed derivatives quantify that local interaction.

How this becomes a building block

Gamma is one entry of a pricing Hessian. Cross sensitivities such as spot–volatility interaction can matter in a joint market shock. In portfolio optimisation, the covariance quadratic form supplies curvature of a mean–variance objective.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

def quadratic_value(x,y): return x*x+x*y+2*y*y

def quadratic_change(x,y,dx,dy):
    linear=(2*x+y)*dx+(x+4*y)*dy
    curvature=dx*dx+dx*dy+2*dy*dy
    return linear,curvature,linear+curvature

print(quadratic_change(1,1,.1,-.2),quadratic_value(1.1,.8)-quadratic_value(1,1))

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