Many financial quantities depend on several variables — for example, an option’s value depends on price, volatility, time, and rates.
To measure how such a multivariable function changes with respect to each variable independently, we use partial derivatives.
Their collection, the gradient, describes the direction and magnitude of the steepest change.
1. Functions of Several Variables
Let $f: \mathbb{R}^n \to \mathbb{R}$ and $\mathbf{x} = (x_1, x_2, \ldots, x_n)$.
The function $f$ assigns a scalar output to multiple inputs.
Examples in finance:
- $V(S, \sigma, t)$ — option value as a function of price, volatility, and time.
- $P(y_1, y_2, \ldots, y_n)$ — bond portfolio value as a function of key interest rates.
- $L(w_1, w_2, \ldots, w_n)$ — portfolio loss as a function of factor exposures.
2. Definition of a Partial Derivative
The partial derivative of $f$ with respect to $x_i$ is
\[\frac{\partial f}{\partial x_i}(\mathbf{x}) = \lim_{h \to 0} \frac{f(x_1, \ldots, x_i + h, \ldots, x_n) - f(x_1, \ldots, x_i, \ldots, x_n)}{h}.\]It measures the instantaneous rate of change in $f$ when only $x_i$ varies, all other variables held fixed.
Example:
For $f(x,y) = x^2 y + 3xy^2$,
\(\frac{\partial f}{\partial x} = 2xy + 3y^2, \quad
\frac{\partial f}{\partial y} = x^2 + 6xy.\)
Interpretation:
Each partial derivative isolates the sensitivity of $f$ to one variable — similar to computing the “Greeks” in finance.
3. Notation and Terminology
- $\frac{\partial f}{\partial x}$ : partial derivative with respect to $x$.
- $\nabla f$ : gradient vector (all partial derivatives).
- $D_i f$ : alternative notation for $\frac{\partial f}{\partial x_i}$.
If $f$ is differentiable in all arguments, the total differential is \(df = \sum_{i=1}^n \frac{\partial f}{\partial x_i} dx_i.\)
This gives the first-order approximation of a small change in $f$.
Example:
If $V(S, \sigma, t)$ is differentiable,
\(dV \approx \frac{\partial V}{\partial S} dS + \frac{\partial V}{\partial \sigma} d\sigma + \frac{\partial V}{\partial t} dt.\)
In finance, this is the infinitesimal P&L approximation.
4. The Gradient Vector
For $f: \mathbb{R}^n \to \mathbb{R}$, the gradient is defined as
\[\nabla f(\mathbf{x}) = \left( \frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \ldots, \frac{\partial f}{\partial x_n} \right).\]Properties:
- Points in the direction of steepest ascent of $f$.
-
Magnitude $ \nabla f $ is the rate of maximal increase. - Orthogonal to level curves (or level surfaces) of $f$.
Geometric Analogy:
In two variables $f(x,y)$, contours of equal $f$ form curves in $(x,y)$-space.
$\nabla f$ is perpendicular to each contour.
Financial Analogy:
$\nabla V = (\Delta, \nu, \Theta, \rho, \dots)$ — vector of Greeks describing sensitivities along each input dimension.
5. Second-Order Partial Derivatives and Hessian Matrix
If all second-order derivatives exist and are continuous,
the matrix of second derivatives is the Hessian:
Clairaut’s Theorem:
If mixed partials are continuous,
\(\frac{\partial^2 f}{\partial x_i \partial x_j} = \frac{\partial^2 f}{\partial x_j \partial x_i}.\)
Interpretation:
The Hessian captures curvature and interaction effects — in finance, this reflects how risk factors jointly affect nonlinear portfolios.
Example:
For $f(x,y)=x^2y+y^3$,
\(H(f)=
\begin{bmatrix}
2y & 2x \\
2x & 6y
\end{bmatrix}.\)
Positive definiteness of $H$ implies local convexity (e.g., convex payoffs like calls).
6. Total Differential and Sensitivity Approximation
The total differential gives a local linear approximation:
\[df \approx \sum_{i=1}^n \frac{\partial f}{\partial x_i} \, dx_i = \nabla f^\top \, d\mathbf{x}.\]Example (Option P&L): \(dV \approx \Delta \, dS + \nu \, d\sigma + \Theta \, dt.\)
This approximation forms the foundation of delta-hedging and risk attribution.
Higher-order terms (e.g., $\frac{1}{2}\Gamma (dS)^2$) appear in second-order Taylor expansions.
7. Chain Rule for Multivariable Functions
If $z = f(x,y)$ and $x = g(t)$, $y = h(t)$, then
\[\frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}.\]Example:
If $f(x,y)=x^2y$, $x(t)=e^t$, $y(t)=t$,
\(\frac{dz}{dt} = 2xy\frac{dx}{dt} + x^2\frac{dy}{dt}
= 2e^t t e^t + e^{2t} = e^{2t}(2t+1).\)
Use in Finance:
When risk factors depend on common drivers (e.g., correlated market variables), total exposure to a driver is obtained via multivariate chain rule.
8. Gradient in Optimization
The gradient vanishes at critical points: \(\nabla f(\mathbf{x}^*) = 0.\)
To determine type:
- $H(f)$ positive definite → local minimum.
- $H(f)$ negative definite → local maximum.
- Indefinite → saddle point.
Applications:
- Portfolio optimization: $\nabla L(w) = 0$ gives optimal weights.
- Calibration: gradients used in gradient-descent or quasi-Newton algorithms for parameter fitting.
- Lagrange multipliers: enforce equality constraints $\nabla f = \lambda \nabla g$.
9. Mixed Sensitivities in Finance
Consider a derivative portfolio value $V(S,\sigma,t)$.
Second-order sensitivities (cross-partials):
These appear in the Taylor expansion: \(dV \approx \Delta \, dS + \nu \, d\sigma + \tfrac{1}{2}\Gamma (dS)^2 + \tfrac{1}{2}\text{Vomma} (d\sigma)^2 + \text{Vanna} \, dS \, d\sigma.\)
Such interactions are crucial for hedging multi-risk portfolios.
10. Directional Derivatives
For a unit vector $\mathbf{u}$,
the directional derivative of $f$ at $\mathbf{x}$ in the direction of $\mathbf{u}$ is
It measures the instantaneous rate of change of $f$ in direction $\mathbf{u}$.
Example:
If $\nabla f(1,2) = (3,4)$ and $\mathbf{u} = (1/\sqrt{2},1/\sqrt{2})$,
then $D_{\mathbf{u}}f = 3/\sqrt{2} + 4/\sqrt{2} = 7/\sqrt{2}$.
Interpretation:
In portfolio space, $\mathbf{u}$ may represent a normalized portfolio change; $D_{\mathbf{u}} f$ is the marginal change in portfolio value along that direction.
11. Summary
- Partial derivatives isolate sensitivities to individual variables.
- The gradient aggregates these sensitivities and points in the direction of steepest ascent.
- The Hessian encodes curvature and cross-effects, essential for risk decomposition and optimization.
- In finance, these tools form the mathematical language of multi-factor risk analysis, portfolio optimization, and option Greeks.